Hypergeometric Functions

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Hypergeometric functions

From Wikipedia, the free encyclopedia

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8 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 9 Dixon’s identity 28 9.1 Statements . . . 28 9.2 q-analogues . . . 28 9.3 References. . . 29

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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

12 Fox H-function 33 12.1 References . . . 33

13 Fox–Wright function 35 13.1 References . . . 35

14 Frobenius solution to the hypergeometric equation 36 14.1 The equation . . . 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

15 General hypergeometric function 51 15.1 References . . . 51

16 Generalized hypergeometric function 52 16.1 Notation . . . 52

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

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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 for2F2 . . . 58

16.7.5 Kummer’s relation . . . 58 16.7.6 Clausen’s formula . . . 58 16.8 Special cases. . . 59 16.8.1 The series0F0 . . . 59 16.8.2 The series1F0 . . . 59 16.8.3 The series0F1 . . . 59 16.8.4 The series1F1 . . . 60 16.8.5 The series2F0 . . . 60 16.8.6 The series2F1 . . . 61 16.8.7 The series3F0 . . . 61 16.8.8 The series3F1 . . . 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 66 18.1 References . . . 66 19 Humbert series 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

20.4 The hypergeometric diﬀerential equation . . . 71

20.4.1 Solutions at the singular points . . . 71

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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

24 Lauricella hypergeometric series 86 24.1 Generalization to n variables. . . 86

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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

26 List of hypergeometric identities 91 26.1 References . . . 91

27 MacRobert E function 92 27.1 Deﬁnition . . . 92

27.2 Relationship with the Meijer G-function . . . 92

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

29 Picard–Fuchs equation 105 29.1 Deﬁnition . . . 105

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CONTENTS vii

29.2 Solutions . . . 105

29.3 Generalization . . . 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

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Chapter 1

Appell series

For generalizations of Lambert series seeAppell–Lerch series.

In mathematics, Appellseriesare a set of fourhypergeometric seriesF1, F2, F3, F4 of twovariablesthat were

introduced byPaul Appell(1880) and that generalizeGauss’s hypergeometric series2F1 of one variable. Appell

established the set ofpartial diﬀerential equationsof which thesefunctionsare 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 F1is deﬁned for |x| < 1, |y| < 1 by the double series:

F1(a, b1, b2, c; x, y) = ∞ ∑ m,n=0 (a)m+n(b1)m(b2)n (c)m+nm! n! xmyn,

where thePochhammer 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 F1can be deﬁned byanalytic continuation.

Similarly, the function F2is deﬁned for |x| + |y| < 1 by the series:

F2(a, b1, b2, c1, c2; x, y) = ∞ ∑ m,n=0 (a)m+n(b1)m(b2)n (c1)m(c2)nm! n! xmyn,

the function F3for |x| < 1, |y| < 1 by the series:

F3(a1, a2, b1, b2, c; x, y) = ∞ ∑ m,n=0 (a1)m(a2)n(b1)m(b2)n (c)m+nm! n! xmyn,

and the function F4for |x|½+ |y|½< 1 by the series:

F4(a, b, c1, c2; x, y) = ∞ ∑ m,n=0 (a)m+n(b)m+n (c1)m(c2)nm! n! xmyn. 1

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2 CHAPTER 1. APPELL SERIES

1.2 Recurrence relations

Like the Gauss hypergeometric series2F1, the Appell double series entailrecurrence relationsamong contiguous

functions. For example, a basic set of such relations for Appell’s F1is given by:

(a−b1−b2)F1(a, b1, b2, c; x, y)−a F1(a+1, b1, b2, c; x, y)+b1F1(a, b1+1, b2, c; x, y)+b2F1(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}

1can be derived from these four.

Similarly, all recurrence relations for Appell’s F3follow 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)−a1F3(a1+1, a2, b1, b2, c+1; x, y)−a2F3(a1, a2+1, b1, b2, c+1; x, y) = 0 ,

c F3(a1, a2, b1, b2, c; x, y)− c F3(a1+ 1, a2, b1, b2, c; x, y) + b1x 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) + b2y 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) + a1x 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) + a2y F3(a1, a2+ 1, b1, b2+ 1, c + 1; x, y) = 0 .

1.3 Derivatives and diﬀerential equations

For Appell’s F1, the followingderivativesresult from the deﬁnition by a double series:

∂ ∂xF1(a, b1, b2, c; x, y) = ab1 c F1(a + 1, b1+ 1, b2, c + 1; x, y) , ∂ ∂yF1(a, b1, b2, c; x, y) = ab2 c F1(a + 1, b1, b2+ 1, c + 1; x, y) .

From its deﬁnition, Appell’s F1is further found to satisfy the following system of second-orderdiﬀerential equations:

( x(1− x) ∂ 2 ∂x2+ y(1− x) ∂2 ∂x∂y + [c− (a + b1+ 1)x] ∂ ∂x− b1y ∂ ∂y − ab1 ) F1(x, y) = 0 , ( y(1− y) ∂ 2 ∂y2+ x(1− y) ∂2 ∂x∂y + [c− (a + b2+ 1)y] ∂ ∂y − b2x ∂ ∂x − ab2 ) F1(x, y) = 0 .

