Hypergeometric Functions

Hypergeometric functions From Wikipedia, the free encyclopedia

Contents 1

2

3

4

5

Appell series

1

1.1

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Derivatives and differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.4

Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.5

Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.6

Related series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.7

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.8

External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Askey scheme

5

2.1

Askey scheme for hypergeometric orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . .

5

2.2

Askey scheme for basic hypergeometric orthogonal polynomials . . . . . . . . . . . . . . . . . . .

5

2.3

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

Askey–Wilson polynomials

7

3.1

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3.2

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Barnes integral

8

4.1

Hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

4.2

Barnes lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

4.3

q-Barnes integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

4.4

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

Basic hypergeometric series

10

5.1

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

5.2

Simple series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

5.3

The q-binomial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

5.4

Ramanujan’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

5.5

Watson’s contour integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

5.6

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

5.7

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

i

ii 6

7

8

9

CONTENTS Bilateral hypergeometric series

13

6.1

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

6.2

Convergence and analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

6.3

Summation formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

6.3.1

Dougall’s bilateral sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

6.3.2

Bailey’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

6.4

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

6.5

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

Binomial transform

15

7.1

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

7.2

Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

7.3

Shift states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

7.4

Ordinary generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

7.5

Euler transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

7.6

Exponential generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

7.7

Integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

7.8

Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

7.9

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

7.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

7.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

Confluent hypergeometric function

20

8.1

Kummer’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

8.1.1

Other equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

8.2

Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

8.3

Asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

8.4

Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

8.4.1

Contiguous relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

8.4.2

Kummer’s transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

8.5

Multiplication theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

8.6

Connection with Laguerre polynomials and similar representations . . . . . . . . . . . . . . . . . .

24

8.7

Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

8.8

Application to continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

8.9

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

8.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

8.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

Dixon’s identity

28

9.1

Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

9.2

q-analogues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

9.3

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

CONTENTS

iii

10 Dougall’s formula

30

11 Elliptic hypergeometric series

31

11.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

11.2 Definitions of additive elliptic hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . .

32

11.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

12 Fox H-function

33

12.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Fox–Wright function

35

13.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Frobenius solution to the hypergeometric equation 14.1 The equation

33

35 36

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

14.2 Solution around x = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

14.3 Analysis of the solution in terms of the difference γ − 1 of the two roots . . . . . . . . . . . . . . .

39

14.3.1 γ not an integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

14.3.2 γ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

14.3.3 γ an integer and γ ≠ 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

14.4 Solution around x = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

14.5 Analysis of the solution in terms of the difference γ − α − β of the two roots . . . . . . . . . . . . .

44

14.5.1 Δ not an integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

14.5.2 Δ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

14.5.3 Δ is a non-zero integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

14.6 Solution around infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

14.7 Analysis of the solution in terms of the difference α − β of the two roots . . . . . . . . . . . . . . .

47

14.7.1 α − β not an integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

14.7.2 α − β = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

14.7.3 α − β an integer and α − β ≠ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

14.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

15 General hypergeometric function 15.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Generalized hypergeometric function

51 51 52

16.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

16.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

16.3 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

16.4 Convergence conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

16.5 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

16.5.1 Euler’s integral transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

16.5.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

16.6 Contiguous function and related identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

iv

CONTENTS 16.7 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

16.7.1 Saalschütz’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

16.7.2 Dixon’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

16.7.3 Dougall’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

16.7.4 Generalization of Kummer’s transformations and identities for 2 F 2 . . . . . . . . . . . . .

58

16.7.5 Kummer’s relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

16.7.6 Clausen’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

16.8 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

16.8.1 The series 0 F 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

16.8.2 The series 1 F 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

16.8.3 The series 0 F 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

16.8.4 The series 1 F 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

16.8.5 The series 2 F 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

16.8.6 The series 2 F 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

16.8.7 The series 3 F 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

16.8.8 The series 3 F 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

16.9 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

16.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

16.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

16.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

17 Gosper’s algorithm

64

17.1 Outline of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

17.2 Relationship to Wilf–Zeilberger pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

17.3 Definite versus indefinite summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

17.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

17.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

18 Horn function 18.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Humbert series

66 66 67

19.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

19.2 Related series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

19.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

20 Hypergeometric function

69

20.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

20.2 The hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

20.3 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

20.4 The hypergeometric differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

20.4.1 Solutions at the singular points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

20.4.2 Kummer’s 24 solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

CONTENTS

v

20.4.3 Q-form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

20.4.4 Schwarz triangle maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

20.4.5 Monodromy group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

20.5 Integral formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

20.5.1 Euler type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

20.5.2 Barnes integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

20.5.3 John transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

20.6 Gauss’ contiguous relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

20.6.1 Gauss’ continued fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

20.7 Transformation formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

20.7.1 Fractional linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

20.7.2 Quadratic transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

20.7.3 Higher order transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

20.8 Values at special points z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

20.8.1 Special values at z = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

20.8.2 Kummer’s theorem (z = −1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

20.8.3 Values at z = 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

20.8.4 Other points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

20.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

20.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

20.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

21 Hypergeometric function of a matrix argument

81

21.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

21.2 Two matrix arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

21.3 Not a typical function of a matrix argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

21.4 The parameter α

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

21.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

21.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

22 Hypergeometric identity

83

22.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

22.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

22.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

22.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

22.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

23 Kampé de Fériet function

85

23.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

23.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

24 Lauricella hypergeometric series 24.1 Generalization to n variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86 86

vi

CONTENTS 24.2 Integral representation of FD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

24.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

24.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

25 Legendre function

88

25.1 Differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

25.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

25.3 Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

25.4 Legendre function as characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

25.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

25.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

26 List of hypergeometric identities 26.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 MacRobert E function

91 91 92

27.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

27.2 Relationship with the Meijer G-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

27.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

27.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

28 Meijer G-function

94

28.1 Definition of the Meijer G-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

28.1.1 Differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

28.2 Relationship between the G-function and the generalized hypergeometric function . . . . . . . . . .

96

28.2.1 Polynomial cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

28.3 Basic properties of the G-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

28.3.1 Derivatives and antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

28.3.2 Recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

28.3.3 Multiplication theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

28.4 Definite integrals involving the G-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

28.4.1 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 28.5 Integral transforms based on the G-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 28.5.1 Narain transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 28.5.2 Wimp transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 28.5.3 Generalized Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 28.5.4 Meijer transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 28.6 Representation of other functions in terms of the G-function . . . . . . . . . . . . . . . . . . . . . 102 28.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 28.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 29 Picard–Fuchs equation

105

29.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

CONTENTS

vii

29.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 29.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 29.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 30 Riemann’s differential equation

107

30.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 30.2 Solutions and relationship with the hypergeometric function . . . . . . . . . . . . . . . . . . . . . 107 30.3 Fractional linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 30.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 30.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 30.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 31 Rogers–Ramanujan identities

110

31.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 31.2 Integer Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 31.3 Modular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 31.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 31.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 31.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 31.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 32 Schwarz’s list

113

32.1 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 32.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 32.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 32.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 32.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 33 Wilson polynomials

115

33.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 33.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 33.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 116 33.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 33.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 33.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Chapter 1

Appell series For generalizations of Lambert series see Appell–Lerch series. In mathematics, Appell series are a set of four hypergeometric series F 1 , F 2 , F 3 , F 4 of two variables that were introduced by Paul Appell (1880) and that generalize Gauss’s hypergeometric series 2 F 1 of one variable. Appell established the set of partial differential equations of which these functions are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variable.

1.1 Definitions The Appell series F 1 is defined for |x| < 1, |y| < 1 by the double series:

F1 (a, b1 , b2 , c; x, y) =

∞ ∑ (a)m+n (b1 )m (b2 )n m n x y , (c)m+n m! n! m,n=0

where the Pochhammer symbol (q)n represents the rising factorial:

(q)n = q (q + 1) · · · (q + n − 1) =

Γ(q + n) , Γ(q)

where the second equality is true for all complex q except q = 0, −1, −2, . . . . For other values of x and y the function F 1 can be defined by analytic continuation. Similarly, the function F 2 is defined for |x| + |y| < 1 by the series:

F2 (a, b1 , b2 , c1 , c2 ; x, y) =

∞ ∑ (a)m+n (b1 )m (b2 )n m n x y , (c1 )m (c2 )n m! n! m,n=0

the function F 3 for |x| < 1, |y| < 1 by the series:

F3 (a1 , a2 , b1 , b2 , c; x, y) =

∞ ∑ (a1 )m (a2 )n (b1 )m (b2 )n m n x y , (c)m+n m! n! m,n=0

and the function F 4 for |x|½ + |y|½ < 1 by the series:

F4 (a, b, c1 , c2 ; x, y) =

∞ ∑

(a)m+n (b)m+n m n x y . (c 1 )m (c2 )n m! n! m,n=0 1

2

CHAPTER 1. APPELL SERIES

1.2 Recurrence relations Like the Gauss hypergeometric series 2 F 1 , the Appell double series entail recurrence relations among contiguous functions. For example, a basic set of such relations for Appell’s F 1 is given by: (a−b1 −b2 )F1 (a, b1 , b2 , c; x, y)−a F1 (a+1, b1 , b2 , c; x, y)+b1 F1 (a, b1 +1, b2 , c; x, y)+b2 F1 (a, b1 , b2 +1, c; x, y) = 0 , c F1 (a, b1 , b2 , c; x, y) − (c − a)F1 (a, b1 , b2 , c + 1; x, y) − a F1 (a + 1, b1 , b2 , c + 1; x, y) = 0 , c F1 (a, b1 , b2 , c; x, y) + c(x − 1)F1 (a, b1 + 1, b2 , c; x, y) − (c − a)x F1 (a, b1 + 1, b2 , c + 1; x, y) = 0 , c F1 (a, b1 , b2 , c; x, y) + c(y − 1)F1 (a, b1 , b2 + 1, c; x, y) − (c − a)y F1 (a, b1 , b2 + 1, c + 1; x, y) = 0 . Any other relation[1] valid for F 1 can be derived from these four. Similarly, all recurrence relations for Appell’s F 3 follow from this set of five:

c F3 (a1 , a2 , b1 , b2 , c; x, y)+(a1 +a2 −c)F3 (a1 , a2 , b1 , b2 , c+1; x, y)−a1 F3 (a1 +1, a2 , b1 , b2 , c+1; x, y)−a2 F3 (a1 , a2 +1, b1 , b2 , c+1 c F3 (a1 , a2 , b1 , b2 , c; x, y) − c F3 (a1 + 1, a2 , b1 , b2 , c; x, y) + b1 x F3 (a1 + 1, a2 , b1 + 1, b2 , c + 1; x, y) = 0 , c F3 (a1 , a2 , b1 , b2 , c; x, y) − c F3 (a1 , a2 + 1, b1 , b2 , c; x, y) + b2 y F3 (a1 , a2 + 1, b1 , b2 + 1, c + 1; x, y) = 0 , c F3 (a1 , a2 , b1 , b2 , c; x, y) − c F3 (a1 , a2 , b1 + 1, b2 , c; x, y) + a1 x F3 (a1 + 1, a2 , b1 + 1, b2 , c + 1; x, y) = 0 , c F3 (a1 , a2 , b1 , b2 , c; x, y) − c F3 (a1 , a2 , b1 , b2 + 1, c; x, y) + a2 y F3 (a1 , a2 + 1, b1 , b2 + 1, c + 1; x, y) = 0 .

1.3 Derivatives and differential equations For Appell’s F 1 , the following derivatives result from the definition by a double series: ∂ ab1 F1 (a, b1 , b2 , c; x, y) = F1 (a + 1, b1 + 1, b2 , c + 1; x, y) , ∂x c ∂ ab2 F1 (a, b1 , b2 , c; x, y) = F1 (a + 1, b1 , b2 + 1, c + 1; x, y) . ∂y c From its definition, Appell’s F 1 is further found to satisfy the following system of second-order differential equations: ( ) ∂2 ∂2 ∂ ∂ x(1 − x) 2 + y(1 − x) + [c − (a + b1 + 1)x] − b1 y − ab1 F1 (x, y) = 0 , ∂x ∂x∂y ∂x ∂y ( ) ∂2 ∂2 ∂ ∂ y(1 − y) 2 + x(1 − y) + [c − (a + b2 + 1)y] − b2 x − ab2 F1 (x, y) = 0 . ∂y ∂x∂y ∂y ∂x Similarly, for F 3 the following derivatives result from the definition: a1 b1 ∂ F3 (a1 , a2 , b1 , b2 , c; x, y) = F3 (a1 + 1, a2 , b1 + 1, b2 , c + 1; x, y) , ∂x c ∂ a2 b2 F3 (a1 , a2 , b1 , b2 , c; x, y) = F3 (a1 , a2 + 1, b1 , b2 + 1, c + 1; x, y) . ∂y c And for F 3 the following system of differential equations is obtained: ( ) ∂2 ∂2 ∂ x(1 − x) 2 + y + [c − (a1 + b1 + 1)x] − a1 b1 F3 (x, y) = 0 , ∂x ∂x∂y ∂x ( ) ∂2 ∂2 ∂ y(1 − y) 2 + x + [c − (a2 + b2 + 1)y] − a2 b2 F3 (x, y) = 0 . ∂y ∂x∂y ∂y

1.4. INTEGRAL REPRESENTATIONS

3

1.4 Integral representations The four functions defined by Appell’s double series can be represented in terms of double integrals involving elementary functions only (Gradshteyn & Ryzhik 1971, § 9.184). However, Émile Picard (1881) discovered that Appell’s F 1 can also be written as a one-dimensional Euler-type integral:

Γ(c) F1 (a, b1 , b2 , c; x, y) = Γ(a)Γ(c − a)

∫

1

ta−1 (1 − t)c−a−1 (1 − xt)−b1 (1 − yt)−b2 dt,

ℜc > ℜa > 0 .

0

This representation can be verified by means of Taylor expansion of the integrand, followed by termwise integration.

1.5 Special cases Picard’s integral representation implies that the incomplete elliptic integrals F and E as well as the complete elliptic integral Π are special cases of Appell’s F 1 : ∫

ϕ

F (ϕ, k) =

√

0

∫

ϕ

E(ϕ, k) = ∫

0

0

1 − k 2 sin θ 2

= sin ϕ F1 ( 12 , 12 , 12 , 32 ; sin2 ϕ, k 2 sin2 ϕ),

|ℜ ϕ| <

√ 1 − k 2 sin2 θ dθ = sin ϕ F1 ( 12 , 12 , − 12 , 32 ; sin2 ϕ, k 2 sin2 ϕ),

π/2

Π(n, k) =

dθ

π , 2

|ℜ ϕ| <

π , 2

π dθ √ = F1 ( 12 , 1, 12 , 1; n, k 2 ) . 2 2 2 (1 − n sin θ) 1 − k sin θ 2

1.6 Related series • Main article: Humbert series

There are seven related series of two variables, Φ1 , Φ2 , Φ3 , Ψ1 , Ψ2 , Ξ1 , and Ξ2 , which generalize Kummer’s confluent hypergeometric function 1 F 1 of one variable and the confluent hypergeometric limit function 0 F 1 of one variable in a similar manner. The first of these was introduced by Pierre Humbert in 1920. • Main article: Lauricella hypergeometric series

Giuseppe Lauricella (1893) defined four functions similar to the Appell series, but depending on many variables rather than just the two variables x and y. These series were also studied by Appell. They satisfy certain partial differential equations, and can also be given in terms of Euler-type integrals and contour integrals.

1.7 References [1] For example, (y − x)F1 (a, b1 + 1, b2 + 1, c, x, y) = y F1 (a, b1 , b2 + 1, c, x, y) − x F1 (a, b1 + 1, b2 , c, x, y)

• Appell, Paul (1880). “Sur les séries hypergéométriques de deux variables et sur des équations différentielles linéaires aux dérivées partielles”. Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French) 90: 296–298 and 731–735. JFM 12.0296.01. (see also “Sur la série F3 (α,α',β,β',γ; x,y)" in C. R. Acad. Sci. 90, pp. 977–980)

4

CHAPTER 1. APPELL SERIES • Appell, Paul (1882). “Sur les fonctions hypergéométriques de deux variables”. Journal de Mathématiques Pures et Appliquées. (3ème série) (in French) 8: 173–216. • Appell, Paul; Kampé de Fériet, Joseph (1926). Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite (in French). Paris: Gauthier–Villars. JFM 52.0361.13. (see p. 14) • Askey, R. A.; Daalhuis, Adri B. Olde (2010), “Appell series”, in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248 • Bateman, H.; Erdélyi, A. (1953). Higher Transcendental Functions, Vol. I (PDF). New York: McGraw–Hill. (see p. 224) • Gradshteyn, Izrail' Solomonovich; Ryzhik, Iosif Moiseevich (1971). Tablitsy integralov, summ, ryadov i proizvedeniy [Tables of integrals, sums, series and products] (in Russian) (5th ed.). Moscow: Nauka. (see Chapter 9.18) • Humbert, Pierre (1920). “Sur les fonctions hypercylindriques”. Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French) 171: 490–492. JFM 47.0348.01. • Lauricella, Giuseppe (1893). “Sulle funzioni ipergeometriche a più variabili”. Rendiconti del Circolo Matematico di Palermo (in Italian) 7: 111–158. doi:10.1007/BF03012437. JFM 25.0756.01. • Picard, Émile (1881). “Sur une extension aux fonctions de deux variables du problème de Riemann relativ aux fonctions hypergéométriques”. Annales scientifiques de l'École Normale Supérieure. (2ème série) (in French) 10: 305–322. JFM 13.0389.01. (see also C. R. Acad. Sci. 90 (1880), pp. 1119–1121 and 1267–1269) • Slater, Lucy Joan (1966). Generalized hypergeometric functions. Cambridge, UK: Cambridge University Press. ISBN 0-521-06483-X. MR 0201688. (there is a 2008 paperback with ISBN 978-0-521-09061-2)

1.8 External links • Aarts, Ronald M., “Lauricella Functions”, MathWorld. • Weisstein, Eric W., “Appell Hypergeometric Function”, MathWorld.

