Hypergeometric functionsFrom Wikipedia, the free encyclopedia

Contents

1 Appell series 11.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Derivatives and dierential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Related series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Askey scheme 52.1 Askey scheme for hypergeometric orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . 52.2 Askey scheme for basic hypergeometric orthogonal polynomials . . . . . . . . . . . . . . . . . . . 52.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 AskeyWilson polynomials 73.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Barnes integral 84.1 Hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Barnes lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 q-Barnes integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Basic hypergeometric series 105.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 Simple series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.3 The q-binomial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.4 Ramanujans identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.5 Watsons contour integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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6 Bilateral hypergeometric series 136.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2 Convergence and analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.3 Summation formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6.3.1 Dougalls bilateral sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3.2 Baileys formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

7 Binomial transform 157.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.3 Shift states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.4 Ordinary generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.5 Euler transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.6 Exponential generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.7 Integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.8 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

8 Conuent hypergeometric function 208.1 Kummers equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8.1.1 Other equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.2 Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.3 Asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.4 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8.4.1 Contiguous relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.4.2 Kummers transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

8.5 Multiplication theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.6 Connection with Laguerre polynomials and similar representations . . . . . . . . . . . . . . . . . . 248.7 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.8 Application to continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

9 Dixons identity 289.1 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.2 q-analogues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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10 Dougalls formula 30

11 Elliptic hypergeometric series 3111.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.2 Denitions of additive elliptic hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . 3211.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

12 Fox H-function 3312.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

13 FoxWright function 3513.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

14 Frobenius solution to the hypergeometric equation 3614.1 The equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3614.2 Solution around x = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3614.3 Analysis of the solution in terms of the dierence 1 of the two roots . . . . . . . . . . . . . . . 39

14.3.1 not an integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3914.3.2 = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3914.3.3 an integer and 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

14.4 Solution around x = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4314.5 Analysis of the solution in terms of the dierence of the two roots . . . . . . . . . . . . . 44

14.5.1 not an integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.5.2 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.5.3 is a non-zero integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

14.6 Solution around innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4514.7 Analysis of the solution in terms of the dierence of the two roots . . . . . . . . . . . . . . . 47

14.7.1 not an integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.7.2 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.7.3 an integer and 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

14.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

15 General hypergeometric function 5115.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

16 Generalized hypergeometric function 5216.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5216.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5316.3 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5416.4 Convergence conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5416.5 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

16.5.1 Eulers integral transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5516.5.2 Dierentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

16.6 Contiguous function and related identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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16.7 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716.7.1 Saalschtzs theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716.7.2 Dixons identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716.7.3 Dougalls formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716.7.4 Generalization of Kummers transformations and identities for 2F2 . . . . . . . . . . . . . 5816.7.5 Kummers relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.7.6 Clausens formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

16.8 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.8.1 The series 0F0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.8.2 The series 1F0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.8.3 The series 0F1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.8.4 The series 1F1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6016.8.5 The series 2F0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6016.8.6 The series 2F1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6116.8.7 The series 3F0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6116.8.8 The series 3F1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

16.9 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6216.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6216.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6216.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

17 Gospers algorithm 6417.1 Outline of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6417.2 Relationship to WilfZeilberger pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6417.3 Denite versus indenite summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6417.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6417.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

18 Horn function 6618.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

19 Humbert series 6719.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6719.2 Related series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6819.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

20 Hypergeometric function 6920.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6920.2 The hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6920.3 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7020.4 The hypergeometric dierential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

20.4.1 Solutions at the singular points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7120.4.2 Kummers 24 solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

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20.4.3 Q-form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7320.4.4 Schwarz triangle maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7320.4.5 Monodromy group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

20.5 Integral formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7420.5.1 Euler type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7420.5.2 Barnes integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7420.5.3 John transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

20.6 Gauss contiguous relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7520.6.1 Gauss continued fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

20.7 Transformation formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7520.7.1 Fractional linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7620.7.2 Quadratic transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7620.7.3 Higher order transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