Similarly, for F3the following derivatives result from the deﬁnition:

∂ ∂xF3(a1, a2, b1, b2, c; x, y) = a1b1 c F3(a1+ 1, a2, b1+ 1, b2, c + 1; x, y) , ∂ ∂yF3(a1, a2, b1, b2, c; x, y) = a2b2 c F3(a1, a2+ 1, b1, b2+ 1, c + 1; x, y) . And for F3the following system of diﬀerential equations is obtained:

( x(1− x) ∂ 2 ∂x2+ y ∂2 ∂x∂y + [c− (a1+ b1+ 1)x] ∂ ∂x− a1b1 ) F3(x, y) = 0 , ( y(1− y) ∂ 2 ∂y2+ x ∂2 ∂x∂y + [c− (a2+ b2+ 1)y] ∂ ∂y− a2b2 ) F3(x, y) = 0 .

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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 functionsonly (Gradshteyn & Ryzhik 1971, § 9.184). However,Émile Picard(1881) discovered that Appell’s F1can also be written as a one-dimensionalEuler-typeintegral:

F1(a, b1, b2, c; x, y) = Γ(c) Γ(a)Γ(c− a) ∫ 1 0 ta−1(1− t)c−a−1(1− xt)−b1₍₁− yt)−b2_dt, ℜ c > ℜ a > 0 .

This representation can be veriﬁed by means ofTaylor expansionof the integrand, followed by termwise integration.

1.5 Special cases

Picard’s integral representation implies that theincomplete elliptic integralsF and E as well as thecomplete elliptic integralΠ are special cases of Appell’s F1:

F (ϕ, k) = ∫ ϕ 0 dθ √ 1− k2_sin2 θ

=sin ϕ F1(1₂,1₂,1₂,3₂;sin2ϕ, k2sin2ϕ), |ℜ ϕ| <

π 2 , E(ϕ, k) = ∫ ϕ 0 √ 1− k2_sin2_θ_{dθ = sin ϕ F} 1(1₂,1₂,−1₂,3₂;sin2ϕ, k2sin2ϕ), |ℜ ϕ| < π 2 , Π(n, k) = ∫ π/2 0 dθ (1− n sin2θ)√1− k2_sin2_θ =π 2 F1( 1 2, 1, 1 2, 1; n, k 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 1F1 of one variable and the confluent hypergeometric limit function 0F1 of one variable in a similar manner. The first of these was introduced byPierre Humbertin1920.

• 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.

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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”, inOlver, 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 Matem-atico 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 withISBN 978-0-521-09061-2)

1.8 External links

• Aarts, Ronald M.,“Lauricella Functions”,MathWorld.

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Chapter 2

Askey scheme

In mathematics, the Askey scheme is a way of organizingorthogonal polynomialsof hypergeometric or basic hyper-geometric type into a hierarchy. For the classical orthogonal polynomials discussed inAndrews & Askey (1985), the Askey scheme was ﬁrst drawn byLabelle (1985) and byAskeyand Wilson (1985), and has since been extended by Koekoek & Swarttouw (1998) andKoekoek, 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:

4F3 Wilson|Racah

1F0 Hermite

2.2 Askey scheme for basic hypergeometric orthogonal polynomials

Koekoek, Lesky & Swarttouw (2010, p.413) give the following scheme for basic hypergeometric orthogonal polyno-mials:

4ϕ3

Askey–Wilson|q-Racah

3ϕ2

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

2ϕ1

2ϕ0/1ϕ1

1ϕ0

Continuous q-Hermite|Stieltjes–Wigert|Discrete q-Hermite I|Discrete q-Hermite II

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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: 10.1007/978-3-642-05014-5,ISBN 978-3-642-05013-8,MR 2656096

• Koornwinder, Tom H. (1988), “Group theoretic interpretations of Askey’s scheme of hypergeometric orthog-onal polynomials”, Orthogorthog-onal 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-le-Duc, 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

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Chapter 3

Askey–Wilson polynomials

In mathematics, the Askey–Wilson polynomials (or q-Wilson polynomials) are a family oforthogonal polynomials introduced byAskeyandWilson(1985) asq-analogsof theWilson polynomials. They include many of the other orthogonal polynomials in 1 variable asspecialorlimiting cases, described in theAskey scheme. Askey–Wilson polynomials are the special case ofMacdonald polynomials(orKoornwinder polynomials) for the non-reducedaﬃne root systemof type (C∨

1, C1), 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)na−n4ϕ3

[

q−n abcdqn−1 aeiθ ae−iθ

ab ac ad ; q, q

]

where φ is abasic hypergeometric functionand x = cos(θ) and (,,,)n is theq-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”, inOlver, 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

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Chapter 4

Barnes integral

In mathematics, a Barnes integral orMellin–Barnes integral is acontour integralinvolving a product ofgamma functions. They were introduced byErnest William Barnes(1908,1910). They are closely related togeneralized 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

Thehypergeometric functionis given as a Barnes integral (Barnes 1908) by

2F1(a, b; c; z) = Γ(c) Γ(a)Γ(b) 1 2πi ∫ i_∞ −i∞ Γ(a + s)Γ(b + s)Γ(−s) Γ(c + s) (−z) s_ds.