Chapter 2

Askey scheme In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in Andrews & Askey (1985), the Askey scheme was first drawn by Labelle (1985) and by Askey and Wilson (1985), and has since been extended by Koekoek & Swarttouw (1998) and Koekoek, Lesky & Swarttouw (2010) to cover basic orthogonal polynomials.

2.1 Askey scheme for hypergeometric orthogonal polynomials Koekoek, Lesky & Swarttouw (2010, p.183) give the following version of the Askey scheme: 4 F3

Wilson | Racah

3 F2

Continuous dual Hahn | Continuous Hahn | Hahn | dual Hahn

2 F1

Meixner–Pollaczek | Jacobi | Pseudo Jacobi | Meixner | Krawtchouk

2 F 0 /1 F 1 1 F0

Laguerre | Bessel | Charlier

Hermite

2.2 Askey scheme for basic hypergeometric orthogonal polynomials Koekoek, Lesky & Swarttouw (2010, p.413) give the following scheme for basic hypergeometric orthogonal polynomials: 4

ϕ3 Askey–Wilson | q-Racah

3

ϕ2 Continuous dual q-Hahn | Continuous q-Hahn | Big q-Jacobi | q-Hahn | dual q-Hahn

2

ϕ1 Al-Salam–Chihara | q-Meixner–Pollaczek | Continuous q-Jacobi | Big q-Laguerre | Little q-Jacobi | q-Meixner | Quantum q-Krawtchouk | q-Krawtchouk | Affine q-Krawtchouk | Dual q-Krawtchouk

2

1

ϕ 0 /1 ϕ 1 Continuous big q-Hermite | Continuous q-Laguerre | Little q-Laguerre | q-Laguerre | q-Bessel | q-Charlier | Al-Salam–Carlitz I | Al-Salam–Carlitz II ϕ0 Continuous q-Hermite | Stieltjes–Wigert | Discrete q-Hermite I | Discrete q-Hermite II 5

6

CHAPTER 2. ASKEY SCHEME

2.3 References • Andrews, George E.; Askey, Richard (1985), “Classical orthogonal polynomials”, in Brezinski, C.; Draux, A.; Magnus, Alphonse P.; Maroni, Pascal; Ronveaux, A., Polynômes orthogonaux et applications. Proceedings of the Laguerre symposium held at Bar-le-Duc, October 15–18, 1984., Lecture Notes in Math. 1171, Berlin, New York: Springer-Verlag, pp. 36–62, doi:10.1007/BFb0076530, ISBN 978-3-540-16059-5, MR 838970 • Askey, Richard; Wilson, James (1985), “Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials”, Memoirs of the American Mathematical Society 54 (319): iv+55, doi:10.1090/memo/0319, ISBN 978-0-8218-2321-7, ISSN 0065-9266, MR 783216 • Koekoek, Roelof; Swarttouw, René F. (1998), The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, 98-17, Delft University of Technology, Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics • Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/9783-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096 • Koornwinder, Tom H. (1988), “Group theoretic interpretations of Askey’s scheme of hypergeometric orthogonal polynomials”, Orthogonal polynomials and their applications (Segovia, 1986), Lecture Notes in Math. 1329, Berlin, New York: Springer-Verlag, pp. 46–72, doi:10.1007/BFb0083353, ISBN 978-3-540-19489-7, MR 973421 • Labelle, Jacques (1985), “Tableau d'Askey”, in Brezinski, C.; Draux, A.; Magnus, Alphonse P.; Maroni, Pascal; Ronveaux, A., Polynômes Orthogonaux et Applications. Proceedings of the Laguerre Symposium held at Bar-leDuc, Lecture Notes in Math. 1171, Berlin, New York: Springer-Verlag, pp. xxxvi–xxxvii, doi:10.1007/BFb0076527, ISBN 978-3-540-16059-5, MR 838967

Chapter 3

Askey–Wilson polynomials In mathematics, the Askey–Wilson polynomials (or q-Wilson polynomials) are a family of orthogonal polynomials introduced by Askey and Wilson (1985) as q-analogs of the Wilson polynomials. They include many of the other orthogonal polynomials in 1 variable as special or limiting cases, described in the Askey scheme. Askey–Wilson polynomials are the special case of Macdonald polynomials (or Koornwinder polynomials) for the non-reduced affine root system of type (C∨ 1, C 1 ), and their 4 parameters a, b, c, d correspond to the 4 orbits of roots of this root system. They are defined by

pn (x; a, b, c, d|q) = (ab, ac, ad; q)n a

−n

[ 4 ϕ3

q −n ab

abcdq n−1 ac

aeiθ ad

ae−iθ

] ; q, q

where φ is a basic hypergeometric function and x = cos(θ) and (,,,)n is the q-Pochhammer symbol. Askey–Wilson functions are a generalization to non-integral values of n.

3.1 See also • Askey scheme

3.2 References • Askey, Richard; Wilson, James (1985), “Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials”, Memoirs of the American Mathematical Society 54 (319): iv+55, doi:10.1090/memo/0319, ISBN 978-0-8218-2321-7, ISSN 0065-9266, MR 783216 • Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719 • Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), “Askey-Wilson class”, in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248 • Koornwinder, Tom H. (2012), “Askey-Wilson polynomial”, Scholarpedia 7 (7): 7761, doi:10.4249/scholarpedia.7761

7

Chapter 4

Barnes integral In mathematics, a Barnes integral or Mellin–Barnes integral is a contour integral involving a product of gamma functions. They were introduced by Ernest William Barnes (1908, 1910). They are closely related to generalized hypergeometric series. The integral is usually taken along a contour which is a deformation of the imaginary axis passing to the left of all poles of factors of the form Γ(a + s) and to the right of all poles of factors of the form Γ(a − s).

4.1 Hypergeometric series The hypergeometric function is given as a Barnes integral (Barnes 1908) by

Γ(c) 1 2 F1 (a, b; c; z) = Γ(a)Γ(b) 2πi

∫

i∞ −i∞

Γ(a + s)Γ(b + s)Γ(−s) (−z)s ds. Γ(c + s)

This equality can be obtained by moving the contour to the right while picking up the residues at s = 0, 1, 2, ... . Given proper convergence conditions, one can relate more general Barnes’ integrals and generalized hypergeometric functions pFq in a similar way.

4.2 Barnes lemmas The first Barnes lemma (Barnes 1908) states

1 2πi

∫

i∞ −i∞

Γ(a + s)Γ(b + s)Γ(c − s)Γ(d − s)ds =

Γ(a + c)Γ(a + d)Γ(b + c)Γ(b + d) . Γ(a + b + c + d)

This is an analogue of Gauss’s 2 F 1 summation formula, and also an extension of Euler’s beta integral. The integral in it is sometimes called Barnes’s beta integral. The second Barnes lemma (Barnes 1910) states

1 2πi =

∫

i∞ −i∞

Γ(a + s)Γ(b + s)Γ(c + s)Γ(1 − d − s)Γ(−s) ds Γ(e + s)

Γ(a)Γ(b)Γ(c)Γ(1 − d + a)Γ(1 − d + b)Γ(1 − d + c) Γ(e − a)Γ(e − b)Γ(e − c)

where e = a + b + c − d + 1. This is an analogue of Saalschütz’s summation formula. 8

4.3. Q-BARNES INTEGRALS

9

4.3 q-Barnes integrals There are analogues of Barnes integrals for basic hypergeometric series, and many of the other results can also be extended to this case (Gasper & Rahman 2004, chapter 4).

4.4 References • Barnes, E.W. (1908). “A new development of the theory of the hypergeometric functions”. Proc. London Math. Soc. s2–6: 141–177. doi:10.1112/plms/s2-6.1.141. JFM 39.0506.01. • Barnes, E.W. (1910). “A transformation of generalised hypergeometric series”. Quarterly Journal of Mathematics 41: 136–140. JFM 41.0503.01. • Gasper, George; Rahman, Mizan (2004). Basic hypergeometric series. Encyclopedia of Mathematics and its Applications 96 (2nd ed.). Cambridge University Press. ISBN 978-0-521-83357-8. MR 2128719.

Chapter 5

Basic hypergeometric series In mathematics, Heine’s basic hypergeometric series, or hypergeometric q-series, are q-analog generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn₊₁/xn is a rational function of n. If the ratio of successive terms is a rational function of qn , then the series is called a basic hypergeometric series. The number q is called the base. The basic hypergeometric series 2 φ1 (qα ,qβ ;qγ ;q,x) was first considered by Eduard Heine (1846). It becomes the hypergeometric series F(α,β;γ;x) in the limit when the base q is 1.

5.1 Definition There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic geometric series ψ. The unilateral basic hypergeometric series is defined as [ j ϕk

a1 b1

a2 b2

... ...

] ∑ ∞ )1+k−j n (a1 , a2 , . . . , aj ; q)n ( aj ; q, z = (−1)n q ( 2 ) zn bk (b , b , . . . , b , q; q) 1 2 k n n=0

where

(a1 , a2 , . . . , am ; q)n = (a1 ; q)n (a2 ; q)n . . . (am ; q)n and where

(a; q)n =

n−1 ∏

(1 − aq k ) = (1 − a)(1 − aq)(1 − aq 2 ) · · · (1 − aq n−1 ).

k=0

is the q-shifted factorial. The most important special case is when j = k+1, when it becomes [ k+1 ϕk

a1 b1

a2 b2

... ...

ak bk

ak+1

] ; q, z =

∞ ∑ (a1 , a2 , . . . , ak+1 ; q)n n z . (b1 , b2 , . . . , bk , q; q)n n=0

This series is called balanced if a1 ...ak+1 = b1 ...bkq. This series is called well poised if a1 q = a2 b1 = ... = a ₊₁bk, and very well poised if in addition a2 = −a3 = qa1 1/2 . The bilateral basic hypergeometric series, corresponding to the bilateral hypergeometric series, is defined as [ a1 j ψk b1

a2 b2

... ...

] ∞ )k−j ∑ n (a1 , a2 , . . . , aj ; q)n ( aj (−1)n q ( 2 ) zn. ; q, z = bk (b , b , . . . , b ; q) 1 2 k n n=−∞ 10

5.2. SIMPLE SERIES

11

The most important special case is when j = k, when it becomes [ k ψk

a1 b1

a2 b2

... ...

] ∞ ∑ (a1 , a2 , . . . , ak ; q)n n ak ; q, z = z . bk (b1 , b2 , . . . , bk ; q)n n=−∞

The unilateral series can be obtained as a special case of the bilateral one by setting one of the b variables equal to q, at least when none of the a variables is a power of q., as all the terms with n Re a > 0, M(a, b, z) can be represented as an integral

M (a, b, z) =

Γ(b) Γ(a)Γ(b − a)

∫

1

ezu ua−1 (1 − u)b−a−1 du. 0

thus M (a, a + b, it) is the characteristic function of the beta distribution. For a with positive real part U can be obtained by the Laplace integral

8.3. ASYMPTOTIC BEHAVIOR

1 U (a, b, z) = Γ(a)

∫

∞

23

e−zt ta−1 (1 + t)b−a−1 dt,

(Re a > 0)

0

The integral defines a solution in the right half-plane Re z > 0. They can also be represented as Barnes integrals

M (a, b, z) =

1 Γ(b) 2πi Γ(a)

∫

i∞

−i∞

Γ(−s)Γ(a + s) (−z)s ds Γ(b + s)

where the contour passes to one side of the poles of Γ(−s) and to the other side of the poles of Γ(a + s).

8.3 Asymptotic behavior If a solution to Kummer’s equation is asymptotic to a power of z as z → ∞, then the power must be −a. This is in fact the case for Tricomi’s solution U(a, b, z). Its asymptotic behavior as z → ∞ can be deduced from the integral representations. If z = x ∈ R, then making a change of variables in the integral followed by expanding the binomial series and integrating it formally term by term gives rise to an asymptotic series expansion, valid as x → ∞:[2]

U (a, b, x) ∼ x

−a

( ) 1 , 2 F0 a, a − b + 1; ; − x

where 2 F0 (·, ·; ; −1/x) is a generalized hypergeometric series (with 1 as leading term), which generally converges nowhere but exists as a formal power series in 1/x. This asymptotic expansion is also valid for complex z instead of real x, with | arg z| < 32 π. The asymptotic behavior of Kummer’s solution for large |z| is: ( M (a, b, z) ∼ Γ(b)

ez z a−b (−z)−a + Γ(a) Γ(b − a)

)

The powers of z are taken using − 23 π < arg z ≤ 12 π .[3] The first term is only needed when Γ(b − a) is infinite (that is, when b − a is a non-positive integer) or when the real part of z is non-negative, whereas the second term is only needed when Γ(a) is infinite (that is, when a is a non-positive integer) or when the real part of z is non-positive. There is always some solution to Kummer’s equation asymptotic to ez z a−b as z → −∞. Usually this will be a combination of both M(a, b, z) and U(a, b, z) but can also be expressed as ez (−1)a−b U (b − a, b, −z) .

8.4 Relations There are many relations between Kummer functions for various arguments and their derivatives. This section gives a few typical examples.

8.4.1

Contiguous relations

Given M(a, b, z), the four functions M(a ± 1, b, z), M(a, b ± 1, z) are called contiguous to M(a, b, z). The function M(a, b, z) can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of a, b, and z. This gives (4 2)=6 relations, given by identifying any two lines on the right hand side of

24

z

CHAPTER 8. CONFLUENT HYPERGEOMETRIC FUNCTION

dM a = z M (a+, b+) = a(M (a+) − M ) dz b = (b − 1)(M (b−) − M ) = (b − a)M (a−) + (a − b + z)M = z(a − b)M (b+)/b + zM

In the notation above, M = M(a, b, z), M(a+) = M(a + 1, b, z), and so on. Repeatedly applying these relations gives a linear relation between any three functions of the form M(a + m, b + n, z) (and their higher derivatives), where m, n are integers. There are similar relations for U.

8.4.2

Kummer’s transformation

Kummer’s functions are also related by Kummer’s transformations:

M (a, b, z) = ez M (b − a, b, −z) U (a, b, z) = z 1−b U (1 + a − b, 2 − b, z)

8.5 Multiplication theorem The following multiplication theorems hold true:

U (a, b, z) = e(1−t)z

∑ (t − 1)i z i

U (a, b + i, zt) i! ( ) ∑ 1− 1 i (1−t)z b−1 t U (a − i, b − i, zt). =e t i! i=0 i=0

8.6 Connection with Laguerre polynomials and similar representations In terms of Laguerre polynomials, Kummer’s functions have several expansions, for example ( ) ∑ xy M a, b, x−1 = (1 − x)a · n

a(n) (b−1) L (y)xn b(n) n

(Erdelyi 1953, 6.12)

8.7 Special cases Functions that can be expressed as special cases of the confluent hypergeometric function include: • Some elementary functions (the left-hand side is not defined when b is a non-positive integer, but the right-hand side is still a solution of the corresponding Kummer equation): M (0, b, z) = 1 U (0, c, z) = 1 M (b, b, z) = ez ∫∞ U (a, a, z) = ez z u−a e−u du (a polynomial if a is a non-positive integer) U (1,b,z) Γ(b−1)

+

M (1,b,z) Γ(b)

= z 1−b ez

8.7. SPECIAL CASES

25

U (a, a + 1, z) = z −a U (−n, −2n, z) for integer n is a Bessel polynomial (see lower down). M (n, b, z) for non-positive integer n is a generalized Laguerre polynomial. • Bateman’s function • Bessel functions and many related functions such as Airy functions, Kelvin functions, Hankel functions. For example, the special case b = 2a the function reduces to a Bessel function:

1 F1 (a, 2a, x)

) ( 1 ( ) ( ) 2 x x ( ) −a = e 2 0 F1 ; a + 12 ; x16 = e 2 x4 2 Γ a + 12 Ia− 21 x2 .

This identity is sometimes also referred to as Kummer’s second transformation. Similarly x ( ) e2 1 U (a, 2a, x) = √ x 2 −a Ka− 1 x2 , π 2

When a is a non-positive integer, this equals 2−a θ−a

(x) 2

where θ is a Bessel polynomial.

• The error function can be expressed as

2 erf(x) = √ π

∫

x

0

( ) 2 2x e−t dt = √ 1 F1 12 , 32 , −x2 . π

• Coulomb wave function • Cunningham functions • Exponential integral and related functions such as the sine integral, logarithmic integral • Hermite polynomials • Incomplete gamma function • Laguerre polynomials • Parabolic cylinder function (or Weber function) • Poisson–Charlier function • Toronto functions • Whittaker functions Mκ,μ(z), Wκ,μ(z) are solutions of Whittaker’s equation that can be expressed in terms of Kummer functions M and U by

z 1 ( ) Mκ,µ (z) = e− 2 z µ+ 2 M µ − κ + 12 , 1 + 2µ; z z 1 ) ( Wκ,µ (z) = e− 2 z µ+ 2 U µ − κ + 12 , 1 + 2µ; z

• The general p-th raw moment (p not necessarily an integer) can be expressed as ( 2 ) p2 ( 1+p ) [ ( ) ( ) ] 2σ Γ 2 µ2 p 1 2 p √ E N µ, σ = 1 F1 − 2 , 2 , − 2σ 2 π ) [ ( )p ( p 1 ) ] ( µ2 2 2 2 p = −2σ E N µ, σ U − 2 , 2 , − 2σ 2 In the second formula the function’s second branch cut can be chosen by multiplying with (−1)p .

26

CHAPTER 8. CONFLUENT HYPERGEOMETRIC FUNCTION

8.8 Application to continued fractions By applying a limiting argument to Gauss’s continued fraction it can be shown that

M (a + 1, b + 1, z) = M (a, b, z)

1 b−a z b(b + 1) 1− a+1 z (b + 1)(b + 2) 1+ b−a+1 z (b + 2)(b + 3) 1− a+2 z (b + 3)(b + 4) 1+ . 1 − ..

and that this continued fraction converges uniformly to a meromorphic function of z in every bounded domain that does not include a pole.

8.9 Notes [1] Campos, LMBC (2001). “On Some Solutions of the Extended Confluent Hypergeometric Differential Equation”. Journal of Computational and Applied Mathematics. Elsevier. [2] Andrews, G.E.; Askey, R.; Roy, R. (2001). Special functions. Cambridge University Press. ISBN 978-0521789882.. [3] This is derived from Abramowitz and Stegun (see reference below), page 508. They give a full asymptotic series. They switch the sign of the exponent in exp(iπa) in the right half-plane but this is unimportant because the term is negligible there or else a is an integer and the sign doesn't matter.

8.10 References • Abramowitz, Milton; Stegun, Irene A., eds. (1965), “Chapter 13”, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 504, ISBN 978-0486612720, MR 0167642. • Chistova, E.A. (2001), “c/c024700”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Daalhuis, Adri B. Olde (2010), “Confluent hypergeometric function”, in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248 • Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz & Tricomi, Francesco G. (1953). Higher transcendental functions. Vol. I. New York–Toronto–London: McGraw–Hill Book Company, Inc. MR 0058756. • Kummer, Ernst Eduard (1837). “De integralibus quibusdam definitis et seriebus infinitis”. Journal für die reine und angewandte Mathematik (in Latin) 17: 228–242. doi:10.1515/crll.1837.17.228. ISSN 0075-4102. • Slater, Lucy Joan (1960). Confluent hypergeometric functions. Cambridge, UK: Cambridge University Press. MR 0107026. • Tricomi, Francesco G. (1947). “Sulle funzioni ipergeometriche confluenti”. Annali di Matematica Pura ed Applicata. Serie Quarta (in Italian) 26: 141–175. doi:10.1007/bf02415375. ISSN 0003-4622. MR 0029451. • Tricomi, Francesco G. (1954). Funzioni ipergeometriche confluenti. Consiglio Nazionale Delle Ricerche Monografie Matematiche (in Italian) 1. Rome: Edizioni cremonese. ISBN 978-88-7083-449-9. MR 0076936.

8.11. EXTERNAL LINKS

8.11 External links • Confluent Hypergeometric Functions in NIST Digital Library of Mathematical Functions • Kummer hypergeometric function on the Wolfram Functions site • Tricomi hypergeometric function on the Wolfram Functions site

27

Chapter 9

Dixon’s identity In mathematics, Dixon’s identity (or Dixon’s theorem or Dixon’s formula) is any of several different but closely related identities proved by A. C. Dixon, some involving finite sums of products of three binomial coefficients, and some evaluating a hypergeometric sum. These identities famously follow from the MacMahon Master theorem, and can now be routinely proved by computer algorithms (Ekhad 1990).

9.1 Statements The original identity, from (Dixon 1891), is a ∑

( (−1)k

k=−a

)3 2a (3a)! = . k+a (a!)3

A generalization, also sometimes called Dixon’s identity, is a ∑

( (−1)k

k=−a

a+b a+k

)(

b+c b+k

)(

c+a c+k

) =

(a + b + c)! a!b!c!

where a, b, and c are non-negative integers (Wilf 1994, p. 156). The sum on the left can be written as the terminating well-poised hypergeometric series (

)( ) b+c c+a 3 F2 (−2a, −a − b, −a − c; 1 + b − a, 1 + c − a; 1) b−a c−a

and the identity follows as a limiting case (as a tends to an integer) of Dixon’s theorem evaluating a well-poised 3 F 2 generalized hypergeometric series at 1, from (Dixon 1902):

3 F2 (a, b, c; 1

+ a − b, 1 + a − c; 1) =

Γ(1 + a/2)Γ(1 + a/2 − b − c)Γ(1 + a − b)Γ(1 + a − c) . Γ(1 + a)Γ(1 + a − b − c)Γ(1 + a/2 − b)Γ(1 + a/2 − c)

This holds for Re(1 + 1 ⁄2 a − b − c) > 0. As c tends to −∞ it reduces to Kummer’s formula for the hypergeometric function 2 F1 at −1. Dixon’s theorem can be deduced from the evaluation of the Selberg integral.

9.2 q-analogues A q-analogue of Dixon’s formula for the basic hypergeometric series in terms of the q-Pochhammer symbol is given by 28

9.3. REFERENCES

[ 4 ϕ3

a

−qa1/2 −a1/2

29

] (aq, aq/bc, qa1/2 /b, qa1/2 /c; q)∞ b c 1/2 ; q, qa /bc = aq/b aq/c (aq/b, aq/c, qa1/2 , qa1/2 /bc; q)∞

where |qa1/2 /bc| < 1.

9.3 References • Dixon, A.C. (1891), “On the sum of the cubes of the coefficients in a certain expansion by the binomial theorem”, Messenger of Mathematics 20: 79–80, JFM 22.0258.01 • Dixon, A.C. (1902), “Summation of a certain series”, Proc. London Math. Soc. 35 (1): 284–291, doi:10.1112/plms/s135.1.284, JFM 34.0490.02 • Ekhad, Shalosh B. (1990), “A very short proof of Dixon’s theorem”, Journal of Combinatorial Theory. Series A 54 (1): 141–142, doi:10.1016/0097-3165(90)90014-N, ISSN 1096-0899, MR 1051787, Zbl 0707.05007 • Gessel, Ira; Stanton, Dennis (1985), “Short proofs of Saalschütz’s and Dixon’s theorems”, Journal of Combinatorial Theory. Series A 38 (1): 87–90, doi:10.1016/0097-3165(85)90026-3, ISSN 1096-0899, MR 773560, Zbl 0559.05008 • Ward, James (1991), “100 years of Dixon’s identity”, Irish Mathematical Society Bulletin (27): 46–54, ISSN 0791-5578, MR 1185413, Zbl 0795.01009 • Wilf, Herbert S. (1994), Generatingfunctionology (2nd ed.), Boston, MA: Academic Press, ISBN 0-12-7519564, Zbl 0831.05001

Chapter 10

Dougall’s formula Dougall’s formula may refer to one of two formulas for hypergeometric series, both named after John Dougall: • Dougall’s formula for the sum of a 7 F 6 hypergeometric series • Dougall’s formula for the sum of a bilateral hypergeometric series

30

Chapter 11

Elliptic hypergeometric series In mathematics, an elliptic hypergeometric series is a series Σcn such that the ratio cn/cn₋₁ is an elliptic function of n, analogous to generalized hypergeometric series where the ratio is a rational function of n, and basic hypergeometric series where the ratio is a periodic function of the complex number n. They were introduced by Frenkel & Turaev (1997) in their study of elliptic 6-j symbols. For surveys of elliptic hypergeometric series see Gasper & Rahman (2004) or Spiridonov (2008).

11.1 Definitions The q-Pochhammer symbol is defined by

(a; q)n =

n−1 ∏

(1 − aq k ) = (1 − a)(1 − aq)(1 − aq 2 ) · · · (1 − aq n−1 ).

k=0

(a1 , a2 , . . . , am ; q)n = (a1 ; q)n (a2 ; q)n . . . (am ; q)n . The modified Jacobi theta function with argument x and nome p is defined by

θ(x; p) = (x, p/x; p)∞ θ(x1 , ..., xm ; p) = θ(x1 ; p)...θ(xm ; p) The elliptic shifted factorial is defined by

(a; q, p)n = θ(a; p)θ(aq; p)...θ(aq n−1 ; p) (a1 , ..., am ; q, p)n = (a1 ; q, p)n · · · (am ; q, p)n The theta hypergeometric series r₊₁Er is defined by

r+1 Er (a1 , ...ar+1 ; b1 , ..., br ; q, p; z) =

∞ ∑ (a1 , ..., ar+1 ; q; p)n n z (q, b1 , ..., br ; q, p)n n=0

The very well poised theta hypergeometric series r₊₁Vr is defined by

r+1 Vr (a1 ; a6 , a7 , ...ar+1 ; q, p; z)

=

∞ ∑ θ(a1 q 2n ; p) (a1 , a6 , a7 , ..., ar+1 ; q; p)n (qz)n θ(a ; p) (q, a q/a , a q/a , ..., a q/a ; q, p) 1 1 6 1 7 1 r+1 n n=0

31

32

CHAPTER 11. ELLIPTIC HYPERGEOMETRIC SERIES

The bilateral theta hypergeometric series rGr is defined by

r Gr (a1 , ...ar ; b1 , ..., br ; q, p; z)

=

∞ ∑ (a1 , ..., ar ; q; p)n n z (b1 , ..., br ; q, p)n n=−∞

11.2 Definitions of additive elliptic hypergeometric series The elliptic numbers are defined by

[a; σ, τ ] =

θ1 (πσa, eπiτ ) θ1 (πσ, eπiτ )

where the Jacobi theta function is defined by

θ1 (x, q) =

∞ ∑

2

(−1)n q (n+1/2) e(2n+1)ix

n=−∞

The additive elliptic shifted factorials are defined by • [a; σ, τ ]n = [a; σ, τ ][a + 1; σ, τ ]...[a + n − 1; σ, τ ] • [a1 , ..., am ; σ, τ ] = [a1 ; σ, τ ]...[am ; σ, τ ] The additive theta hypergeometric series r₊₁er is defined by ∞ ∑ [a1 , ..., ar+1 ; σ; τ ]n n z r+1 er (a1 , ...ar+1 ; b1 , ..., br ; σ, τ ; z) = [1, b1 , ..., br ; σ, τ ]n n=0

The additive very well poised theta hypergeometric series r₊₁vr is defined by

r+1 vr (a1 ; a6 , ...ar+1 ; σ, τ ; z)

=

∞ ∑ [a1 + 2n; σ, τ ] [a1 , a6 , ..., ar+1 ; σ, τ ]n zn [a ; σ, τ ] [1, 1 + a − a , ..., 1 + a − a ; σ, τ ] 1 1 6 1 r+1 n n=0

11.3 References • Frenkel, Igor B.; Turaev, Vladimir G. (1997), “Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions”, The Arnold-Gelfand mathematical seminars, Boston, MA: Birkhäuser Boston, pp. 171–204, ISBN 978-0-8176-3883-2, MR 1429892 • Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719 • Spiridonov, V. P. (2002), “Theta hypergeometric series”, Asymptotic combinatorics with application to mathematical physics (St. Petersburg, 2001), NATO Sci. Ser. II Math. Phys. Chem. 77, Dordrecht: Kluwer Acad. Publ., pp. 307–327, arXiv:math/0303204, MR 2000728 • Spiridonov, V. P. (2003), “Theta hypergeometric integrals”, Rossiĭskaya Akademiya Nauk. Algebra i Analiz 15 (6): 161–215, arXiv:math/0303205, doi:10.1090/S1061-0022-04-00839-8, MR 2044635

• Spiridonov, V. P. (2008), “Essays on the theory of elliptic hypergeometric functions”, Rossiĭskaya Akademiya Nauk. Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk 63 (3): 3–72, doi:10.1070/RM2008v063n03A MR 2479997 • Warnaar, S. Ole (2002), “Summation and transformation formulas for elliptic hypergeometric series”, Constructive Approximation. an International Journal for Approximations and Expansions 18 (4): 479–502, arXiv:math/0001006, doi:10.1007/s00365-002-0501-6, MR 1920282

Chapter 12

Fox H-function In mathematics, the Fox H-function H(x) is a generalization of the Meijer G-function introduced by Charles Fox (1961). It is defined by a Mellin–Barnes integral

m,n Hp,q

[ (a , A ) (a2 , A2 ) . . . z 1 1 (b1 , B1 ) (b2 , B2 ) . . .

∏m ∏n ] ∫ ( j=1 Γ(bj + Bj s))( j=1 Γ(1 − aj − Aj s)) 1 (ap , Ap ) ∏ ∏ z −s ds = (bq , Bq ) 2πi L ( qj=m+1 Γ(1 − bj − Bj s))( pj=n+1 Γ(aj + Aj s))

where L is a certain contour separating the poles of the two factors in the numerator. Another generalization of Fox H-function is given by Innayat Hussain AA (1987). For a further generalization of this function, useful in Physics and Statistics, see Rathie (1997). The special case for which the Fox H-function reduces to the Meijer G-function is Aj = Bk = C, C > 0 for j = 1...p and k = 1...q (Srivastava 1984, p. 50): [ (a , C) m,n Hp,q z 1 (b1 , C)

(a2 , C) (b2 , C)

... ...

] ( 1 m,n a1 , . . . , ap (ap , C) = Gp,q (bq , C) b1 , . . . , bq C

) 1/C z .

12.1 References • Fox, Charles (1961), “The G and H functions as symmetrical Fourier kernels”, Transactions of the American Mathematical Society 98: 395–429, ISSN 0002-9947, JSTOR 1993339, MR 0131578 • Innayat-Hussain, AA (1987), “New properties of hypergeometric series derivable from Feynman integrals. I: Transformation and reduction formulae”, J. Phys. A: Math. Gen. 20: 4109–4117. • Innayat-Hussain, AA (1987), “New properties of hypergeometric series derivable from Feynman integrals. II: A generalization of the H-function”, J. Phys. A: Math. Gen. 20: 4119–4128. • Mathai, A. M.; Saxena, Ram Kishore (1978), The H-function with applications in statistics and other disciplines, Halsted Press [John Wiley & Sons], New York-London-Sidney, ISBN 978-0-470-26380-8, MR 513025 • Mathai, A. M.; Saxena, Ram Kishore; Haubold, Hans J. (2010), The H-function, Berlin, New York: SpringerVerlag, ISBN 978-1-4419-0915-2, MR 2562766 • Rathie, Arjun K. (1997), “A new generalization of generalized hypergeometric function”, Le Matematiche LII: 297–310. 33

34

CHAPTER 12. FOX H-FUNCTION • Srivastava, H. M.; Gupta, K. C.; Goyal, S. P. (1982), The H-functions of one and two variables, New Delhi: South Asian Publishers Pvt. Ltd., MR 691138 • Srivastava, H. M.; Manocha, H. L. (1984). A treatise on generating functions. ISBN 0-470-20010-3.

Chapter 13

Fox–Wright function In mathematics, the Fox–Wright function (also known as Fox–Wright Psi function or just Wright function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric function pFq(z) based on an idea of E. Maitland Wright (1935): [ p Ψq

(a1 , A1 ) (b1 , B1 )

(a2 , A2 ) . . . (b2 , B2 ) . . .

] ∑ ∞ Γ(a1 + A1 n) · · · Γ(ap + Ap n) z n (ap , Ap ) ;z = . (bq , Bq ) Γ(b1 + B1 n) · · · Γ(bq + Bq n) n! n=0

(a2 , A2 ) . . . (b2 , B2 ) . . .

] ∞ Γ(b1 ) · · · Γ(bq ) ∑ Γ(a1 + A1 n) · · · Γ(ap + Ap n) z n (ap , Ap ) ;z = (bq , Bq ) Γ(a1 ) · · · Γ(ap ) n=0 Γ(b1 + B1 n) · · · Γ(bq + Bq n) n!

Its normalisation

∗ p Ψq

[

(a1 , A1 ) (b1 , B1 )

becomes pFq(z) for A₁...p = B₁...q = 1. The Fox–Wright function is a special case of the Fox H-function (Srivastava & Manocha 1984, p. 50): [ p Ψq

(a1 , A1 ) (b1 , B1 )

(a2 , A2 ) . . . (b2 , B2 ) . . .

] [ (1 − a1 , A1 ) (ap , Ap ) 1,p ; z = Hp,q+1 −z (bq , Bq ) (0, 1)

(1 − a2 , A2 ) ... (1 − ap , Ap ) (1 − b1 , B1 ) (1 − b2 , B2 ) ...

13.1 References • Wright, E. M. (1935). “The asymptotic expansion of the generalized hypergeometric function”. Proc. London Math. Soc. 10 (4): 286–293. doi:10.1112/jlms/s1-10.40.286. • Srivastava, H.M.; Manocha, H.L. (1984). A treatise on generating functions. ISBN 0-470-20010-3. • Miller, A. R.; Moskowitz, I.S. (1995). “Reduction of a Class of Fox–Wright Psi Functions for Certain Rational Parameters”. Computers Math. Applic. 30 (11): 73–82.

35

(1 − bq ,

Chapter 14

Frobenius solution to the hypergeometric equation In the following we solve the second-order differential equation called the hypergeometric differential equation using Frobenius method, named after Ferdinand Georg Frobenius. This is a method that uses the series solution for a differential equation, where we assume the solution takes the form of a series. This is usually the method we use for complicated ordinary differential equations. The solution of the hypergeometric differential equation is very important. For instance, Legendre’s differential equation can be shown to be a special case of the hypergeometric differential equation. Hence, by solving the hypergeometric differential equation, one may directly compare its solutions to get the solutions of Legendre’s differential equation, after making the necessary substitutions. For more details, please check the hypergeometric differential equation. We shall prove that this equation has three singularities, namely at x = 0, x = 1 and around infinity. However, as these will turn out to be regular singular points, we will be able to assume a solution on the form of a series. Since this is a second-order differential equation, we must have two linearly independent solutions. The problem however will be that our assumed solutions may or not be independent, or worse, may not even be defined (depending on the value of the parameters of the equation). This is why we shall study the different cases for the parameters and modify our assumed solution accordingly.