This equality can be obtained by moving the contour to the right while picking up theresiduesat s = 0, 1, 2, ... . Given proper convergence conditions, one can relate more general Barnes’ integrals andgeneralized hypergeometric functionspFq 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’s2F1summation 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) Γ(e + s) ds = 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 ofSaalschütz’s summation formula.

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4.3. Q-BARNES INTEGRALS 9

4.3 q-Barnes integrals

There are analogues of Barnes integrals forbasic 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 Mathe-matics41: 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.

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Chapter 5

Basic hypergeometric series

Inmathematics, Heine’s basic hypergeometric series, or hypergeometric q-series, areq-analoggeneralizations of generalized hypergeometric series, and are in turn generalized byelliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn₊₁/xn is arational functionof 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 series2φ1(qα,qβ;qγ;q,x) was ﬁrst considered byEduard 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

jϕk [ a1 a2 . . . aj b1 b2 . . . bk ; q, z ] = ∞ ∑ n=0 (a1, a2, . . . , aj; q)n (b1, b2, . . . , bk, q; q)n ( (−1)nq(n2) )1+k−j zn where (a1, a2, . . . , am; q)n= (a1; q)n(a2; q)n. . . (am; q)n and where (a; q)n = n_∏₋₁ k=0 (1− aqk) = (1− a)(1 − aq)(1 − aq2)· · · (1 − aqn−1).

is theq-shifted factorial. The most important special case is when j = k+1, when it becomes

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

This series is called balanced if a1...ak+1 = b1...bkq. This series is called well poised if a1q = a2b1= ... = a ₊₁bk,

and very well poised if in addition a2= −a3= qa11/2.

The bilateral basic hypergeometric series, corresponding to thebilateral hypergeometric series, is deﬁned as

jψk [ a1 a2 . . . aj b1 b2 . . . bk ; q, z ] = ∞ ∑ n=_−∞ (a1, a2, . . . , aj; q)n (b1, b2, . . . , bk; q)n ( (−1)nq(n2) )k−j zn. 10

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5.2. SIMPLE SERIES 11

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

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

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<0 then vanish.

5.2 Simple series

Some simple series expressions include

z 1− q 2ϕ1 [ q q q2 ; q, z ] = z 1− q+ z2 1− q2+ z3 1− q3 + . . . and z 1− q1/2 2ϕ1 [ q q1/2 q3/2 ; q, z ] = z 1− q1/2 + z2 1− q3/2 + z3 1− q5/2 + . . . and 2ϕ1 [ q − 1 −q ; q, z ] = 1 + 2z 1 + q + 2z2 1 + q2 + 2z3 1 + q3 + . . . .

5.3 The q-binomial theorem

The q-binomial theorem (ﬁrst published in 1811 byHeinrich August Rothe)[1][2]_{states that}

1ϕ0(a; q, z) = (az; q)_∞ (z; q)_∞ = ∞ ∏ n=0 1− aqnz 1− qn_z

which follows by repeatedly applying the identity

1ϕ0(a; q, z) =

1− az

1− z 1ϕ0(a; q, qz).

The special case of a = 0 is closely related to theq-exponential.

5.4 Ramanujan’s identity

Ramanujangave the identity

1ψ1 [ a b; q, z ] = ∞ ∑ n=_−∞ (a; q)n (b; q)n

zn =(b/a, q, q/az, az; q)∞ (b, b/az, q/a, z; q)_∞

valid for |q| < 1 and |b/a| < |z| < 1. Similar identities for 6ψ6have been given by Bailey. Such identities can be

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12 CHAPTER 5. BASIC HYPERGEOMETRIC SERIES

∞

∑

n=−∞

qn(n+1)/2zn= (q; q)_∞(−1/z; q)_∞(−zq; q)_∞. Ken Onogives a relatedformal power series

A(z; q)def= 1 1 + z ∞ ∑ n=0 (z; q)n (−zq; q)n zn= ∞ ∑ n=0 (−1)nz2nqn2.

5.5 Watson’s contour integral

As an analogue of theBarnes integralfor the hypergeometric series, Watson showed that

2ϕ1(a, b; c; q, z) = −1 2πi (a, b; q)_∞ (q, c; q)_∞ ∫ i∞ −i∞ (qqs, cqs; q)_∞ (aqs_{, bq}s_{; q)} ∞ π(−z)s sin πs ds where the poles of (aqs_{, bq}s_{; q)}

∞lie to the left of the contour and the remaining poles lie to the right. There is a

similar contour integral for r₊₁φr. This contour integral gives an analytic continuation of the basic hypergeometric function in z.

5.6 Notes

[1] Bressoud, D. M. (1981), “Some identities for terminating q-series”, Mathematical Proceedings of the Cambridge

Philosoph-ical Society 89 (2): 211–223,doi:10.1017/S0305004100058114,MR 600238.