14.1 The equation Solve the hypergeometric equation around all singularities: x(1 − x)y ′′ + {γ − (1 + α + β)x} y ′ − αβy = 0

14.2 Solution around x = 0 Let P0 (x) = −αβ, P1 (x) = γ − (1 + α + β)x, P2 (x) = x(1 − x) Then

P2 (0) = P2 (1) = 0. 36

14.2. SOLUTION AROUND X = 0

37

Hence, x = 0 and x = 1 are singular points. Let’s start with x = 0. To see if it is regular, we study the following limits: (x − a)P1 (x) (x − 0)(γ − (1 + α + β)x) x(γ − (1 + α + β)x) = lim = lim =γ x→a x→0 x→0 P2 (x) x(1 − x) x(1 − x) (x − a)2 P0 (x) (x − 0)2 (−αβ) x2 (−αβ) lim = lim = lim =0 x→a x→0 x→0 x(1 − x) P2 (x) x(1 − x) lim

Hence, both limits exist and x = 0 is a regular singular point. Therefore, we assume the solution takes the form ∞ ∑

y=

ar xr+c

r=0

with a0 ≠ 0. Hence,

y′ =

∞ ∑

ar (r + c)xr+c−1

r=0

y ′′ =

∞ ∑

ar (r + c)(r + c − 1)xr+c−2 .

r=0

Substituting these into the hypergeometric equation, we get

x

∞ ∑

ar (r+c)(r+c−1)x

r+c−2

−x

2

r=0

∞ ∑

r+c−2

ar (r+c)(r+c−1)x

+γ

r=0

∞ ∑

r+c−1

ar (r+c)x

−(1+α+β)x

r=0

∞ ∑

r+c−1

ar (r+c)x

r=0

−αβ

∞ ∑

r=

That is, ∞ ∑

r+c−1

ar (r+c)(r+c−1)x

−

r=0

∞ ∑

r+c

ar (r+c)(r+c−1)x

+γ

r=0

∞ ∑

r+c−1

ar (r+c)x

−(1+α+β)

r=0

∞ ∑

r+c

ar (r+c)x

−αβ

r=0

In order to simplify this equation, we need all powers to be the same, equal to r + c − 1, the smallest power. Hence, we switch the indices as follows: ∞ ∑

ar (r + c)(r + c − 1)xr+c−1 −

r=0

∞ ∑

ar−1 (r + c − 1)(r + c − 2)xr+c−1 + γ

r=1

− (1 + α + β)

∞ ∑

∞ ∑

ar (r + c)xr+c−1

r=0

ar−1 (r + c − 1)xr+c−1 − αβ

r=1

∞ ∑

ar−1 xr+c−1 = 0

r=1

Thus, isolating the first term of the sums starting from 0 we get

a0 (c(c − 1) + γc)xc−1 +

∞ ∑

ar (r + c)(r + c − 1)xr+c−1 −

r=1

+γ

∞ ∑

ar (r + c)xr+c−1 − (1 + α + β)

r=1

∞ ∑

ar−1 (r + c − 1)(r + c − 2)xr+c−1

r=1 ∞ ∑ r=1

ar−1 (r + c − 1)xr+c−1 − αβ

∞ ∑

ar−1 xr+c−1 = 0

r=1

Now, from the linear independence of all powers of x, that is, of the functions 1, x, x2 , etc., the coefficients of xk vanish for all k. Hence, from the first term, we have

a0 (c(c − 1) + γc) = 0

∞ ∑ r=0

ar xr+c =

38

CHAPTER 14. FROBENIUS SOLUTION TO THE HYPERGEOMETRIC EQUATION

which is the indicial equation. Since a0 ≠ 0, we have

c(c − 1 + γ) = 0. Hence,

c1 = 0, c2 = 1 − γ Also, from the rest of the terms, we have

((r + c)(r + c − 1) + γ(r + c))ar + (−(r + c − 1)(r + c − 2) − (1 + α + β)(r + c − 1) − αβ)ar−1 = 0 Hence, (r + c − 1)(r + c − 2) + (1 + α + β)(r + c − 1) + αβ ar−1 (r + c)(r + c − 1) + γ(r + c) (r + c − 1)(r + c + α + β − 1) + αβ = ar−1 (r + c)(r + c + γ − 1)

ar =

But (r + c − 1)(r + c + α + β − 1) + αβ = (r + c − 1)(r + c + α − 1) + (r + c − 1)β + αβ = (r + c − 1)(r + c + α − 1) + β(r + c + α − 1) Hence, we get the recurrence relation

ar =

(r + c + α − 1)(r + c + β − 1) ar−1 , for r ≥ 1. (r + c)(r + c + γ − 1)

Let’s now simplify this relation by giving ar in terms of a0 instead of ar₋₁. From the recurrence relation (note: below, expressions of the form (u)r refer to the Pochhammer symbol).

(c + α)(c + β) a0 (c + 1)(c + γ) (c + α + 1)(c + β + 1) (c + α + 1)(c + α)(c + β)(c + β + 1) (c + α)2 (c + β)2 a2 = a1 = a0 = a0 (c + 2)(c + γ + 1) (c + 2)(c + 1)(c + γ)(c + γ + 1) (c + 1)2 (c + γ)2 (c + α + 2)(c + β + 2) (c + α)2 (c + α + 2)(c + β)2 (c + β + 2) (c + α)3 (c + β)3 a3 = a2 = a0 = a0 (c + 3)(c + γ + 2) (c + 1)2 (c + 3)(c + γ)2 (c + γ + 2) (c + 1)3 (c + γ)3 a1 =

As we can see,

ar =

(c + α)r (c + β)r a0 , for r ≥ 0 (c + 1)r (c + γ)r

Hence, our assumed solution takes the form

y = a0

∞ ∑ (c + α)r (c + β)r r=0

(c + 1)r (c + γ)r

xr+c .

We are now ready to study the solutions corresponding to the different cases for c1 − c2 = γ − 1 (this reduces to studying the nature of the parameter γ: whether it is an integer or not).

14.3. ANALYSIS OF THE SOLUTION IN TERMS OF THE DIFFERENCE Γ − 1 OF THE TWO ROOTS

39

14.3 Analysis of the solution in terms of the difference γ − 1 of the two roots 14.3.1

γ not an integer

Then y1 = y|c ₌ ₀ and y2 = y|c ₌ ₁ ₋ ᵧ. Since ∞ ∑ (c + α)r (c + β)r

y = a0

r=0

(c + 1)r (c + γ)r

xr+c ,

we have

y1 = a0

∞ ∑ (α)r (β)r r=0

y2 = a0

(1)r (γ)r

xr = a0 · 2 F1 (α, β; γ; x)

∞ ∑ (α + 1 − γ)r (β + 1 − γ)r r=0

(1 − γ + 1)r (1 − γ + γ)r

= a0 x1−γ

xr+1−γ

∞ ∑ (α + 1 − γ)r (β + 1 − γ)r

(1)r (2 − γ)r

r=0

xr

= a0 x1−γ 2 F1 (α − γ + 1, β − γ + 1; 2 − γ; x) Hence, y = A′ y1 + B ′ y2 . Let A′ a0 = a and B′ a0 = B. Then

y = A2 F1 (α, β; γ; x) + Bx1−γ 2 F1 (α − γ + 1, β − γ + 1; 2 − γ; x)

14.3.2

γ=1

Then y1 = y|c ₌ ₀. Since γ = 1, we have

y = a0

∞ ∑ (c + α)r (c + β)r

(c + 1)2r

r=0

xr+c .

Hence,

y1 = a0

∞ ∑ (α)r (β)r

(1)r (1)r

xr = a02 F1 (α, β; 1; x)

r=0 ∂y y2 = . ∂c c=0

To calculate this derivative, let

Mr =

(c + α)r (c + β)r . (c + 1)2r

Then ( ln(Mr ) = ln

(c + α)r (c + β)r (c + 1)2r

) = ln(c + α)r + ln(c + β)r − 2 ln(c + 1)r

40

CHAPTER 14. FROBENIUS SOLUTION TO THE HYPERGEOMETRIC EQUATION

But

ln(c + α)r = ln ((c + α)(c + α + 1) · · · (c + α + r − 1)) =

r−1 ∑

ln(c + α + k).

k=0

Hence,

ln(Mr ) =

r−1 ∑

ln(c + α + k) +

k=0

=

r−1 ∑

r−1 ∑

ln(c + β + k) − 2

k=0

r−1 ∑

ln(c + 1 + k)

k=0

(ln(c + α + k) + ln(c + β + k) − 2 ln(c + 1 + k))

k=0

Differentiating both sides of the equation with respect to c, we get: r−1 (

∑ 1 ∂Mr = Mr ∂c

k=0

1 1 2 + − c+α+k c+β+k c+1+k

) .

Hence, ) r−1 ( 1 (c + α)r (c + β)r ∑ 1 2 ∂Mr = + − . ∂c (c + 1)2r c+α+k c+β+k c+1+k k=0

Now,

y = a0 xc

∞ ∑ (c + α)r (c + β)r

(c + 1)2r

r=0

xr = a0 xc

∞ ∑

Mr xr .

r=0

Hence, ( {r−1 ( )}) ∞ ∞ ∑ ∑ ∂y (c + α)r (c + β)r r (c + α)r (c + β)r ∑ 1 1 2 c c = a0 x ln(x) x + a0 x + − xr 2 2 ∂c (c + 1) (c + 1) c + α + k c + β + k c + 1 + k r r r=0 r=0 k=0 ( )) ∞ r−1 ( ∑ ∑ (c + α) (c + β) 1 1 2 r r = a0 x c ln x + + − xr . (c + 1)r )2 c+α+k c+β+k c+1+k r=0 k=0

For c = 0, we get

y2 = a0

∞ ∑ (α)r (β)r r=0

(1)2r

( ln x +

r−1 ( ∑ k=0

1 1 2 + − α+k β+k 1+k

)) xr .

Hence, y = C′y1 + D′y2 . Let C′a0 = C and D′a0 = D. Then

y = C 2 F1 (α, β; 1; x) + D

∞ ∑ (α)r (β)r r=0

(1)2r

( ln(x) +

r−1 ( ∑ k=0

1 1 2 + − α+k β+k 1+k

)) xr

14.3. ANALYSIS OF THE SOLUTION IN TERMS OF THE DIFFERENCE Γ − 1 OF THE TWO ROOTS

14.3.3

41

γ an integer and γ ≠ 1

γ≤0 From the recurrence relation

ar =

(r + c + α − 1)(r + c + β − 1) ar−1 , (r + c)(r + c + γ − 1)

we see that when c = 0 (the smaller root), a₁ ₋ ᵧ → ∞. Hence, we must make the substitution a0 = b0 (c - ci), where ci is the root for which our solution is infinite. Hence, we take a0 = b0 c and our assumed solution takes the new form

yb = b0 xc

∞ ∑ c(c + α)r (c + β)r r=0

(c + 1)r (c + γ)r

xr

Then y1 = yb|c ₌ ₀. As we can see, all terms before c(c + α)1−γ (c + β)1−γ 1−γ x (c + 1)1−γ (c + γ)1−γ vanish because of the c in the numerator. Starting from this term however, the c in the numerator vanishes. To see this, note that

(c + γ)1−γ = (c + γ)(c + γ + 1) · · · c. Hence, our solution takes the form (

(α)1−γ (β)1−γ 1−γ (α)2−γ (β)2−γ 2−γ (α)3−γ (β)3−γ y1 = b0 x + x + x3−γ + · · · (1)1−γ (γ)−γ (1)2−γ (γ)−γ (1) (1)3−γ (γ)−γ (1)(2) ∞ ∑ b0 (α)r (β)r = xr . (γ)−γ r=1−γ (1)r (1)r+γ−1

)

Now, ∂yb y2 = . ∂c c=1−γ To calculate this derivative, let

Mr =

c(c + α)r (c + β)r . (c + 1)r (c + γ)r

Then following the method in the previous case, we get

∂Mr c(c + α)r (c + β)r = ∂c (c + 1)r (c + γ)r

{

r−1 (

1 ∑ + c

k=0

1 1 1 1 + − − c+α+k c+β+k c+1+k c+γ+k

Now,

yb = b0

∞ ∑ c(c + α)r (c + β)r r=0

(c + 1)r (c + γ)r

xr+c = b0 xc

∞ ∑ r=0

Mr xr .

)} .

42

CHAPTER 14. FROBENIUS SOLUTION TO THE HYPERGEOMETRIC EQUATION

Hence,

∂y = b0 x c ∂c

( ln(x)

∞ ∑ c(c + α)r (c + β)r r=0

(c + 1)r (c + γ)r

r

x +

∞ ∑ c(c + α)r (c + β)r r=0

(c + 1)r (c + γ)r

{

r−1 (

1 ∑ + c

k=0

1 1 1 1 + − − c+α+k c+β+k c+1+k c+γ

Hence, ∞ ∑ ∂y c(c + α)r (c + β)r = b0 xc ∂c (c + 1)r (c + γ)r r=0

(

r−1 (

1 ∑ ln(x) + + c

k=0

1 1 1 1 + − − c+α+k c+β+k c+1+k c+γ+k

)) xr .

At c = 1 − γ, we get y2 . Hence, y = E′ y1 + F′ y2 . Let E′ b0 = E and F′ b0 = F. Then ∞ ∞ ∑ ∑ (α)r (β)r (1 − γ)(α + 1 − γ)r (β + 1 − γ)r E xr + F x1−γ y= (γ)−γ r=1−γ (1)r (1)r+γ−1 (2 − γ)r (1)r r=0 r−1 (

∑ 1 + + 1−γ

k=0

(

1 1 1 1 + − − α+k+1−γ β+k+1−γ 2+k−γ 1+k

ln(x)+ )) xr .

γ>1 From the recurrence relation

ar =

(r + c + α − 1)(r + c + β − 1) ar−1 , (r + c)(r + c + γ − 1)

we see that when c = 1 - γ (the smaller root), aᵧ₋₁ → ∞. Hence, we must make the substitution a0 = b0 (c − ci), where ci is the root for which our solution is infinite. Hence, we take a0 = b0 (c + γ - 1) and our assumed solution takes the new form:

yb = b0 xc

∞ ∑ (c + γ − 1)(c + α)r (c + β)r r=0

(c + 1)r (c + γ)r

xr .

Then y1 = yb|c ₌ ₁ - ᵧ. All terms before (c + γ − 1)(c + α)γ−1 (c + β)γ−1 γ−1 x (c + 1)γ−1 (c + γ)γ−1 vanish because of the c + γ - 1 in the numerator. Starting from this term, however, the c + γ - 1 in the numerator vanishes. To see this, note that

(c + 1)γ−1 = (c + 1)(c + 2) · · · (c + γ − 1). Hence, our solution takes the form (

(α + 1 − γ)γ−1 (β + 1 − γ)γ−1 γ−1 (α + 1 − γ)γ (β + 1 − γ)γ γ x + x + ··· (2 − γ)γ−2 (1)γ−1 (2 − γ)γ−2 (1)(1)γ ∞ ∑ b0 (α + 1 − γ)r (β + 1 − γ)r r = x1−γ x . (2 − γ)γ−2 (1)r (1)r+1−γ r=γ−1

y1 = b0 x1−γ

)

14.4. SOLUTION AROUND X = 1

43

Now, ∂yb y2 = ∂c c=0 To calculate this derivative, let

Mr =

(c + γ − 1)(c + α)r (c + β)r . (c + 1)r (c + γ)r

Then following the method in the second case above,

(c + γ − 1)(c + α)r (c + β)r ∂Mr = ∂c (c + 1)r (c + γ)r

(

r−1 (

∑ 1 + c+γ−1

k=0

1 1 1 1 + − − c+α+k c+β+k c+1+k c+γ+k

))

Now,

yb = b0

∞ ∑ (c + γ − 1)(c + α)r (c + β)r

(c + 1)r (c + γ)r

r=0

xr+c = b0 xc

∞ ∑

Mr xr .

r=0

Hence,

∂y = b0 x c ∂c

( ln(x)

∞ ∑ (c + γ − 1)(c + α)r (c + β)r r=0

(c + 1)r (c + γ)r

r−1 ( ∑

r

x +

∞ ∑ (c + γ − 1)(c + α)r (c + β)r r=0

(c + 1)r (c + γ)r )} )

{

1 + c+γ−1

1 1 1 1 + − − xr c+α+k c+β+k c+1+k c+γ+k k=0 ( }) ∞ r−1 { ∑ ∑ 1 1 1 1 (c + γ − 1)(c + α) (c + β) 1 r r = b0 x c ln(x) + + + − − (c + 1)r (c + γ)r c+γ−1 c+α+k c+β+k c+1+k c+γ+k r=0 +

k=0

At c = 0 we get y2 . Hence, y = G′y1 + H′y2 . Let G′b0 = E and H′b0 = F. Then ∞ ∑ (α + 1 − γ)r (β + 1 − γ)r r G 1−γ x x + y= (2 − γ)γ−2 (1)r (1)r+1−γ r=γ−1 ( }) ∞ r−1 { ∑ ∑ (1 − γ)(α + 1 − γ)r (β + 1 − γ)r 1 1 1 1 1 +H ln(x) + + + − − xr . (2 − γ) (1) γ − 1 α + k β + k 1 + k γ + k r r r=0 k=0

14.4 Solution around x = 1 Let us now study the singular point x = 1. To see if it is regular, (x − 1)(γ − (1 + α + β)x) −(γ − (1 + α + β)x) (x − a)P1 (x) = lim = lim =1+α+β−γ x→a x→1 x→1 P2 (x) x(1 − x) x (x − a)2 P0 (x) (x − 1)2 (−αβ) (x − 1)αβ lim = lim = lim =0 x→a x→1 x→1 P2 (x) x(1 − x) x lim

Hence, both limits exist and x = 1 is a regular singular point. Now, instead of assuming a solution on the form

44

y=

CHAPTER 14. FROBENIUS SOLUTION TO THE HYPERGEOMETRIC EQUATION

∞ ∑

ar (x − 1)r+c ,

r=0

we will try to express the solutions of this case in terms of the solutions for the point x = 0. We proceed as follows: we had the hypergeometric equation

x(1 − x)y ′′ + (γ − (1 + α + β)x)y ′ − αβy = 0. Let z = 1 − x. Then

dy dz dy dy = × =− = −y ′ dx dz ( dx) dz ( ) ( ) d2 y d dy d dy d dy dz d2 y = = − = − × = = y ′′ dx2 dx dx dx dz dz dz dx dz 2 Hence, the equation takes the form

z(1 − z)y ′′ + (α + β − γ + 1 − (1 + α + β)z)y ′ − αβy = 0. Since z = 1 − x, the solution of the hypergeometric equation at x = 1 is the same as the solution for this equation at z = 0. But the solution at z = 0 is identical to the solution we obtained for the point x = 0, if we replace each γ by α + β − γ + 1. Hence, to get the solutions, we just make this substitution in the previous results. Note also that for x = 0, c1 = 0 and c2 = 1 − γ. Hence, in our case, c1 = 0 while c2 = γ − α − β. Let us now write the solutions. In the following we replaced each z by 1 - x.

14.5 Analysis of the solution in terms of the difference γ − α − β of the two roots To simplify notation from now on denote γ − α − β by Δ, therefore γ = Δ + α + β.

14.5.1

Δ not an integer

{ } y = A {2 F1 (α, β; −∆ + 1; 1 − x)} + B (1 − x)∆ 2 F1 (∆ + β, ∆ + α; ∆ + 1; 1 − x)

14.5.2

Δ=0 {

y = C {2 F1 (α, β; 1; 1 − x)} + D

∞ ∑ (α)r (β)r r=0

14.5.3

(1)2r

( ln(1 − x) +

r−1 ( ∑ k=0

1 1 2 + − α+k β+k 1+k

))

} (1 − x)r

Δ is a non-zero integer

Δ>0  ∞  ∑ 1 (α)r (β)r (1 − x)r + y=E  (−∆ + 1)∆−1  (1)r (1)r−∆ r=1−∆−α−β { ( )) ∞ r−1 ( ∑ ∑ 1 (∆)(∆ + α)r (∆ + β)r 1 1 1 1 ∆ + F (1 − x) ln(1 − x) + + + − − (∆ + 1)r (1)r ∆ ∆+α+k ∆+β+k ∆+1+k 1+k r=0  

k=0

14.6. SOLUTION AROUND INFINITY

45

Δ 1.

25.3 Integral representations The Legendre functions can be written as contour integrals. For example (needs clarification: how is Pλ (x) related to Pλµ (x) ?) 1 Pλ (z) = 2πi

∫ 1,z

(t2 − 1)λ dt 2λ (t − z)λ+1

where the contour circles around the points 1 and z in the positive direction and does not circle around −1. For real x, we have

Ps (x) =

1 2π

∫

π

( x+

−π

∫ )s )s √ √ 1 1( dt x2 − 1 cos θ dθ = x + x2 − 1(2t − 1) √ , π 0 t(1 − t)

s∈C

25.4 Legendre function as characters The real integral representation of Ps are very useful in the study of harmonic analysis on L1 (G//K) where G//K is the double coset space of SL(2, R) (see Zonal spherical function). Actually the Fourier transform on L1 (G//K) is given by

L1 (G//K) ∋ f 7→ fˆ where ∫

∞

fˆ(s) =

f (x)Ps (x)dx, 1

−1 ≤ ℜ(s) ≤ 0

90

CHAPTER 25. LEGENDRE FUNCTION

25.5 References • Abramowitz, Milton; Stegun, Irene A., eds. (1965), “Chapter 8”, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 332, ISBN 978-0486612720, MR 0167642. • Courant, Richard; Hilbert, David (1953), Methods of Mathematical Physics, Volume 1, New York: Interscience Publisher, Inc. • Dunster, T. M. (2010), “Legendre and Related Functions”, in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248 • Ivanov, A.B. (2001), “L/l058030”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Snow, Chester (1952) [1942], Hypergeometric and Legendre functions with applications to integral equations of potential theory, National Bureau of Standards Applied Mathematics Series, No. 19, Washington, D.C.: U. S. Government Printing Office, MR 0048145 • Whittaker, E. T.; Watson, G. N. (1963), A Course in Modern Analysis, Cambridge University Press, ISBN 978-0-521-58807-2

25.6 External links • Legendre function P on the Wolfram functions site. • Legendre function Q on the Wolfram functions site. • Associated Legendre function P on the Wolfram functions site. • Associated Legendre function Q on the Wolfram functions site.

Chapter 26

List of hypergeometric identities Below is a list of hypergeometric identities. • Hypergeometric function lists identities for the Gaussian hypergeometric function • Generalized hypergeometric function lists identities for more general hypergeometric functions • Bailey’s list is a list of the hypergeometric function identities in Bailey (1935) given by Koepf (1995). • Wilf–Zeilberger pair is a method for proving hypergeometric identities

26.1 References • Bailey, W. N. (1935), Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Cambridge University Press, MR 0185155 • Koepf, Wolfram (1995), “Algorithms for m-fold hypergeometric summation”, Journal of Symbolic Computation 20 (4): 399–417, doi:10.1006/jsco.1995.1056, ISSN 0747-7171, MR 1384455

91

Chapter 27

MacRobert E function In mathematics, the E-function was introduced by Thomas Murray MacRobert (1937–38) to extend the generalized hypergeometric series pFq(·) to the case p > q + 1. The underlying objective was to define a very general function that includes as particular cases the majority of the special functions known until then. However, this function had no great impact on the literature as it can always be expressed in terms of the Meijer G-function, while the opposite is not true, so that the G-function is of a still more general nature.

27.1 Definition There are several ways to define the MacRobert E-function; the following definition is in terms of the generalized hypergeometric function: • when p ≤ q and x ≠ 0, or p = q + 1 and |x| > 1: (

a E p bq

) ∏p ( Γ(aj ) ap x = ∏j=1 F p q q bq Γ(bj ) j=1

) − x−1

• when p ≥ q + 2, or p = q + 1 and |x| < 1: ) p ∏p ∗ ∑ ap j=1 Γ(aj − ah ) ∏ E x = Γ(ah ) xah q bq Γ(b − a ) j h j=1 (

( q+1 Fp−1

h=1

) ah , 1 + ah − b1 , . . . , 1 + ah − bq p−q (−1) x . 1 + ah − a1 , . . . , ∗, . . . , 1 + ah − ap

The asterisks here remind us to ignore the contribution with index j = h as follows: In the product this amounts to replacing Γ(0) with 1, and in the argument of the hypergeometric function this amounts to shortening the vector length from p to p − 1. Evidently, this definition covers all values of p and q.

27.2 Relationship with the Meijer G-function The MacRobert E-function can always be expressed in terms of the Meijer G-function: (

a E p bq

) ( 1, bq x = G p, 1 q+1, p ap

) x

where the parameter values are unrestricted, i.e. this relation holds without exception.

27.3 References • Andrews, L. C. (1985). Special Functions for Engineers and Applied Mathematicians. New York: MacMillan. ISBN 0-02-948650-5. 92

27.4. EXTERNAL LINKS

93

• Erdélyi, A.; Magnus, W.; Oberhettinger, F. & Tricomi, F. G. (1953). Higher Transcendental Functions (PDF). Vol. 1. New York: McGraw–Hill. (see § 5.2, “Definition of the E-Function”, p. 203) • Gradshteyn, Izrail' Solomonovich & Ryzhik, Iosif Moiseevich (1971). Tablitsy integralov, summ, ryadov i proizvedeniy [Tables of integrals, sums, series and products] (in Russian) (5th ed.). Moscow: Nauka. (see Chapter 9.4) • MacRobert, T. M. (1937–38). “Induction proofs of the relations between certain asymptotic expansions and corresponding generalised hypergeometric series”. Proc. Roy. Soc. Edinburgh 58: 1–13. JFM 64.0337.01. • MacRobert, T. M. (1962). “Barnes integrals as a sum of E-functions”. Mathematische Annalen 147 (3): 240– 243. doi:10.1007/bf01470741. Zbl 0100.28601.

27.4 External links • Weisstein, Eric W., “MacRobert’s E-Function”, MathWorld.

Chapter 28

Meijer G-function In mathematics, the G-function was introduced by Cornelis Simon Meijer (1936) as a very general function intended to include most of the known special functions as particular cases. This was not the only attempt of its kind: the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meijer’s G-function was able to include those as particular cases as well. The first definition was made by Meijer using a series; nowadays the accepted and more general definition is via a line integral in the complex plane, introduced in its full generality by Arthur Erdélyi in 1953. With the modern definition, the majority of the established special functions can be represented in terms of the Meijer G-function. A notable property is the closure of the set of all G-functions not only under differentiation but also under indefinite integration. In combination with a functional equation that allows to liberate from a G-function G(z) any factor zρ that is a constant power of its argument z, the closure implies that whenever a function is expressible as a G-function of a constant multiple of some constant power of the function argument, f(x) = G(cxγ ), the derivative and the antiderivative of this function are expressible so too. The wide coverage of special functions also lends power to uses of Meijer’s G-function other than the representation and manipulation of derivatives and antiderivatives. Thus, the definite integral over the positive real axis of any function g(x) that can be written as a product G1 (cxγ )·G2 (dxδ ) of two G-functions with rational γ/δ equals just another G-function, and generalizations of integral transforms like the Hankel transform and the Laplace transform and their inverses result when suitable G-function pairs are employed as transform kernels. A still more general function, which introduces additional parameters into Meijer’s G-function, is Fox’s H-function.

28.1 Definition of the Meijer G-function A general definition of the Meijer G-function is given by the following line integral in the complex plane (Bateman & Erdélyi 1953, § 5.3-1): ( m,n Gp,q

a1 , . . . , ap b1 , . . . , bq

∏m ∏n ) ∫ j=1 Γ(bj − s) j=1 Γ(1 − aj + s) z = 1 ∏q ∏ z s ds, 2πi L j=m+1 Γ(1 − bj + s) pj=n+1 Γ(aj − s)

where Γ denotes the gamma function. This integral is of the so-called Mellin–Barnes type, and may be viewed as an inverse Mellin transform. The definition holds under the following assumptions: • 0 ≤ m ≤ q and 0 ≤ n ≤ p, where m, n, p and q are integer numbers • ak − bj ≠ 1, 2, 3, ... for k = 1, 2, ..., n and j = 1, 2, ..., m, which implies that no pole of any Γ(bj − s), j = 1, 2, ..., m, coincides with any pole of any Γ(1 − ak + s), k = 1, 2, ..., n • z≠0 Note that for historical reasons the first lower and second upper index refer to the top parameter row, while the second lower and first upper index refer to the bottom parameter row. One often encounters the following more synthetic notation using vectors: 94

28.1. DEFINITION OF THE MEIJER G-FUNCTION

( m,n Gp,q

) ( a1 , . . . , ap m,n ap z = G p,q b1 , . . . , bq bq

95

) z .

Implementations of the G-function in computer algebra systems typically employ separate vector arguments for the four (possibly empty) parameter groups a1 ... an, an₊₁ ... ap, b1 ... bm, and bm₊₁ ... bq, and thus can omit the orders p, q, n, and m as redundant. The L in the integral represents the path to be followed while integrating. Three choices are possible for this path: 1. L runs from −i∞ to +i∞ such that all poles of Γ(bj − s), j = 1, 2, ..., m, are on the right of the path, while all poles of Γ(1 − ak + s), k = 1, 2, ..., n, are on the left. The integral then converges for |arg z| < δ π, where δ = m + n − 12 (p + q); an obvious prerequisite for this is δ > 0. The integral additionally converges for |arg z| = δ π ≥ 0 if (q − p) (σ + 1 ⁄2 ) > Re(ν) + 1, where σ represents Re(s) as the integration variable s approaches both +i∞ and −i∞, and where ν=

q ∑ j=1

bj −

p ∑

aj .

j=1

As a corollary, for |arg z| = δ π and p = q the integral converges independent of σ whenever Re(ν) < −1. 2. L is a loop beginning and ending at +∞, encircling all poles of Γ(bj − s), j = 1, 2, ..., m, exactly once in the negative direction, but not encircling any pole of Γ(1 − ak + s), k = 1, 2, ..., n. Then the integral converges for all z if q > p ≥ 0; it also converges for q = p > 0 as long as |z| < 1. In the latter case, the integral additionally converges for |z| = 1 if Re(ν) < −1, where ν is defined as for the first path. 3. L is a loop beginning and ending at −∞ and encircling all poles of Γ(1 − ak + s), k = 1, 2, ..., n, exactly once in the positive direction, but not encircling any pole of Γ(bj − s), j = 1, 2, ..., m. Now the integral converges for all z if p > q ≥ 0; it also converges for p = q > 0 as long as |z| > 1. As noted for the second path too, in the case of p = q the integral also converges for |z| = 1 when Re(ν) < −1. The conditions for convergence are readily established by applying Stirling’s asymptotic approximation to the gamma functions in the integrand. When the integral converges for more than one of these paths, the results of integration can be shown to agree; if it converges for only one path, then this is the only one to be considered. In fact, numerical path integration in the complex plane constitutes a practicable and sensible approach to the calculation of Meijer G-functions. As a consequence of this definition, the Meijer G-function is an analytic function of z with possible exception of the origin z = 0 and of the unit circle |z| = 1.

28.1.1

Differential equation

The G-function satisfies the following linear differential equation of order max(p,q):  (−1)p−m−n

 ) ∏ ) p ( q ( ∏ d d z − aj + 1 − − bj  G(z) = 0. z z dz dz j=1 j=1

For a fundamental set of solutions of this equation in the case of p ≤ q one may take: ( 1,p Gp,q

) a1 , . . . , a p (−1)p−m−n+1 z , bh , b1 , . . . , bh−1 , bh+1 , . . . , bq

h = 1, 2, . . . , q,

96

CHAPTER 28. MEIJER G-FUNCTION

and similarly in the case of p ≥ q: ( q,1 Gp,q

) ah , a1 , . . . , ah−1 , ah+1 , . . . , ap q−m−n+1 (−1) z , b1 , . . . , bq

h = 1, 2, . . . , p.

These particular solutions are analytic except for a possible singularity at z = 0 (as well as a possible singularity at z = ∞), and in the case of p = q also an inevitable singularity at z = (−1)p−m−n . As will be seen presently, they can be identified with generalized hypergeometric functions pFq₋₁ of argument (−1)p−m−n z that are multiplied by a power zbh , and with generalized hypergeometric functions qFp₋₁ of argument (−1)q−m−n z−1 that are multiplied by a power zah−1 , respectively.

28.2 Relationship between the G-function and the generalized hypergeometric function If the integral converges when evaluated along the second path introduced above, and if no confluent poles appear among the Γ(bj − s), j = 1, 2, ..., m, then the Meijer G-function can be expressed as a sum of residues in terms of generalized hypergeometric functions pFq₋₁ (Slater’s theorem): ( m,n Gp,q

ap bq (

× p Fq−1

∏n ) ∑ m ∏m ∗ bh j=1 Γ(bj − bh ) j=1 Γ(1 + bh − aj ) z z = ∏ ∏ × p q Γ(aj − bh ) Γ(1 + bh − bj ) h=1

j=m+1

j=n+1

) 1 + bh − ap p−m−n (−1) z . (1 + bh − bq )∗

For the integral to converge along the second path one must have either p < q, or p = q and |z| < 1, and for the poles to be distinct no pair among the bj, j = 1, 2, ..., m, may differ by an integer or zero. The asterisks in the relation remind us to ignore the contribution with index j = h as follows: In the product this amounts to replacing Γ(0) with 1, and in the argument of the hypergeometric function, if we recall the meaning of the vector notation,

1 + bh − bq = (1 + bh − b1 ), . . . , (1 + bh − bj ), . . . , (1 + bh − bq ), this amounts to shortening the vector length from q to q−1. Note that when m = 0, the second path does not contain any pole, and so the integral must vanish identically, ( 0,n Gp,q

) ap z = 0, bq

if either p < q, or p = q and |z| < 1. Similarly, if the integral converges when evaluated along the third path above, and if no confluent poles appear among the Γ(1 − ak + s), k = 1, 2, ..., n, then the G-function can be expressed as: ( m,n Gp,q

ap bq (

× q Fp−1

∏m ) ∑ n ∏n ∗ ah −1 j=1 Γ(ah − aj ) j=1 Γ(1 − ah + bj ) z z = ∏ ∏ × p q Γ(1 − ah + aj ) Γ(ah − bj ) h=1

1 − a h + bq (1 − ah + ap )∗

j=n+1

j=m+1

) (−1)q−m−n z −1 .

For this, either p > q, or p = q and |z| > 1 are required, and no pair among the ak, k = 1, 2, ..., n, may differ by an integer or zero. For n = 0 one consequently has: ( m,0 Gp,q

ap bq

) z = 0,

28.3. BASIC PROPERTIES OF THE G-FUNCTION

97

if either p > q, or p = q and |z| > 1. On the other hand, any generalized hypergeometric function can readily be expressed in terms of the Meijer Gfunction: ( p Fq

) ( Γ(bq ) 1, p 1 − ap ap z = G bq Γ(ap ) p, q+1 0, 1 − bq

) ( 1, bq − z = Γ(bq ) G p, 1 Γ(ap ) q+1, p ap

) − z −1 ,

where we have made use of the vector notation:

Γ(ap ) =

p ∏

Γ(aj ).

j=1

This holds unless a nonpositive integer value of at least one of its parameters a reduces the hypergeometric function to a finite polynomial, in which case the gamma prefactor of either G-function vanishes and the parameter sets of the G-functions violate the requirement ak − bj ≠ 1, 2, 3, ... for k = 1, 2, ..., n and j = 1, 2, ..., m from the definition above. Apart from this restriction, the relationship is valid whenever the generalized hypergeometric series pFq(z) converges, i. e. for any finite z when p ≤ q, and for |z| < 1 when p = q + 1. In the latter case, the relation with the G-function automatically provides the analytic continuation of pFq(z) to |z| ≥ 1 with a branch cut from 1 to ∞ along the real axis. Finally, the relation furnishes a natural extension of the definition of the hypergeometric function to orders p > q + 1. By means of the G-function we can thus solve the generalized hypergeometric differential equation for p > q + 1 as well.

28.2.1

Polynomial cases

To express polynomial cases of generalized hypergeometric functions in terms of Meijer G-functions, a linear combination of two G-functions is needed in general: ∏q ∏p ) −h, ap j=m+1 Γ(bj ) j=n+1 Γ(1 − aj ) ∏ ∏ × z = h! n m bq j=1 Γ(aj ) j=1 Γ(1 − bj ) [ ( ) ( 1 − ap , h + 1 h + 1, 1 − ap m+1, n m, n+1 p−m−n h × Gp+1, (−1) z + (−1) G q+1 p+1, q+1 0, 1 − bq 1 − bq , 0 (

p+1 Fq

)] (−1)p−m−n z ,

where h = 0, 1, 2, ... equals the degree of the polynomial p₊₁Fq(z). The orders m and n can be chosen freely in the ranges 0 ≤ m ≤ q and 0 ≤ n ≤ p, which allows to avoid that specific integer values or integer differences among the parameters a and b of the polynomial give rise to divergent gamma functions in the prefactor or to a conflict with the definition of the G-function. Note that the first G-function vanishes for n = 0 if p > q, while the second G-function vanishes for m = 0 if p < q. Again, the formula can be verified by expressing the two G-functions as sums of residues; no cases of confluent poles permitted by the definition of the G-function need be excluded here.