[2] Benaoum, H. B., "h-analogue of Newton’s binomial formula”, Journal of Physics A: Mathematical and General 31 (46): L751–L754,arXiv:math-ph/9812011,doi:10.1088/0305-4470/31/46/001.

5.7 References

• Andrews, G. E. (2010),“q-Hypergeometric and Related Functions”, inOlver, 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

• W.N. Bailey, Generalized Hypergeometric Series, (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.

• William Y. C. Chen and Amy Fu,Semi-Finite Forms of Bilateral Basic Hypergeometric Series(2004)

• Gwynneth H. Coogan andKen Ono,A q-series identity and the Arithmetic of Hurwitz Zeta Functions, (2003) Proceedings of theAmerican Mathematical Society131, pp. 719–724

• Sylvie Corteel and Jeremy Lovejoy, Frobenius Partitions and the Combinatorics of Ramanujan’s 1ψ1

Summa-tion

• Fine, Nathan J. (1988),Basic hypergeometric series and applications, Mathematical Surveys and Monographs 27, Providence, R.I.: American Mathematical Society,ISBN 978-0-8218-1524-3,MR 956465

• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 (2nd ed.),Cambridge University Press,doi:10.2277/0521833574,ISBN 978-0-521-83357-8, MR 2128719

• Heine, Eduard (1846), "Über die Reihe 1+(qα−1)(qβ−1) (q−1)(qγ₋₁₎ x+

(qα−1)(qα+1−1)(qβ−1)(qβ+1−1) (q−1)(q2_−1)(qγ_−1)(qγ+1₋₁₎ x

2₊_{· · · ", Journal}

für die reine und angewandte Mathematik 32: 210–212

• Eduard Heine, Theorie der Kugelfunctionen, (1878) 1, pp 97–125.

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Chapter 6

Bilateral hypergeometric series

In mathematics, a bilateral hypergeometric series is a series Σan summed over all integers n, and such that the ratio an/an₊₁

of two terms is a rational function of n. The deﬁnition of thegeneralized hypergeometric seriesis similar, except that the terms with negative n must vanish; the bilateral series will in general have inﬁnite numbers of non-zero terms for both positive and negative n.

The bilateral hypergeometric series fails to converge for most rational functions, though it can be analytically continued to a function deﬁned for most rational functions. There are several summation formulas giving its values for special values where it does converge.

6.1 Deﬁnition

The bilateral hypergeometric series pHp is deﬁned by

pHp(a1, . . . , ap; b1, . . . , bp; z) =pHp ( a1 . . . ap b1 . . . bp ; z ) = ∞ ∑ n=−∞ (a1)n(a2)n. . . (ap)n (b1)n(b2)n. . . (bp)n zn where

(a)n= a(a + 1)(a + 2)· · · (a + n − 1)

is therising factorialorPochhammer symbol.

Usually the variable z is taken to be 1, in which case it is omitted from the notation. It is possible to deﬁne the series pHq with diﬀerent p and q in a similar way, but this either fails to converge or can be reduced to the usual hypergeomtric series by changes of variables.

6.2 Convergence and analytic continuation

Suppose that none of the variables a or b are integers, so that all the terms of the series are ﬁnite and non-zero. Then the terms with n<0 diverge if |z| <1, and the terms with n>0 diverge if |z| >1, so the series cannot converge unless |z|=1. When |z|=1, the series converges if

ℜ(b1+· · · bn− a1− · · · − an) > 1.

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14 CHAPTER 6. BILATERAL HYPERGEOMETRIC SERIES

The bilateral hypergeometric series can be analytically continued to a multivalued meromorphic function of several variables whose singularities are branch points at z = 0 and z=1 and simple poles at ai = −1, −2,... and bi = 0, 1, 2, ... This can be done as follows. Suppose that none of the a or b variables are integers. The terms with n positive converge for |z| <1 to a function satisfying an inhomogeneous linear equation with singularities at z = 0 and z=1, so can be continued to a multivalued function with these points as branch points. Similarly the terms with n negative converge for |z| >1 to a function satisfying an inhomogeneous linear equation with singularities at z = 0 and z=1, so can also be continued to a multivalued function with these points as branch points. The sum of these functions gives the analytic continuation of the bilateral hypergeometric series to all values of z other than 0 and 1, and satisfies a linear differential equationin z similar to the hypergeometric differential equation.

6.3 Summation formulas

6.3.1 Dougall’s bilateral sum

2H2(a, b; c, d; 1) = ∞ ∑ −∞ (a)n(b)n (c)n(d)n =Γ(d)Γ(c)Γ(1− a)Γ(1 − b)Γ(c + d − a − b − 1) Γ(c− a)Γ(c − b)Γ(d − a)Γ(d − b) (Dougall 1907)

This is sometimes written in the equivalent form

∞ ∑ n=_−∞ Γ(a + n)Γ(b + n) Γ(c + n)Γ(d + n) = π2 sin(πa) sin(πb) Γ(c + d− a − b − 1) Γ(c− a)Γ(d − a)Γ(c − b)Γ(d − b).

6.3.2 Bailey’s formula

(Bailey 1959) gave the following generalization of Dougall’s formula:

3H3(a, b, f + 1; d, e, f ; 1) = ∞ ∑ −∞ (a)n(b)n(f + 1)n (d)n(e)n(f )n = λΓ(d)Γ(e)Γ(1− a)Γ(1 − b)Γ(d + e − a − b − 2) Γ(d− a)Γ(d − b)Γ(e − a)Γ(e − b) where

λ = f−1[(f− a)(f − b) − (1 + f − d)(1 + f − e)] .

6.4 See also

• basic bilateral hypergeometric series

6.5 References

• Bailey, W. N.(1959), “On the sum of a particular bilateral hypergeometric series3H3", The Quarterly

Jour-nal of Mathematics. Oxford. Second Series 10: 92–94,doi:10.1093/qmath/10.1.92,ISSN 0033-5606,MR 0107727

• Dougall, J. (1907), “On Vandermonde’s Theorem and Some More General Expansions”, Proc. Edinburgh Math. Soc. 25: 114–132,doi:10.1017/S0013091500033642

• Slater, Lucy Joan(1966), Generalized hypergeometric functions, Cambridge, UK: Cambridge University Press, ISBN 0-521-06483-X,MR 0201688(there is a 2008 paperback withISBN 978-0-521-09061-2)

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Chapter 7

Binomial transform

Incombinatorics, the binomial transform is asequence transformation(i.e., a transform of asequence) that com-putes itsforward diﬀerences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the sequence associated with itsordinary generating function.

7.1 Deﬁnition

The binomial transform, T, of a sequence, {an}, is the sequence {sn} deﬁned by

sn= n ∑ k=0 (−1)k ( n k ) ak.

Formally, one may write (Ta)n = sn for the transformation, where T is an inﬁnite-dimensionaloperatorwith matrix elements Tnk: sn= (T a)n = ∞ ∑ k=0 Tnkak.

The transform is aninvolution, that is,

T T = 1

or, using index notation,

∞

∑

k=0

TnkTkm = δnm

where δnmis theKronecker delta. The original series can be regained by

an= n ∑ k=0 (−1)k ( n k ) sk.

The binomial transform of a sequence is just the nthforward diﬀerencesof the sequence, with odd diﬀerences carrying a negative sign, namely:

s0= a0

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16 CHAPTER 7. BINOMIAL TRANSFORM s1=−(△a)0=−a1+ a0 s2= (△2a)0=−(−a2+ a1) + (−a1+ a0) = a2− 2a1+ a0 .. . sn= (−1)n(△na)0

where Δ is theforward diﬀerence operator.

Some authors deﬁne the binomial transform with an extra sign, so that it is not self-inverse:

tn = n ∑ k=0 (−1)n−k ( n k ) ak whose inverse is an= n ∑ k=0 ( n k ) tk.

7.2 Example

Binomial transforms can be seen in diﬀerence tables. Consider the following:

The top line 0, 1, 10, 63, 324, 1485,... (a sequence deﬁned by (2n2_{+ n)3}n − 2_{) is the (noninvolutive version of the)}

binomial transform of the diagonal 0, 1, 8, 36, 128, 400,... (a sequence deﬁned by n2₂n − 1_).

7.3 Shift states

The binomial transform is theshift operatorfor theBell numbers. That is,

Bn+1= n ∑ k=0 ( n k ) Bk

where the Bn are the Bell numbers.

7.4 Ordinary generating function

The transform connects thegenerating functionsassociated with the series. For theordinary generating function, let

f (x) = ∞ ∑ n=0 anxn and g(x) = ∞ ∑ n=0 snxn then g(x) = (T f )(x) = 1 1− xf ( x x− 1 ) .

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7.5. EULER TRANSFORM 17

7.5 Euler transform

The relationship between the ordinary generating functions is sometimes called the Euler transform. It commonly makes its appearance in one of two diﬀerent ways. In one form, it is used to accelerate the convergenceof an alternating series. That is, one has the identity

∞ ∑ n=0 (−1)nan = ∞ ∑ n=0 (−1)n∆ n_a 0 2n+1

which is obtained by substituting x=1/2 into the last formula above. The terms on the right hand side typically become much smaller, much more rapidly, thus allowing rapid numerical summation.

The Euler transform can be generalized (Borisov B. and Shkodrov V., 2007):

∞ ∑ n=0 (−1)n ( n + p n ) an= ∞ ∑ n=0 (−1)n ( n + p n ) ∆n_a 0 2n+p+1 where p = 0, 1, 2,...

The Euler transform is also frequently applied to theEuler hypergeometric integral 2F1. Here, the Euler transform

takes the form:

2F1(a, b; c; z) = (1− z)−b2F1 ( c− a, b; c; z z− 1 ) .

The binomial transform, and its variation as the Euler transform, is notable for its connection to thecontinued fraction representation of a number. Let 0 < x < 1 have the continued fraction representation

x = [0; a1, a2, a3,· · · ] then x 1− x = [0; a1− 1, a2, a3,· · · ] and x 1 + x = [0; a1+ 1, a2, a3,· · · ].