28.3 Basic properties of the G-function As can be seen from the definition of the G-function, if equal parameters appear among the a and b determining the factors in the numerator and the denominator of the integrand, the fraction can be simplified, and the order of the function thereby be reduced. Whether the order m or n will decrease depends of the particular position of the parameters in question. Thus, if one of the ak, k = 1, 2, ..., n, equals one of the bj, j = m + 1, ..., q, the G-function lowers its orders p, q and n: ( m,n Gp,q

) ( a1 , a2 , . . . , ap a2 , . . . , ap m, n−1 z = G p−1, q−1 b , . . . , b b1 , . . . , bq−1 , a1 1 q−1

) z ,

n, p, q ≥ 1.

For the same reason, if one of the ak, k = n + 1, ..., p, equals one of the bj, j = 1, 2, ..., m, then the G-function lowers its orders p, q and m:

98

CHAPTER 28. MEIJER G-FUNCTION

( m,n Gp,q

) ( a1 , . . . , ap−1 , b1 a1 , . . . , ap−1 m−1, n z = G p−1, q−1 b 1 , b2 , . . . , b q b2 , . . . , bq

) z ,

m, p, q ≥ 1.

Starting from the definition, it is also possible to derive the following properties: ) ) ( m,n ap + ρ z z = Gp,q bq + ρ z , ) ) ( ( ′ ′ m, n+1 α, ap , α m, n+1 α , ap , α α′ −α ′ Gp+2, q Gp+2, q z = (−1) z , n ≤ p, α − α ∈ Z, bq bq ) ) ( ( ap ap m+1, n m+1, n β ′ −β z , m ≤ q, β ′ − β ∈ Z, z = (−1) Gp, q+2 Gp, q+2 β, bq , β ′ β ′ , bq , β ) ) ( ( α, ap ap , α m, n+1 m+1, n β−α z , m ≤ q, β − α = 0, 1, 2, . . . , Gp+1, q+1 z = (−1) Gp+1, q+1 bq , β β, bq ) ( ) ( m,n ap n,m 1 − bq −1 Gp,q z = G z , q,p bq 1 − ap ( ) ( ) h1+ν+(p−q)/2 hm, hn a1 /h, . . . , (a1 + h − 1)/h, . . . , ap /h, . . . , (ap + h − 1)/h zh m,n ap Gp,q z = Ghp, hq , bq b1 /h, . . . , (b1 + h − 1)/h, . . . , bq /h, . . . , (bq + h − 1)/h hh(q−p) (2π)(h−1)δ The abbreviations ν and δ were introduced in the definition of the G-function above. (

ρ

m,n Gp,q

28.3.1

ap bq

Derivatives and antiderivatives

Concerning derivatives of the G-function, one finds these relationships: ) [ ( )] ( d 1−a1 m,n ap −a1 m,n a1 − 1, a2 , . . . , ap z Gp,q z =z Gp,q z , n ≥ 1, bq bq dz ) [ ( )] ( d 1−ap m,n ap −ap m,n a1 , . . . , ap−1 , ap − 1 z Gp,q z = −z Gp,q z , n < p. bq bq dz ) [ ( )] ( d ap m,n ap m,n z , m ≥ 1, z −b1 Gp,q z = −z −1−b1 Gp,q bq b1 + 1, b2 , . . . , bq dz ) [ ( )] ( d ap −bq m,n ap −1−bq m,n z , m < q, z Gp,q z =z Gp,q b b , . . . , b , b + 1 dz q 1 q−1 q From these four, equivalent relations can be deduced by simply evaluating the derivative on the left-hand side and manipulating a bit. One obtains for example: ( ) ( d m,n ap m,n a1 − 1, a2 , . . . , ap G z = Gp,q z bq dz p,q bq

) ( m,n ap z + (a1 − 1) Gp,q bq

) z ,

n ≥ 1.

Moreover, for derivatives of arbitrary order h, one has ) ) ( ) ( ( dh 0, ap ap , 0 m, n+1 m+1, n m,n ap h G z = Gp+1, q+1 z = (−1) Gp+1, q+1 z z , bq , h h, bq dz h p,q bq ( ) ( ) ( h ap , 1 − h −1 1 − h, ap m+1, n m, n+1 m,n ap −1 h h d G z = Gp+1, q+1 z = (−1) Gp+1, q+1 z 1, bq bq , 1 dz h p,q bq h

) −1 z ,

which hold for h < 0 as well, thus allowing to obtain the antiderivative of any G-function as easily as the derivative. By choosing one or the other of the two results provided in either formula, one can always prevent the set of parameters in the result from violating the condition ak − bj ≠ 1, 2, 3, ... for k = 1, 2, ..., n and j = 1, 2, ..., m that is imposed by the definition of the G-function. Note that each pair of results becomes unequal in the case of h < 0. From these relationships, corresponding properties of the Gauss hypergeometric function and of other special functions can be derived.

h ∈ N.

28.4. DEFINITE INTEGRALS INVOLVING THE G-FUNCTION

28.3.2

99

Recurrence relations

By equating different expressions for the first-order derivatives, one arrives at the following 3-term recurrence relations among contiguous G-functions: ) ) ) ( ( ap m,n a1 − 1, a2 , . . . , ap m,n a1 , . . . , ap−1 , ap − 1 = G +G z z p,q p,q z , 1 ≤ n < p, bq b1 , . . . , b q b1 , . . . , bq ) ) ( ) ( ( a1 , . . . , a p a1 , . . . , a p m,n ap m,n m,n z , 1 ≤ m < q, (b1 − bq ) Gp,q z = Gp,q z + Gp,q bq b1 + 1, b2 , . . . , bq b1 , . . . , bq−1 , bq + 1 ) ) ( ) ( ( a1 , . . . , ap m,n ap m,n a1 − 1, a2 , . . . , ap m,n (b1 −a1 +1) Gp,q z = Gp,q z +Gp,q b1 + 1, b2 , . . . , bq z , n ≥ 1, m ≥ 1, bq b1 , . . . , bq ) ) ( ) ( ( a1 , . . . , ap m,n ap m,n a1 , . . . , ap−1 , ap − 1 m,n z , n < p, m < q. (ap −bq −1) Gp,q z = Gp,q z +Gp,q bq b1 , . . . , b q b1 , . . . , bq−1 , bq + 1 (

m,n (ap −a1 ) Gp,q

Similar relations for the diagonal parameter pairs a1 , bq and b1 , ap follow by suitable combination of the above. Again, corresponding properties of hypergeometric and other special functions can be derived from these recurrence relations.

28.3.3

Multiplication theorems

Provided that z ≠ 0, the following relationships hold: ( m,n Gp,q

( m,n Gp,q

( m,n Gp,q

( m,n Gp,q

ap bq ap bq ap bq ap bq

( ) ∞ ∑ (1 − w)h m,n ap wz = wb1 G p,q b1 + h, b2 , . . . , bq h!

) z ,

m ≥ 1,

h=0

) ( ) ∞ ∑ (w − 1)h m,n ap z , m < q, wz = wbq Gp,q b , . . . , b , b + h h! 1 q−1 q h=0 ) ) ( ∞ ∑ z (1 − w)h m,n a1 − h, a2 , . . . , ap 1−a1 Gp,q w =w z , n ≥ 1, bq h! h=0 ) ) ( ∞ ∑ z (w − 1)h m,n a1 , . . . , ap−1 , ap − h 1−ap Gp,q z , n < p. w =w bq h! h=0

These follow by Taylor expansion about w = 1, with the help of the basic properties discussed above. The radii of convergence will be dependent on the value of z and on the G-function that is expanded. The expansions can be regarded as generalizations of similar theorems for Bessel, hypergeometric and confluent hypergeometric functions.

28.4 Definite integrals involving the G-function Among definite integrals involving an arbitrary G-function one has: ∫

(

∞

x

s−1

m,n Gp,q

0

∏m ∏n ) η −s j=1 Γ(bj + s) j=1 Γ(1 − aj − s) ap ∏p ηx dx = ∏q . bq j=m+1 Γ(1 − bj − s) j=n+1 Γ(aj + s)

Note that the restrictions under which this integral exists have been omitted here. It is, of course, no surprise that the Mellin transform of a G-function should lead back to the integrand appearing in the definition above. Euler-type integrals for the G-function are given by: ∫

1

x 0

−α

( (1 − x)

α−β−1

m,n Gp,q

ap bq

) ( zx dx = Γ(α − β) G m, n+1 α, ap p+1, q+1 b , β q

) z ,

100 ∫

∞

CHAPTER 28. MEIJER G-FUNCTION

m,n x−α (x − 1)α−β−1 Gp,q

1

(

ap bq

) ) ( zx dx = Γ(α − β) G m+1, n ap , α z . p+1, q+1 β, b q

Here too, the restrictions under which the integrals exist have been omitted. Note that, in view of their effect on the G-function, these integrals can be used to define the operation of fractional integration for a fairly large class of functions (Erdélyi–Kober operators). A result of fundamental importance is that the product of two arbitrary G-functions integrated over the positive real axis can be represented by just another G-function (convolution theorem): ∫

∞

0

=

1 η

) ( ) ap µ,ν cσ ηx G ωx dx = σ,τ bq dτ ) ( n+µ, m+ν −b1 , . . . , −bm , cσ , −bm+1 , . . . , −bq ω Gq+σ, p+τ = −a1 , . . . , −an , dτ , −an+1 , . . . , −ap η ) ( m+ν, n+µ a1 , . . . , an , −dτ , an+1 , . . . , ap η Gp+τ, . q+σ b1 , . . . , bm , −cσ , bm+1 , . . . , bq ω (

m,n Gp,q

1 ω Again, the restrictions under which the integral exists have been omitted here. Note how the Mellin transform of the result merely assembles the gamma factors from the Mellin transforms of the two functions in the integrand. Many of the amazing definite integrals listed in tables or produced by computer algebra systems are nothing but special cases of this formula. =

The convolution formula can be derived by substituting the defining Mellin–Barnes integral for one of the G-functions, reversing the order of integration, and evaluating the inner Mellin-transform integral. The preceding Euler-type integrals follow analogously.

28.4.1

Laplace transform

Using the above convolution integral and basic properties one can show that: ∫

∞

e

−ωx

x

−α

( m,n Gp,q

0

) ( ap m, n+1 α, ap α−1 ηx dx = ω Gp+1, q bq bq

) η ω ,

where Re(ω) > 0. This is the Laplace transform of a function G(ηx) multiplied by a power x−α ; if we put α = 0 we get the Laplace transform of the G-function. As usual, the inverse transform is then given by:

m, n x−α Gp, q+1

(

) ( ∫ c+i∞ 1 ap ωx α−1 m,n ap ηx = e ω G p,q bq , α bq 2πi c−i∞

) η ω dω,

where c is a real positive constant that places the integration path to the right of any pole in the integrand. Another formula for the Laplace transform of a G-function is: ∫

∞

m,n e−ωx Gp,q

0

(

) ( 1 1 ap 2 m, n+2 0, 2 , ap √ dx = G ηx bq bq πω p+2, q

) 4η ω2 ,

where again Re(ω) > 0. Details of the restrictions under which the integrals exist have been omitted in both cases.

28.5 Integral transforms based on the G-function In general, two functions k(z,y) and h(z,y) are called a pair of transform kernels if, for any suitable function f(z) or any suitable function g(z), the following two relationships hold simultaneously: ∫

∫

∞

g(z) =

k(z, y) f (y) dy, 0

f (z) =

∞

h(z, y) g(y) dy. 0

The pair of kernels is said to be symmetric if k(z,y) = h(z,y).

28.5. INTEGRAL TRANSFORMS BASED ON THE G-FUNCTION

28.5.1

101

Narain transform

Roop Narain (1962, 1963a, 1963b) showed that the functions: ( k(z, y) = 2γ (zy)

γ−1/2

m, p Gp+q, m+n

γ−1/2

n, q Gp+q, m+n

( h(z, y) = 2γ (zy)

ap , bq cm , dn

) (zy)2γ ,

−bq , −ap −dn , −cm

) (zy)2γ

are an asymmetric pair of transform kernels, where γ > 0, n − p = m − q > 0, and: p ∑

aj +

j=1

q ∑

bj =

j=1

m ∑

cj +

j=1

n ∑

dj ,

j=1

along with further convergence conditions. In particular, if p = q, m = n, aj + bj = 0 for j = 1, 2, ..., p and cj + dj = 0 for j = 1, 2, ..., m, then the pair of kernels becomes symmetric. The well-known Hankel transform is a symmetric special case of the Narain transform (γ = 1, p = q = 0, m = n = 1, c1 = −d1 = ν ⁄2 ).

28.5.2

Wimp transform

Jet Wimp (1964) showed that these functions are an asymmetric pair of transform kernels: ( m, n+2 k(z, y) = Gp+2, q

) 1 − ν + iz, 1 − ν − iz, ap y , bq

] i −νπi [ πy ye e A(ν + iy, ν − iy | zeiπ ) − e−πy A(ν − iy, ν + iy | zeiπ ) , π where the function A(·) is defined as: h(z, y) =

( q−m, p−n+1 A(α, β | z) = Gp+2, q

28.5.3

) −an+1 , −an+2 , . . . , −ap , α, −a1 , −a2 , . . . , −an , β z . −bm+1 , −bm+2 , . . . , −bq , −b1 , −b2 , . . . , −bm

Generalized Laplace transform

The Laplace transform can be generalized in close analogy with Narain’s generalization of the Hankel transform: ∫

(

∞

g(s) = 2γ

γ+ρ−1/2

(st)

q+1, 0 Gp, q+1

0

f (t) =

γ πi

∫

ap 0, bq (

c+i∞ 1, p (ts)γ−ρ−1/2 Gp, q+1

c−i∞

) (st)2γ f (t) dt,

) −ap 2γ − (ts) g(s) ds, 0, −bq

where γ > 0, p ≤ q, and: ∑ ∑ ρ = (q + 1 − p) aj − bj , 2γ j=1 j=1 p

q

and where the constant c > 0 places the second integration path to the right of any pole in the integrand. For γ = 1 ⁄2 , ρ = 0 and p = q = 0, this corresponds to the familiar Laplace transform.

102

CHAPTER 28. MEIJER G-FUNCTION

28.5.4

Meijer transform

Two particular cases of this generalization were given by C.S. Meijer in 1940 and 1941. The case resulting for γ = 1, ρ = −ν, p = 0, q = 1 and b1 = ν may be written (Meijer 1940):

g(s) =

∫ √ 2/π

∞

(st)1/2 Kν (st) f (t) dt,

0

1 f (t) = √ 2π i

∫

c+i∞

(ts)1/2 Iν (ts) g(s) ds, c−i∞

and the case obtained for γ = 1 ⁄2 , ρ = −m − k, p = q = 1, a1 = m − k and b1 = 2m may be written (Meijer 1941a): ∫

∞

g(s) =

(st)−k−1/2 e−st/2 Wk+1/2, m (st) f (t) dt,

0

Γ(1 − k + m) f (t) = 2πi Γ(1 + 2m)

∫

c+i∞

(ts)k−1/2 ets/2 Mk−1/2, m (ts) g(s) ds. c−i∞

Here Iν and Kν are the modified Bessel functions of the first and second kind, respectively, Mk,m and Wk,m are the Whittaker functions, and constant scale factors have been applied to the functions f and g and their arguments s and t in the first case.

28.6 Representation of other functions in terms of the G-function The following list shows how the familiar elementary functions result as special cases of the Meijer G-function: ( ) 1 H(1 − |x|) = x , ∀x 0 ( ) 0,1 1 H(|x| − 1) = G1,1 x , ∀x 0 ( ) 1,0 − x e = G0,1 −x , ∀x 0 ( ) √ − x2 1,0 , ∀x cos x = π G0,2 0, 12 4 ( ) √ −π π − x2 1,0 sin x = π G0,2 < arg x ≤ 1 4 , , 0 2 2 2 ( ) √ x2 − 1,0 , ∀x cosh x = π G0,2 1 − 0, 2 4 ) ( √ x2 − 1,0 sinh x = − πi G0,2 1 − , −π < arg x ≤ 0 4 2, 0 ) ( −i 1,2 1, 1 2 −π < arg x ≤ 0 arcsin x = √ G2,2 1 − x , 2 π 2, 0 ) ( 1 1,2 12 , 1 2 −π π arctan x = G2,2 < arg x ≤ 1 x , , 0 2 2 2 2 (1 ) π 1 2,1 2 , 1 2 −π < arg x ≤ arccot x = G2,2 x , 1 2 2 2 2, 0 ) ( 1,2 1, 1 ln(1 + x) = G2,2 x , ∀x 1, 0 1,0 G1,1

28.7. REFERENCES

103

Here, H denotes the Heaviside step function. The subsequent list shows how some higher functions can be expressed in terms of the G-function: ) 1 γ(α, x) = x , ∀x α, 0 ) ( 1 2,0 Γ(α, x) = G1,2 x , ∀x α, 0 ) ( −π π − x2 1,0 , < arg x ≤ Jν (x) = G0,2 ν −ν , 4 2 2 2 2 2) ( −ν−1 π −π 2,0 2 x , Yν (x) = G1,3 < arg x ≤ ν −ν −ν−1 , , 4 2 2 2 2 2 ) ( x2 − 1,0 Iν (x) = i−ν G0,2 , −π < arg x ≤ 0 ν −ν − 4 2, 2 ( ) 1 2,0 −π π − x2 Kν (x) = G0,2 ν −ν , < arg x ≤ , 2 4 2 2 2 2 ( ) 0, 1 − a, . . . , 1 − a 1, n+1 Φ(x, n, a) = Gn+1, −x , ∀x, n = 0, 1, 2, . . . n+1 0, −a, . . . , −a ( ) 0, −a, . . . , −a 1, n+1 Φ(x, −n, a) = Gn+1, − x , ∀x, n = 0, 1, 2, . . . n+1 0, 1 − a, . . . , 1 − a (

1,1 G1,2

Even the derivatives of γ(α,x) and Γ(α,x) with respect to α can be expressed in terms of the Meijer G-function. Here, γ and Γ are the lower and upper incomplete gamma functions, Jν and Yν are the Bessel functions of the first and second kind, respectively, Iν and Kν are the corresponding modified Bessel functions, and Φ is the Lerch transcendent.

28.7 References • Andrews, L. C. (1985). Special Functions for Engineers and Applied Mathematicians. New York: MacMillan. ISBN 0-02-948650-5. • Askey, R. A.; Daalhuis, Adri B. Olde (2010), “Meijer G-function”, in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248 • Bateman, H.; Erdélyi, A. (1953). Higher Transcendental Functions, Vol. I (PDF). New York: McGraw–Hill. (see § 5.3, “Definition of the G-Function”, p. 206) • Beals, Richard; Szmigielski, Jacek (2013). “Meijer G-Functions: A Gentle Introduction,” (PDF). Notices of the American Mathematical Society 60 (7). • Brychkov, Yu. A.; Prudnikov, A. P. (2001), “Meijer transform”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Gradshteyn, Izrail' Solomonovich; Ryzhik, Iosif Moiseevich (1971). Tablitsy integralov, summ, ryadov i proizvedeniy [Tables of integrals, sums, series and products] (in Russian) (5th ed.). Moscow: Nauka. (see Chapter 9.3) • Klimyk, A. U. (2001), “Meijer G-functions”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Luke, Yudell L. (1969). The Special Functions and Their Approximations, Vol. I. New York: Academic Press. ISBN 0-12-459901-X. (see Chapter V, “The Generalized Hypergeometric Function and the G-Function”, p. 136) • Meijer, C. S. (1936). "Über Whittakersche bzw. Besselsche Funktionen und deren Produkte”. Nieuw Archief voor Wiskunde (2) (in German) 18 (4): 10–39. JFM 62.0421.02.

104

CHAPTER 28. MEIJER G-FUNCTION

• Meijer, C. S. (1940). "Über eine Erweiterung der Laplace-Transformation – I, II”. Proceedings of the Section of Sciences, Koninklijke Akademie van Wetenschappen (Amsterdam) (in German) 43: 599–608 and 702–711. JFM 66.0523.01. • Meijer, C. S. (1941a). “Eine neue Erweiterung der Laplace-Transformation – I, II”. Proceedings of the Section of Sciences, Koninklijke Akademie van Wetenschappen (Amsterdam) (in German) 44: 727–737 and 831–839. JFM 67.0396.01. m,n • Meijer, C. S. (1941b). “Multiplikationstheoreme für die Funktion Gp,q (z) ". Proceedings of the Section of Sciences, Koninklijke Akademie van Wetenschappen (Amsterdam) (in German) 44: 1062–1070. JFM 67.1016.01.

• Narain, Roop (1962). “The G-functions as unsymmetrical Fourier kernels – I” (PDF). Proceedings of the American Mathematical Society 13 (6): 950–959. doi:10.1090/S0002-9939-1962-0144157-5. MR 0144157. • Narain, Roop (1963a). “The G-functions as unsymmetrical Fourier kernels – II” (PDF). Proceedings of the American Mathematical Society 14 (1): 18–28. doi:10.1090/S0002-9939-1963-0145263-2. MR 0145263. • Narain, Roop (1963b). “The G-functions as unsymmetrical Fourier kernels – III” (PDF). Proceedings of the American Mathematical Society 14 (2): 271–277. doi:10.1090/S0002-9939-1963-0149210-9. MR 0149210. • Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. (1990). Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach. ISBN 2-88124-682-6. (see § 8.2, “The Meijer G-function”, p. 617) • Slater, Lucy Joan (1966). Generalized hypergeometric functions. Cambridge, UK: Cambridge University Press. ISBN 0-521-06483-X. (there is a 2008 paperback with ISBN 978-0-521-09061-2) • Wimp, Jet (1964). “A Class of Integral Transforms”. Proceedings of the Edinburgh Mathematical Society (Series 2) 14: 33–40. doi:10.1017/S0013091500011202. MR 0164204. Zbl 0127.05701.

28.8 External links • Weisstein, Eric W., “Meijer G-Function”, MathWorld. • Gradshteyn-Ryzhik (German Wikipedia)

Chapter 29

Picard–Fuchs equation In mathematics, the Picard–Fuchs equation, named after Émile Picard and Lazarus Fuchs, is a linear ordinary differential equation whose solutions describe the periods of elliptic curves.

29.1 Definition Let

j=

g23

g23 − 27g32

be the j-invariant with g2 and g3 the modular invariants of the elliptic curve in Weierstrass form: y 2 = 4x3 − g2 x − g3 . Note that the j-invariant is an isomorphism from the Riemann surface H/Γ to the Riemann sphere C ∪ {∞} ; where H is the upper half-plane and Γ is the modular group. The Picard–Fuchs equation is then d2 y 1 dy 31j − 4 + + y = 0. dj 2 j dj 144j 2 (1 − j)2 Written in Q-form, one has d2 f 1 − 1968j + 2654208j 2 + f = 0. 2 dj 4j 2 (1 − 1728j)2

29.2 Solutions This equation can be cast into the form of the hypergeometric differential equation. It has two linearly independent solutions, called the periods of elliptic functions. The ratio of the two periods is equal to the period ratio τ, the standard coordinate on the upper-half plane. However, the ratio of two solutions of the hypergeometric equation is also known as a Schwarz triangle map. The Picard–Fuchs equation can be cast into the form of Riemann’s differential equation, and thus solutions can be directly read off in terms of Riemann P-functions. One has  

0 1/6 y(j) = P  −1/6

1 1/4 3/4

∞ 0 0

  j

 105

106

CHAPTER 29. PICARD–FUCHS EQUATION

At least four methods to find the j-function inverse can be given. Dedekind defines the j-function by its Schwarz derivative in his letter to Borchardt. As a partial fraction, it reveals the geometry of the fundamental domain:

2Sτ (j) =

1 − 41 1 − 19 1 − 14 − 19 3 8 23 + + = + 2+ 2 2 2 (1 − j) j j(1 − j) 4(1 − j) 9j 36j(1 − j)

where (Sƒ)(x) is the Schwarzian derivative of ƒ with respect to x.

29.3 Generalization In algebraic geometry this equation has been shown to be a very special case of a general phenomenon, the Gauss– Manin connection.

29.4 References • Adlaj, Semjon (2011). “An inverse of the modular invariant”. arXiv:1110.3274 [math.NT]. • J. Harnad and J. McKay, Modular solutions to equations of generalized Halphen type, Proc. R. Soc. London A 456 (2000), 261–294, (Provides a readable introduction, some history, references, and various interesting identities and relations between solutions) • J. Harnad, Picard–Fuchs Equations, Hauptmoduls and Integrable Systems, Chapter 8 (Pgs. 137–152) of Integrability: The Seiberg–Witten and Witham Equation (Eds. H.W. Braden and I.M. Krichever, Gordon and Breach, Amsterdam (2000)). (Provides further examples of Picard–Fuchs equations satisfied by modular functions of genus 0, including non-triangular ones, and introduces Inhomogeneous Picard–Fuchs equations as special solutions to isomonodromic deformation equations of Painlevé type.)

Chapter 30

Riemann’s differential equation In mathematics, Riemann’s differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere on the Riemann sphere, rather than merely at 0, 1, and ∞. The equation is also known as the Papperitz equation.[1] The hypergeometric differential equation is a second-order linear differential equation which has three regular singular points, 0, 1 and ∞ . That equation admits two linearly independent solutions; near a singularity zs , the solutions take the form xs f (x) , where x = z − zs is a local variable, and f is locally holomorphic with f (0) ̸= 0 . The real number s is called the exponent of the solution at zs . Let α, β and γ be the exponents of one solution solution at 0, 1 and &infin respectively; and let α', β' and γ' be that of the other. Then α + α′ + β + β ′ + γ + γ ′ = 1. By applying suitable changes of variable, it is possible to transform the hypergeometric equation: Applying Möbius transformations will adjust the positions of the RSPs, while other transformations (see below) can change the exponents at the RSPs (Regular singular points?), subject to the exponents adding up to 1.

30.1 Definition The differential equation is given by d2 w dz 2

[ +

[ +

1−α−α′ z−a

+

1−β−β ′ z−b

+

1−γ−γ ′ z−c

]

dw dz

] αα′ (a − b)(a − c) ββ ′ (b − c)(b − a) γγ ′ (c − a)(c − b) w + + = 0. z−a z−b z−c (z − a)(z − b)(z − c)

The regular singular points are a, b, and c. The exponents of the solutions at these RSPs are, respectively, α; α′, β; β′, and γ; γ′. As before, the exponents are subject to the condition α + α′ + β + β ′ + γ + γ ′ = 1.

30.2 Solutions and relationship with the hypergeometric function The solutions are denoted by the Riemann P-symbol (also known as the Papperitz symbol)  a w(z) = P α  ′ α

b β β′

c γ γ′

  z

 107

108

CHAPTER 30. RIEMANN’S DIFFERENTIAL EQUATION

The standard hypergeometric function may be expressed as   ∞ 1  0  0 a 0 z F (a, b; c; z) = P 2 1   1−c b c−a−b The P-functions obey a number of identities; one of them allows a general P-function to be expressed in terms of the hypergeometric function. It is  a P α  ′ α

b β β′

 

c γ γ′

z



( =

z−a z−b

)α (

z−c z−b

)γ

 

0 0 P  ′ α −α

∞ α+β+γ α + β′ + γ

1 0 ′ γ −γ

 

(z−a)(c−b) (z−b)(c−a) 

In other words, one may write the solutions in terms of the hypergeometric function as ( w(z) =

z−a z−b

)α (

z−c z−b

)γ

( ) ′ ′ (z − a)(c − b) F α + β + γ, α + β + γ; 1 + α − α ; 2 1 (z − b)(c − a)

The full complement of Kummer's 24 solutions may be obtained in this way; see the article hypergeometric differential equation for a treatment of Kummer’s solutions.

30.3 Fractional linear transformations The P-function possesses a simple symmetry under the action of fractional linear transformations known as Möbius transformations (that are the conformal remappings of the Riemann sphere), or equivalently, under the action of the group GL(2, C). Given arbitrary complex numbers A, B, C, D such that AD − BC ≠ 0, define the quantities

u=

Az + B Cz + D

and

η=

Aa + B Ca + D

Ab + B Cb + D

and

θ=

Ac + B Cc + D

and

ζ=

then one has the simple relation  a P α  ′ α

b β β′

c γ γ′

 

 η z =P α   ′ α

ζ β β′

expressing the symmetry.

30.4 See also • Complex differential equation • Method of Frobenius • Monodromy

θ γ γ′

  u



30.5. NOTES

109

30.5 Notes [1] Siklos, Stephen. “The Papperitz equation”. Retrieved 21 April 2014.

30.6 References • Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover: New York, 1972) • Chapter 15 Hypergeometric Functions • Section 15.6 Riemann’s Differential Equation

Chapter 31

Rogers–Ramanujan identities In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series, first discovered and proved by Leonard James Rogers (1894). They were subsequently rediscovered (without a proof) by Srinivasa Ramanujan some time before 193. Ramanujan had no proof, but rediscovered Rogers’s paper in 1917, and they then published a joint new proof (Rogers & Ramanujan 1919). Issai Schur (1917) independently rediscovered and proved the identities.

31.1 Definition The Rogers–Ramanujan identities are ∑∞ q n2 G(q) = n=0 (q;q) = n A003114 in OEIS)

1 (q;q 5 )∞ (q 4 ;q 5 )∞

= 1 + q + q 2 + q 3 + 2q 4 + 2q 5 + 3q 6 + · · · (sequence

1 (q 2 ;q 5 )∞ (q 3 ;q 5 )∞

= 1 + q 2 + q 3 + q 4 + q 5 + 2q 6 + · · · (sequence A003106

and ∑∞ H(q) = n=0 in OEIS).

2

q n +n (q;q)n

=

Here, (·; ·)n denotes the q-Pochhammer symbol.

31.2 Integer Partitions Consider the following: 2

qn (q;q)n

•

is the generating function for partitions with exactly n parts such that adjacent parts have diffference at least 2.

•

1 (q;q 5 )∞ (q 4 ;q 5 )∞

is the generating function for partitions such that each part is congruent to either 1 or 4 modulo

5.

2

q n +n (q;q)n

•

is the generating function for partitions with exactly n parts such that adjacent parts have difference at least 2 and such that the smallest part is at least 2.

•

1 (q 2 ;q 5 )∞ (q 3 ;q 5 )∞

is the generating function for partitions such that each part is congruent to either 2 or 3 modulo

5.

The Rogers-Ramanujan identities could be now interpreted in the following way. Let n be a non-negative integer. 110

31.3. MODULAR FUNCTIONS

111

1. The number of partitions of n such that the adjacent parts differ by at least 2 is the same as the number of partitions of n such that each part is congruent to either 1 or 4 modulo 5. 2. The number of partitions of n such that the adjacent parts differ by at least 2 and such that the smallest part is at least 2 is the same as the number of partitions of n such that each part is congruent to either 2 or 3 modulo 5.

31.3 Modular functions If q = e2πiτ , then q−1/60 G(q) and q11/60 H(q) are modular functions of τ.

31.4 Applications The Rogers–Ramanujan identities appeared in Baxter’s solution of the hard hexagon model in statistical mechanics. Ramanujan’s continued fraction is

1+

q 1+

q2 q3 1+ 1+···

=

G(q) . H(q)

31.5 See also • Rogers polynomials

31.6 References • Rogers, L. J.; Ramanujan, Srinivasa (1919), “Proof of certain identities in combinatory analysis.”, Cambr. Phil. Soc. Proc. 19: 211–216, Reprinted as Paper 26 in Ramanujan’s collected papers • Rogers, L. J. (1892), “On the expansion of some infinite products”, Proc. London Math. Soc. 24 (1): 337–352, doi:10.1112/plms/s1-24.1.337, JFM 25.0432.01 • Rogers, L. J. (1893), “Second Memoir on the Expansion of certain Infinite Products”, Proc. London Math. Soc. 25 (1): 318–343, doi:10.1112/plms/s1-25.1.318 • Rogers, L. J. (1894), “Third Memoir on the Expansion of certain Infinite Products”, Proc. London Math. Soc. 26 (1): 15–32, doi:10.1112/plms/s1-26.1.15 • Issai Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche, (1917) Sitzungsberichte der Berliner Akademie, pp. 302–321. • W.N. Bailey, Generalized Hypergeometric Series, (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge. • George Gasper and Mizan Rahman, Basic Hypergeometric Series, 2nd Edition, (2004), Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. ISBN 0-521-83357-4. • Bruce C. Berndt, Heng Huat Chan, Sen-Shan Huang, Soon-Yi Kang, Jaebum Sohn, Seung Hwan Son, The Rogers-Ramanujan Continued Fraction, J. Comput. Appl. Math. 105 (1999), pp. 9–24. • Cilanne Boulet, Igor Pak, A Combinatorial Proof of the Rogers-Ramanujan and Schur Identities, Journal of Combinatorial Theory, Ser. A, vol. 113 (2006), 1019–1030. • Slater, L. J. (1952), “Further identities of the Rogers-Ramanujan type”, Proceedings of the London Mathematical Society. Second Series 54 (2): 147–167, doi:10.1112/plms/s2-54.2.147, ISSN 0024-6115, MR 0049225

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31.7 External links • Weisstein, Eric W., “Rogers-Ramanujan Identities”, MathWorld. • Weisstein, Eric W., “Rogers-Ramanujan Continued Fraction”, MathWorld.

Chapter 32

Schwarz’s list In the mathematical theory of special functions, Schwarz’s list or the Schwartz table is the list of 15 cases found by Hermann Schwarz (1873, p. 323) when hypergeometric functions can be expressed algebraically. More precisely, it is a listing of parameters determining the cases in which the hypergeometric equation has a finite monodromy group, or equivalently has two independent solutions that are algebraic functions. It lists 15 cases, divided up by the isomorphism class of the monodromy group (excluding the case of a cyclic group), and was first derived by Schwarz by methods of complex analytic geometry. Correspondingly the statement is not directly in terms of the parameters specifying the hypergeometric equation, but in terms of quantities used to describe certain spherical triangles. The wider importance of the table, for general second-order differential equations in the complex plane, was shown by Felix Klein, who proved a result to the effect that cases of finite monodromy for such equations and regular singularities could be attributed to changes of variable (complex analytic mappings of the Riemann sphere to itself) that reduce the equation to hypergeometric form. In fact more is true: Schwarz’s list underlies all second-order equations with regular singularities on compact Riemann surfaces having finite monodromy, by a pullback from the hypergeometric equation on the Riemann sphere by a complex analytic mapping, of degree computable from the equation’s data.[1][2] The numbers λ, μ, ν are (up to a sign) the differences 1 − c, c − a − b, a − b of the exponents of the hypergeometric differential equation at the three singular points 0, 1, ∞. They are rational numbers if and only if a, b and c are, a point that matters in arithmetic rather than geometric approaches to the theory.

32.1 Further work An extension of Schwarz’s results was given by T. Kimura, who dealt with cases where the identity component of the differential Galois group of the hypergeometric equation is a solvable group.[3][4] A general result connecting the differential Galois group G and the monodromy group Γ states that G is the Zariski closure of Γ — this theorem is attributed in the book of Matsuda to Michio Kuga. By general differential Galois theory, the resulting KimuraSchwarz table classifies cases of integrability of the equation by algebraic functions and quadratures. Another relevant list is that of K. Takeuchi, who classified the (hyperbolic) triangle groups that are arithmetic groups (85 examples).[5] Émile Picard sought to extend the work of Schwarz in complex geometry, by means of a generalized hypergeometric function, to construct cases of equations where the monodromy was a discrete group in the projective unitary group PU(1, n). Pierre Deligne and George Mostow used his ideas to construct lattices in the projective unitary group. This work recovers in the classical case the finiteness of Takeuchi’s list, and by means of a characterisation of the lattices they construct that are arithmetic groups, provided new examples of non-arithmetic lattices in PU(1, n).[6] Baldassari applied the Klein universality, to discuss algebraic solutions of the Lamé equation by means of the Schwarz list.[7]

32.2 See also • Schwarz triangle 113

114

CHAPTER 32. SCHWARZ’S LIST

32.3 Notes [1] A modern treatment is in F. Baldassarri, B. Dwork, On second order linear differential equations with algebraic solutions, Amer. J. Math. 101 (1) (1979) 42–76. [2] http://archive.numdam.org/ARCHIVE/GAU/GAU_1986-1987__14_/GAU_1986-1987__14__A12_0/GAU_1986-1987_ _14__A12_0.pdf, pp.5-6. [3] http://fe.math.kobe-u.ac.jp/FE/Free/vol12/fe12-18.pdf [4] http://www.intlpress.com/MAA/p/2001/8_1/MAA-8-1-113-120.pdf at p. 116 for the formulation. [5] http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.jmsj/1240433796 [6] http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1986__63_/PMIHES_1986__63__5_0/PMIHES_1986__63_ _5_0.pdf [7] F. Baldassarri, On algebraic solutions of Lamé’s differential equation, J. Differential Equations 41 (1) (1981) 44–58. Correction in Algebraic Solutions of the Lamé Equation, Revisited (PDF), by Robert S. Maier.