7.6 Exponential generating function

For theexponential generating function, let

f (x) = ∞ ∑ n=0 an xn n! and g(x) = ∞ ∑ n=0 sn xn n!

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18 CHAPTER 7. BINOMIAL TRANSFORM

then

g(x) = (T f )(x) = exf (−x).

TheBorel transformwill convert the ordinary generating function to the exponential generating function.

7.7 Integral representation

When the sequence can be interpolated by acomplex analyticfunction, then the binomial transform of the sequence can be represented by means of aNörlund–Rice integralon the interpolating function.

7.8 Generalizations

Prodinger gives a related,modular-liketransformation: letting

un= n ∑ k=0 ( n k ) ak(−c)n−kbk gives U (x) = 1 cx + 1B ( ax cx + 1 )

where U and B are the ordinary generating functions associated with the series{un} and {bn} , respectively.

The rising k-binomial transform is sometimes deﬁned as

n ∑ j=0 ( n j ) jkaj.

The falling k-binomial transform is

n ∑ j=0 ( n j ) jn−kaj

Both are homomorphisms of thekernelof theHankel transform of a series. In the case where the binomial transform is deﬁned as

n ∑ i=0 (−1)n−i ( n i ) ai= bn.

Let this be equal to the function J(a)n = bn.

If a newforward diﬀerencetable is made and the ﬁrst elements from each row of this table are taken to form a new sequence{bn} , then the second binomial transform of the original sequence is,

J2(a)n = n ∑ i=0 (−2)n−i ( n i ) ai.

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7.9. SEE ALSO 19

If the same process is repeated k times, then it follows that,

Jk(a)n= bn= n ∑ i=0 (−k)n−i ( n i ) ai.

Its inverse is,

J−k(b)n= an= n ∑ i=0 kn−i ( n i ) bi.

This can be generalized as,

Jk(a)n= bn= (E− k)na0

where E is theshift operator. Its inverse is J−k(b)n= an= (E + k)nb0.

7.9 See also

• Newton series • Hankel matrix • Möbius transform • Stirling transform • Euler summation

• List of factorial and binomial topics

7.10 References

• John H. Conway and Richard K. Guy, 1996, The Book of Numbers

• Donald E. Knuth, The Art of Computer Programming Vol. 3, (1973) Addison-Wesley, Reading, MA. • Helmut Prodinger, 1992,Some information about the Binomial transform

• Michael Z. Spivey and Laura L. Steil, 2006,The k-Binomial Transforms and the Hankel Transform

• Borisov B. and Shkodrov V., 2007, Divergent Series in the Generalized Binomial Transform, Adv. Stud. Cont. Math., 14 (1): 77-82

7.11 External links

• Binomial Transform,

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Chapter 8

Conﬂuent hypergeometric function

Inmathematics, a confluenthypergeometric functionis a solution of a confluent hypergeometric equation, which is a degenerate form of ahypergeometric differential equationwhere two of the threeregular singularitiesmerge into anirregular singularity. (The term "confluent" refers to the merging of singular points of families of differential equations; “confluere” is Latin for “to flow together”.) There are several common standard forms of confluent hyper-geometric functions:

• Kummer’s (confluent hypergeometric) function M(a, b, z), introduced byKummer(1837), is a solution to Kummer’s differential equation. There is a different and unrelatedKummer’s functionbearing the same name.

• Tricomi’s (conﬂuent hypergeometric) function U(a, b, z) introduced byFrancesco Tricomi(1947), some-times denoted by Ψ(a; b; z), is another solution to Kummer’s equation.

• Whittaker functions(forEdmund Taylor Whittaker) are solutions to Whittaker’s equation.

• Coulomb wave functionsare solutions to the Coulomb wave equation. The Kummer functions, Whittaker functions, and Coulomb wave functions are essentially the same, and diﬀer from each other only by elementary functions and change of variables.

8.1 Kummer’s equation

Kummer’s equation may be written as:

zd

2_w

dz2 + (b− z)

dw

dz − aw = 0,

with a regular singular point at z = 0 and an irregular singular point at z = ∞ . It has two (usually)linearly independentsolutions M(a, b, z) and U(a, b, z).

Kummer’s function (of the ﬁrst kind) M is ageneralized hypergeometric seriesintroduced in (Kummer 1837), given by: M (a, b, z) = ∞ ∑ n=0 a(n)_zn b(n)_n! =1F1(a; b; z), where: a(0)= 1, a(n)= a(a + 1)(a + 2)· · · (a + n − 1) , 20

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8.1. KUMMER’S EQUATION 21

is therising factorial. Another common notation for this solution is Φ(a, b, z). Considered as a function of a, b, or z with the other two held constant, this deﬁnes anentire functionof a or z, except when b = 0, −1, −2, ... As a function of b it isanalyticexcept for poles at the non-positive integers.

Some values of a and b yield solutions that can be expressed in terms of other known functions. See#Special cases. When a is a non-positive integer then Kummer’s function (if it is defined) is a (generalized)Laguerre polynomial. Just as the confluent differential equation is a limit of thehypergeometric differential equationas the singular point at 1 is moved towards the singular point at ∞, the confluent hypergeometric function can be given as a limit of the hypergeometric function

M (a, c, z) = lim

b→∞2F1(a, b; c; z/b)

and many of the properties of the conﬂuent hypergeometric function are limiting cases of properties of the hyperge-ometric function.

Since Kummer’s equation is second order there must be another, independent, solution. For this we can usually use the Tricomi conﬂuent hypergeometric function U(a, b, z) introduced byFrancesco Tricomi(1947), and sometimes denoted by Ψ(a; b; z). The function U is deﬁned in terms of Kummer’s function M by

U (a, b, z) = Γ(1− b)

Γ(a− b + 1)M (a, b, z) +

Γ(b− 1) Γ(a) z

1_−b_{M (a}_{− b + 1, 2 − b, z).}

This is undeﬁned for integer b, but can be extended to integer b by continuity. Unlike Kummer’s function which is anentire functionof z, U(z) usually has asingularityat zero. But see#Special casesfor some examples where it is an entire function (polynomial).

Note that if

Γ(b− 1)

Γ(a) = 0,

which can occur if a is a non-positive integer, then U(a, b, z) and M(a, b, z) are not independent and another solution is needed. Also when b is a non-positive integer we need another solution because then M(a, b, z) is not deﬁned. For instance, if a = b = 0, Kummer’s function is undeﬁned, but two independent solutions are w(z) = U (0, 0, z) = 1 and w(z) = exp(z). For a = 0 but at other values of b, we have the two solutions:

U (0, b, z) = 1 w(z) =

∫ z −∞

u−beudu

When b = 1 this second solution is theexponential integralEi(z). See#Special casesfor solutions to some other cases.

8.1.1 Other equations

Conﬂuent Hypergeometric Functions can be used to solve the Extended Conﬂuent Hypergeometric Equation whose general form is given as:

zd_dz2w2 + (b− z) dw dz − ( ∑M m=0amz m_)w[1]

{NB that for M=0 (or when the summation involves just one term), it reduces to the conventional Conﬂuent Hyper-geometric Equation}

Thus Confluent Hypergeometric Functions can be used to solve “most” second-order ordinary differential equations whose variable coefficients are all linear functions of z; because they can be transformed to the Extended Confluent Hypergeometric Equation. Consider the equation:

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22 CHAPTER 8. CONFLUENT HYPERGEOMETRIC FUNCTION (A + Bz)d 2_w dz2 + (C + Dz) dw dz + (E + F z)w = 0

First we move theregular singular pointto 0 by using the substitution of A + Bz ↦ z which converts the equation to:

zd

2_w

dz2 + (C + Dz)

dw

dz + (E + F z)w = 0

with new values of C, D, E, and F. Next we use the substitution:

z7→ √ 1

D2_{− 4F}z

and multiply the equation by the same factor, we get:

zd 2_w dz2 + ( C +√ D D2− 4Fz ) dw dz + ( E √ D2− 4F + F D2− 4Fz ) w = 0 whose solution is exp ( − ( 1 +√ D D2_{− 4F} ) z 2 ) w(z),

where w(z) is a solution to Kummer’s equation with

a = ( 1 +√ D D2_{− 4F} ) C 2 − E √ D2_{− 4F}, b = C.

Note that the square root may give an imaginary (or complex) number. If it is zero, another solution must be used, namely

exp(−1 2Dz

) w(z),

where w(z) is aconﬂuent hypergeometric limit functionsatisfying

zw′′(z) + Cw′(z) +(E−1₂CD)w(z) = 0.

As noted lower down, even theBessel equationcan be solved using conﬂuent hypergeometric functions.

8.2 Integral representations

If Re b > 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 0

ezuua−1(1− u)b−a−1du.

thus M (a, a + b, it) is thecharacteristic functionof thebeta distribution. For a with positive real part U can be obtained by theLaplace integral

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8.3. ASYMPTOTIC BEHAVIOR 23 U (a, b, z) = 1 Γ(a) ∫ _∞ 0 e−ztta−1(1 + t)b−a−1dt, (Re a > 0)

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

M (a, b, z) = 1 2πi Γ(b) Γ(a) ∫ i∞ −i∞ Γ(−s)Γ(a + s) Γ(b + s) (−z) s_ds

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). Itsasymptoticbehavior 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 thebinomial seriesand integrating it formally term by term gives rise to anasymptotic seriesexpansion, valid as x → ∞:[2]

U (a, b, x)∼ x−a2F0 ( a, a− b + 1; ; −1 x ) ,

where2F0(·, ·; ; −1/x) is ageneralized hypergeometric series(with 1 as leading term), which generally converges

nowhere but exists as aformal power seriesin 1/x. Thisasymptotic expansionis also valid for complex z instead of real x, with| arg z| < 3

2π.

The asymptotic behavior of Kummer’s solution for large |z| is:

M (a, b, z)∼ Γ(b) ( ez_za−b Γ(a) + (−z)−a Γ(b− a) )

The powers of z are taken using−3

2π <arg z≤ 1 2π.