32.4 References • Matsuda, Michihiko (1985), Lectures on algebraic solutions of hypergeometric differential equations (PDF), Lectures in Mathematics 15, Tokyo: Kinokuniya Company Ltd., MR 1104881 • Schwarz, H. A. (1873), “Ueber diejenigen Fälle in welchen die Gaussichen hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt”, Journal für die reine und angewandte Mathematik 75: 292–335, ISSN 0075-4102

32.5 External links • Towards a nonlinear Schwarz’s list (PDF)

Chapter 33

Wilson polynomials In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by James A. Wilson (1980) that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials. They are defined in terms of the generalized hypergeometric function and the Pochhammer symbols by ( 2

pn (t ) = (a + b)n (a + c)n (a + d)n4 F3

−n a + b + c + d + n − 1 a+b a+c

) a−t a+t ;1 . a+d

33.1 See also • Askey-Wilson polynomials are a q-analogue of Wilson polynomials.

33.2 References • Wilson, James A. (1980), “Some hypergeometric orthogonal polynomials”, SIAM Journal on Mathematical Analysis 11 (4): 690–701, doi:10.1137/0511064, ISSN 0036-1410, MR 579561 • Koornwinder, T.H. (2001), “Wilson polynomials”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

115

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33.3 Text and image sources, contributors, and licenses 33.3.1

Text

• Appell series Source: https://en.wikipedia.org/wiki/Appell_series?oldid=655474160 Contributors: Michael Hardy, Giftlite, Almit39, Rgdboer, R.e.b., MvH, Headbomb, Connor Behan, HowiAuckland, Yobot, Kilom691, Citation bot 1, DrilBot, ServiceAT and Anonymous: 52 • Askey scheme Source: https://en.wikipedia.org/wiki/Askey_scheme?oldid=642631054 Contributors: Rjwilmsi, R.e.b., Headbomb, HowiAuckland, Citation bot, Stamptrader and Anonymous: 1 • Askey–Wilson polynomials Source: https://en.wikipedia.org/wiki/Askey%E2%80%93Wilson_polynomials?oldid=660607349 Contributors: Giftlite, PWilkinson, Rjwilmsi, R.e.b., CBM, Headbomb, Beeblebrox, Yobot, Specfunfan, Duoduoduo, Suslindisambiguator and Anonymous: 1 • Barnes integral Source: https://en.wikipedia.org/wiki/Barnes_integral?oldid=598499779 Contributors: Michael Hardy, Giftlite, R.e.b., Widdma, Headbomb, Tasar, Addbot, Kilom691, AnomieBOT, Omnipaedista, RjwilmsiBot, EmausBot and Anonymous: 6 • Basic hypergeometric series Source: https://en.wikipedia.org/wiki/Basic_hypergeometric_series?oldid=542551880 Contributors: Giftlite, Alberto da Calvairate~enwiki, Oleg Alexandrov, Linas, R.e.b., Mathbot, RJChapman, Headbomb, Twsx, David Eppstein, VolkovBot, Addbot, Sławomir Biały, Ionutzmovie, Citation bot 1 and Anonymous: 3 • Bilateral hypergeometric series Source: https://en.wikipedia.org/wiki/Bilateral_hypergeometric_series?oldid=553562150 Contributors: Edward, Michael Hardy, Charles Matthews, Giftlite, Rjwilmsi, MZMcBride, R.e.b., Headbomb, David Eppstein, Reedy Bot, Addbot, Smallman12q, Ionutzmovie, Citation bot 1 and Anonymous: 3 • Binomial transform Source: https://en.wikipedia.org/wiki/Binomial_transform?oldid=664439869 Contributors: Michael Hardy, Fredrik, Ruakh, Giftlite, Linas, Mandarax, Nneonneo, Bo Jacoby, Gilliam, Michael Ross, Derek farn, A. Pichler, CRGreathouse, CmdrObot, Bonás, LordAnubisBOT, STBotD, LokiClock, Bluetryst, JerroldPease-Atlanta, DumZiBoT, Tre2, Addbot, Bte99, Xqbot, Locobot, Auclairde, D'ohBot, Maltenfort, Kallikanzarid, ZéroBot, Homk, YFdyh-bot, TedDokos and Anonymous: 18 • Confluent hypergeometric function Source: https://en.wikipedia.org/wiki/Confluent_hypergeometric_function?oldid=664660862 Contributors: Michael Hardy, Stevenj, Charles Matthews, Robinh, Giftlite, Eric Kvaalen, PAR, Oleg Alexandrov, Linas, Rjwilmsi, MarSch, R.e.b., Mohawkjohn, FlaBot, Bgwhite, Closedmouth, SmackBot, RDBury, InverseHypercube, Betacommand, A. Pichler, Karho.Yau, Headbomb, Valandil211, Baccyak4H, Ekotkie, DavidCBryant, Yecril, VolkovBot, Tesi1700, StewartMH, Addbot, AstroDave, Ckk253, Aichilee, LilHelpa, Sławomir Biały, Citation bot 1, Max139, Episanty, Slawekb, Helpful Pixie Bot, Kjohnsson, Phleg1, Cesarlegendre and Anonymous: 31 • Dixon’s identity Source: https://en.wikipedia.org/wiki/Dixon’{}s_identity?oldid=597075539 Contributors: Michael Hardy, Giftlite, Igorpak, R.e.b., Mhym, Headbomb, Unara, Deltahedron, K9re11 and Anonymous: 1 • Dougall’s formula Source: https://en.wikipedia.org/wiki/Dougall’{}s_formula?oldid=553562025 Contributors: Rjwilmsi, R.e.b., David Eppstein, Omnipaedista and Citation bot 1 • Elliptic hypergeometric series Source: https://en.wikipedia.org/wiki/Elliptic_hypergeometric_series?oldid=632503187 Contributors: Giftlite, R.e.b., Chris the speller, Makyen, Headbomb, HowiAuckland, Citation bot 1, Helpful Pixie Bot, CitationCleanerBot, Adam.stinchcombe and Anonymous: 1 • Fox H-function Source: https://en.wikipedia.org/wiki/Fox_H-function?oldid=671784416 Contributors: Michael Hardy, Giftlite, R.e.b., CBM, R.Zorn, Arjunkrathie, CitationCleanerBot and Anonymous: 8 • Fox–Wright function Source: https://en.wikipedia.org/wiki/Fox%E2%80%93Wright_function?oldid=652949699 Contributors: Michael Hardy, Giftlite, Pichpich, Yobot, R.Zorn, SoledadKabocha, K9re11 and Anonymous: 2 • Frobenius solution to the hypergeometric equation Source: https://en.wikipedia.org/wiki/Frobenius_solution_to_the_hypergeometric_ equation?oldid=615050737 Contributors: Michael Hardy, TakuyaMurata, Cobaltbluetony, Ukexpat, Killing Vector, Ryan Reich, Rjwilmsi, Krishnavedala, Carlosguitar, SmackBot, Khazar, CBM, Xxanthippe, MostafaSabry, Ettrig, Yobot, LilHelpa, Helpful Pixie Bot, KLBot2, Vkpd11, Smokey martini and Anonymous: 4 • General hypergeometric function Source: https://en.wikipedia.org/wiki/General_hypergeometric_function?oldid=447602719 Contributors: R.e.b., Headbomb and Omnipaedista • Generalized hypergeometric function Source: https://en.wikipedia.org/wiki/Generalized_hypergeometric_function?oldid=668014607 Contributors: Michael Hardy, Ahoerstemeier, Cyp, Stevenj, Charles Matthews, Dysprosia, Jitse Niesen, McKay, Robbot, Humus sapiens, Robinh, Wile E. Heresiarch, Giftlite, BenFrantzDale, Marcika, Waltpohl, Almit39, Ben Standeven, Nabla, Crisófilax, Jonon, Delius, Msh210, PAR, Linas, Rjwilmsi, Ligulem, R.e.b., Sodin, DVdm, WriterHound, Algebraist, YurikBot, RussBot, NymphadoraTonks, Schmock, C h fleming, Bo Jacoby, SmackBot, RDBury, Reedy, InverseHypercube, Eskimbot, Thomas Bliem, Colonies Chris, Hgrosser, Ligulembot, Makyen, ThePI, Don Warren, A. Pichler, JohnCD, Zahlentheorie, Dogaroon, Quadricode, Sketchjoy, R'n'B, TomyDuby, JonMcLoone, DavidCBryant, Jarry1250, Yecril, VolkovBot, JohnBlackburne, Asympt, Abelzaal, SieBot, Dagoberto.salazar, COBot, JL-Bot, JP.Martin-Flatin, Rathemis, Christian.fritz, SilvonenBot, AndersBot, Lightbot, Rory-Mulvaney, Luckas-bot, Quadrescence, Timeroot, Citation bot, LilHelpa, Specfunfan, Omnipaedista, Citation bot 4, AperiodicOrder, Tinabeana, Humanist bd, RjwilmsiBot, LucasBrown, R.Zorn, Sameenahmedkhan, BG19bot, Arjunkrathie, Xdb11112 and Anonymous: 61 • Gosper’s algorithm Source: https://en.wikipedia.org/wiki/Gosper’{}s_algorithm?oldid=649833160 Contributors: Michael Hardy, Andreas Kaufmann, Gareth McCaughan, Alaibot, Dawnseeker2000, MatthewVanitas, Yobot, Apollo, D.Lazard, Helpful Pixie Bot, SoSivr and Anonymous: 4 • Horn function Source: https://en.wikipedia.org/wiki/Horn_function?oldid=626902004 Contributors: Charles Matthews, Colonel Cow, RJFJR, Rjwilmsi, R.e.b., Chrisbaird.ma, SmackBot, Scoty6776, David Eppstein, Optigan13, Lightbot, Yobot, Omnipaedista, Trappist the monk, Suslindisambiguator, ServiceAT, Brad7777 and Anonymous: 1 • Humbert series Source: https://en.wikipedia.org/wiki/Humbert_series?oldid=658862493 Contributors: R.e.b., Headbomb, HowiAuckland, Kilom691, Doraemonpaul, Sven Manguard and Anonymous: 14

33.3. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

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• Hypergeometric function Source: https://en.wikipedia.org/wiki/Hypergeometric_function?oldid=673523599 Contributors: Michael Hardy, Smack, Giftlite, Anythingyouwant, Fintor, Ben Standeven, PWilkinson, Eric Kvaalen, Killing Vector, Oleg Alexandrov, Linas, R.e.b., Mathbot, DVdm, Dmharvey, Bhny, Tavilis, Bo Jacoby, SmackBot, InverseHypercube, BahramH, Chris the speller, Kjetil1001, Ligulembot, MvH, A. Pichler, CRGreathouse, Ntsimp, Headbomb, Lovibond, ColdPhage, Bongwarrior, Connor Behan, GuidoGer, Ttwo, TomyDuby, Policron, Yecril, Malik Shabazz, Camrn86, Dmcq, Mild Bill Hiccup, Addbot, MostafaSabry, Delaszk, Luckas-bot, Yobot, Kilom691, AnomieBOT, Andrewrp, Citation bot, Specfunfan, Soku56, Alidev, Duoduoduo, RjwilmsiBot, EmausBot, WikitanvirBot, R.Zorn, Slawekb, MichaelPenk, Suslindisambiguator, D.Lazard, Petrb, MelbourneStar, Mathcop, ServiceAT, Pengsun.ustc, Decaluwe.t, Fsedit, Arjunkumarrathie, Qzykcc, EFZR090440, CitingAllArticles, JudgeDeadd, Meeples10 and Anonymous: 73 • Hypergeometric function of a matrix argument Source: https://en.wikipedia.org/wiki/Hypergeometric_function_of_a_matrix_argument? oldid=569797033 Contributors: Michael Hardy, Chris the speller, Alaibot, David Eppstein, Plamenkoev, Melcombe and Anonymous: 5 • Hypergeometric identity Source: https://en.wikipedia.org/wiki/Hypergeometric_identity?oldid=669156551 Contributors: Michael Hardy, Stevenj, Jurgen~enwiki, Charles Matthews, Romanm, (:Julien:), Sam Hocevar, Paul August, Crisófilax, Grutness, Tbsmith, Linas, BD2412, Kbdank71, R.e.b., Mathbot, Scythe33, RussBot, LokiClock, Geometry guy, Paolo1979~enwiki, Dtchkff, Dinnertimeok, Charvest, Miracle Pen, D.Lazard, JudgeDeadd and Anonymous: 2 • Kampé de Fériet function Source: https://en.wikipedia.org/wiki/Kamp%C3%A9_de_F%C3%A9riet_function?oldid=597074356 Contributors: Giftlite, R.e.b., InverseHypercube, Olin, Headbomb, Yobot, Double sharp, RjwilmsiBot, Suslindisambiguator, DoctorKubla, K9re11 and Anonymous: 3 • Lauricella hypergeometric series Source: https://en.wikipedia.org/wiki/Lauricella_hypergeometric_series?oldid=632375983 Contributors: Michael Hardy, SebastianHelm, Jitse Niesen, Paul August, Rgdboer, R.e.b., Encyclops, HowiAuckland, TXiKiBoT, Addbot, Citation bot 1, DrilBot, TuHan-Bot, CeraBot and Anonymous: 23 • Legendre function Source: https://en.wikipedia.org/wiki/Legendre_function?oldid=652885848 Contributors: The Anome, Klemen Kocjancic, Linas, R.e.b., Krishnavedala, Roy Brumback, Julius Sahara, Headbomb, HowiAuckland, Mild Bill Hiccup, Addbot, Kaktus Kid, Louperibot, Miracle Pen, Logexp, LaguerreLegendre and Anonymous: 5 • List of hypergeometric identities Source: https://en.wikipedia.org/wiki/List_of_hypergeometric_identities?oldid=626965704 Contributors: Michael Hardy, Charles Matthews, Giftlite, Fuzzy, Linas, Kbdank71, Rjwilmsi, R.e.b., RDBury, Syrcatbot, LadyofShalott, A. Pichler, Albmont, R'n'B, Geometry guy, Ncsinger, Tide rolls, Xqbot, Citation bot 1, Trappist the monk, Diannaa, RjwilmsiBot, Michael assis, ChrisGualtieri and Anonymous: 17 • MacRobert E function Source: https://en.wikipedia.org/wiki/MacRobert_E_function?oldid=607161906 Contributors: Michael Hardy, Giftlite, MSGJ, Bender235, Rjwilmsi, R.e.b., Addbot, Yobot, Helpful Pixie Bot and Anonymous: 15 • Meijer G-function Source: https://en.wikipedia.org/wiki/Meijer_G-function?oldid=671036711 Contributors: Michael Hardy, Charles Matthews, Giftlite, Michael Devore, Rgdboer, R.e.b., Alejo2083, Planetneutral, NawlinWiki, Lambiam, CRGreathouse, Tulasiramreddy.atr, Headbomb, Applrpn, Unbuttered Parsnip, DumZiBoT, Addbot, Luckas-bot, Yobot, LilHelpa, FrescoBot, Suslindisambiguator, Helpful Pixie Bot, Mvasef, Arcandam, Mogism, Pzinn and Anonymous: 136 • Picard–Fuchs equation Source: https://en.wikipedia.org/wiki/Picard%E2%80%93Fuchs_equation?oldid=595967893 Contributors: Michael Hardy, Charles Matthews, Giftlite, Almit39, Oleg Alexandrov, Linas, GeometryJim, Titus III, Jac16888, Jharnad, GirasoleDE, HartmutMonien, Addbot, Yobot, Anne Bauval, DrilBot and Anonymous: 7 • Riemann’s differential equation Source: https://en.wikipedia.org/wiki/Riemann’{}s_differential_equation?oldid=655999968 Contributors: Gareth Owen, Michael Hardy, Smack, Charles Matthews, Alberto da Calvairate~enwiki, Oleg Alexandrov, Linas, Jftsang, BD2412, MarSch, Sodin, Bhny, Incnis Mrsi, Thumperward, Myasuda, TomyDuby, Yecril, LokiClock, Addbot, Delaszk, AnomieBOT, Pythagoras0, ZéroBot and Anonymous: 5 • Rogers–Ramanujan identities Source: https://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_identities?oldid=664807165 Contributors: The Anome, Michael Hardy, Charles Matthews, Gandalf61, Giftlite, Linas, Kbdank71, R.e.b., Mathbot, RexNL, Niels Olson, Cydebot, Headbomb, David Eppstein, Duncan.Hull, Yobot, Citation bot 1, Svkanade, Difu Wu, Jiffles1, Jowa fan, EmausBot, John of Reading, Mogism, Cerabot~enwiki, Mark viking, Garfield Garfield and Anonymous: 6 • Schwarz’s list Source: https://en.wikipedia.org/wiki/Schwarz’{}s_list?oldid=670167709 Contributors: Michael Hardy, Charles Matthews, Giftlite, Tomruen, R.e.b., MvH, CBM, Magioladitis, R'n'B, Yobot and Thecheesykid • Wilson polynomials Source: https://en.wikipedia.org/wiki/Wilson_polynomials?oldid=627092014 Contributors: Michael Hardy, Charles Matthews, Giftlite, R.e.b., Yobot, Pcap, Trappist the monk and Anonymous: 1

33.3.2

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33.3.3

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