[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_za_−b_{as z → −∞. Usually this will be a}

com-bination 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

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24 CHAPTER 8. CONFLUENT HYPERGEOMETRIC FUNCTION

zdM dz = z

a

bM (a+, b+) = a(M (a+)− M) = (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) = ezM (b− a, b, −z) U (a, b, z) = z1−bU (1 + a− b, 2 − b, z)

8.5 Multiplication theorem

The followingmultiplication theoremshold true:

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

8.6 Connection with Laguerre polynomials and similar representations

In terms ofLaguerre polynomials, Kummer’s functions have several expansions, for example

M ( a, b,_xxy₋₁ ) = (1− x)a·∑_na_b(n)(n)L (b−1) n (y)xn(Erdelyi 1953, 6.12)

8.7 Special cases

Functions that can be expressed as special cases of the conﬂuent hypergeometric function include:

• Someelementary 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−ae−udu(a polynomial if a is a non-positive integer) U (1,b,z)

Γ(b₋₁₎ +

M (1,b,z) Γ(b) = z

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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 ageneralized Laguerre polynomial. • Bateman’s function

• Bessel functionsand many related functions such asAiry functions,Kelvin functions,Hankel functions. For example, the special case b = 2a the function reduces to aBessel function:

1F1(a, 2a, x) = e x 2 ₀_F₁ ( ; a + 1₂;x₁₆2 ) = ex2 (x 4 )1 2 −a_Γ(_{a +}1 2 ) I_a₋1 2 (_x 2 ) .

This identity is sometimes also referred to asKummer’ssecond transformation. Similarly

U (a, 2a, x) = e x 2 √ πx 1 2 −aK a−1₂ (_x 2 ) ,

When a is a non-positive integer, this equals 2−aθ_−a(x₂)where θ is aBessel polynomial. • Theerror functioncan be expressed as

erf(x) = √2 π ∫ x 0 e−t2dt = √2x π 1F1 (₁ 2, 3 2,−x 2)_.

• Coulomb wave function • Cunningham functions

• Exponential integraland related functions such as thesine integral,logarithmic integral • Hermite polynomials

• Incomplete gamma function • Laguerre polynomials

• Parabolic cylinder function(or Weber function) • Poisson–Charlier function

• Toronto functions

• Whittaker functionsMκ,μ(z), Wκ,μ(z) are solutions ofWhittaker’s equationthat can be expressed in terms of Kummer functions M and U by

Mκ,µ(z) = e− z 2_zµ+ 1 2_M(_µ− κ +1 2, 1 + 2µ; z ) Wκ,µ(z) = e− z 2_zµ+ 1 2_U(_µ− κ +1 2, 1 + 2µ; z )

• The general p-th raw moment (p not necessarily an integer) can be expressed as

E[N (µ, σ2)p]₌ ( 2σ2)p2 _Γ(1+p 2 ) √ π 1F1 ( −p 2, 1 2,− µ2 2σ2 ) E[N(µ, σ2)p ] =(−2σ2) p 2 _U ( −p 2, 1 2,− µ2 2σ2 )

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26 CHAPTER 8. CONFLUENT HYPERGEOMETRIC FUNCTION

8.8 Application to continued fractions

By applying a limiting argument toGauss’s continued fractionit can be shown that

M (a + 1, b + 1, z) M (a, b, z) = 1 1− b− a b(b + 1)z 1 + a + 1 (b + 1)(b + 2)z 1− b− a + 1 (b + 2)(b + 3)z 1 + a + 2 (b + 3)(b + 4)z 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”, inOlver, 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 transcen-dental 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

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8.11. EXTERNAL LINKS 27

8.11 External links

• Conﬂuent Hypergeometric Functionsin NIST Digital Library of Mathematical Functions • Kummer hypergeometric functionon the Wolfram Functions site

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Chapter 9

Dixon’s identity

Inmathematics, Dixon’s identity (or Dixon’s theorem or Dixon’s formula) is any of several different but closely related identities proved byA. C. Dixon, some involving finite sums of products of threebinomial coefficients, and some evaluating ahypergeometric sum. These identities famously follow from theMacMahon Master theorem, and can now be routinely proved by computer algorithms (Ekhad 1990).

9.1 Statements

The original identity, from (Dixon 1891), is

a ∑ k=−a (−1)k ( 2a k + a )3 = (3a)! (a!)3.

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

a ∑ k=_−a (−1)k ( 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 b− a )( c + a c− a )

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

and the identity follows as a limiting case (as a tends to an integer) of Dixon’s theorem evaluating a well-poised3F2 generalized hypergeometric seriesat 1, from (Dixon 1902):

3F2(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_⁄

2a − b − c) > 0. As c tends to −∞ it reduces toKummer’s formulafor the hypergeometric

function2F1at −1. Dixon’s theorem can be deduced from the evaluation of theSelberg integral.

9.2 q-analogues

A q-analogue of Dixon’s formula for thebasic hypergeometric seriesin terms of theq-Pochhammer symbolis given by

Hypergeometric Functions