20.8 Values at special points z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7620.8.1 Special values at z = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7620.8.2 Kummers theorem (z = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7720.8.3 Values at z = 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7720.8.4 Other points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

20.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7820.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7920.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

21 Hypergeometric function of a matrix argument 8121.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8121.2 Two matrix arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8121.3 Not a typical function of a matrix argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8121.4 The parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8221.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8221.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

22 Hypergeometric identity 8322.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8322.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8322.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8422.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8422.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

23 Kamp de Friet function 8523.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8523.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

24 Lauricella hypergeometric series 8624.1 Generalization to n variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

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24.2 Integral representation of FD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8724.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8724.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

25 Legendre function 8825.1 Dierential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8825.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8925.3 Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8925.4 Legendre function as characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8925.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9025.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

26 List of hypergeometric identities 9126.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

27 MacRobert E function 9227.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9227.2 Relationship with the Meijer G-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9227.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9227.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

28 Meijer G-function 9428.1 Denition of the Meijer G-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

28.1.1 Dierential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9528.2 Relationship between the G-function and the generalized hypergeometric function . . . . . . . . . . 96

28.2.1 Polynomial cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.3 Basic properties of the G-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

28.3.1 Derivatives and antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9828.3.2 Recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9928.3.3 Multiplication theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

28.4 Denite integrals involving the G-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9928.4.1 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

28.5 Integral transforms based on the G-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10028.5.1 Narain transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10128.5.2 Wimp transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10128.5.3 Generalized Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10128.5.4 Meijer transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

28.6 Representation of other functions in terms of the G-function . . . . . . . . . . . . . . . . . . . . . 10228.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10328.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

29 PicardFuchs equation 10529.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

CONTENTS vii

29.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10529.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10629.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

30 Riemanns dierential equation 10730.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10730.2 Solutions and relationship with the hypergeometric function . . . . . . . . . . . . . . . . . . . . . 10730.3 Fractional linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10830.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10830.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10930.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

31 RogersRamanujan identities 11031.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11031.2 Integer Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11031.3 Modular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11131.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11131.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11131.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11131.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

32 Schwarzs list 11332.1 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11332.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11332.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

33 Wilson polynomials 11533.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11533.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11533.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 116

33.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11633.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11733.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Chapter 1

Appell series

For generalizations of Lambert series see AppellLerch series.

In mathematics, Appell series are a set of four hypergeometric series F1, F2, F3, F4 of two variables that wereintroduced by Paul Appell (1880) and that generalize Gausss hypergeometric series 2F1 of one variable. Appellestablished the set of partial dierential equations of which these functions are solutions, and found various reductionformulas and expressions of these series in terms of hypergeometric series of one variable.

1.1 DenitionsThe Appell series F1 is dened for |x| < 1, |y| < 1 by the double series:

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

1Xm;n=0

(a)m+n(b1)m(b2)n(c)m+nm!n!

xmyn ;

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

(q)n = q (q + 1) (q + n 1) = (q + n)(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 F1 can be dened by analytic continuation.Similarly, the function F2 is dened for |x| + |y| < 1 by the series:

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

1Xm;n=0

(a)m+n(b1)m(b2)n(c1)m(c2)nm!n!

xmyn ;

the function F3 for |x| < 1, |y| < 1 by the series:

F3(a1; a2; b1; b2; c;x; y) =1X

m;n=0

(a1)m(a2)n(b1)m(b2)n(c)m+nm!n!

xmyn ;

and the function F4 for |x| + |y| < 1 by the series:

F4(a; b; c1; c2;x; y) =1X

m;n=0

(a)m+n(b)m+n(c1)m(c2)nm!n!

xmyn :

1

2 CHAPTER 1. APPELL SERIES

1.2 Recurrence relationsLike the Gauss hypergeometric series 2F1, the Appell double series entail recurrence relations among contiguousfunctions. For example, a basic set of such relations for Appells F1 is given by:

(ab1b2)F1(a; b1; b2; c;x; y)aF1(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) aF1(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)xF1(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 F1 can be derived from these four.Similarly, all recurrence relations for Appells F3 follow from this set of ve:

c F3(a1; a2; b1; b2; c;x; y)+(a1+a2c)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) + b1xF3(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) + a1xF3(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 dierential equationsFor Appells F1, the following derivatives result from the denition by a double series:

@

@xF1(a; b1; b2; c;x; y) =

ab1cF1(a+ 1; b1 + 1; b2; c+ 1;x; y) ;

@

@yF1(a; b1; b2; c;x; y) =

ab2cF1(a+ 1; b1; b2 + 1; c+ 1;x; y) :

From its denition, Appells F1 is further found to satisfy the following system of second-order dierential 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 F3 the following derivatives result from the denition:

@

@xF3(a1; a2; b1; b2; c;x; y) =

a1b1c

F3(a1 + 1; a2; b1 + 1; b2; c+ 1;x; y) ;

@

@yF3(a1; a2; b1; b2; c;x; y) =

a2b2c

F3(a1; a2 + 1; b1; b2 + 1; c+ 1;x; y) :

And for F3 the following system of dierential 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 :

1.4. INTEGRAL REPRESENTATIONS 3

1.4 Integral representationsThe four functions dened by Appells double series can be represented in terms of double integrals involvingelementary functions only (Gradshteyn & Ryzhik 1971, 9.184). However, mile Picard (1881) discovered thatAppells F1 can also be written as a one-dimensional Euler-type integral:

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

(a)(c a)Z 10

ta1(1 t)ca1(1 xt)b1(1 yt)b2 dt; < c > < a > 0 :

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

1.5 Special casesPicards integral representation implies that the incomplete elliptic integrals F and E as well as the complete ellipticintegral are special cases of Appells F1:

F (; k) =

Z 0

dp1 k2 sin2

= sinF1( 12 ; 12 ; 12 ; 32 ; sin2 ; k2 sin2 ); j

4 CHAPTER 1. APPELL SERIES

Appell, Paul (1882). Sur les fonctions hypergomtriques de deux variables. Journal deMathmatiques Pureset Appliques. (3me srie) (in French) 8: 173216.

Appell, Paul; Kamp de Friet, Joseph (1926). Fonctions hypergomtriques et hypersphriques; Polynmesd'Hermite (in French). Paris: GauthierVillars. 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 UniversityPress, ISBN 978-0521192255, MR 2723248

Bateman, H.; Erdlyi, A. (1953). Higher Transcendental Functions, Vol. I (PDF). New York: McGrawHill.(see p. 224)

Gradshteyn, Izrail' Solomonovich; Ryzhik, Iosif Moiseevich (1971). Tablitsy integralov, summ, ryadov iproizvedeniy [Tables of integrals, sums, series and products] (in Russian) (5th ed.). Moscow: Nauka. (seeChapter 9.18)

Humbert, Pierre (1920). Sur les fonctions hypercylindriques. Comptes rendus hebdomadaires des sances del'Acadmie des sciences (in French) 171: 490492. JFM 47.0348.01.

Lauricella, Giuseppe (1893). Sulle funzioni ipergeometriche a pi variabili. Rendiconti del Circolo Matem-atico di Palermo (in Italian) 7: 111158. doi:10.1007/BF03012437. JFM 25.0756.01.

Picard, mile (1881). Sur une extension aux fonctions de deux variables du problme de Riemann relativ auxfonctions hypergomtriques. Annales scientiques de l'cole Normale Suprieure. (2me srie) (in French)10: 305322. JFM 13.0389.01. (see also C. R. Acad. Sci. 90 (1880), pp. 11191121 and 12671269)

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 hyper-geometric type into a hierarchy. For the classical orthogonal polynomials discussed in Andrews & Askey (1985), theAskey scheme was rst drawn by Labelle (1985) and by Askey and Wilson (1985), and has since been extended byKoekoek & Swarttouw (1998) and Koekoek, Lesky & Swarttouw (2010) to cover basic orthogonal polynomials.

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

4F3 Wilson | Racah

3F2 Continuous dual Hahn | Continuous Hahn | Hahn | dual Hahn

2F1 MeixnerPollaczek | Jacobi | Pseudo Jacobi | Meixner | Krawtchouk

2F0/1F1 Laguerre | Bessel | Charlier

1F0 Hermite

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

4 3AskeyWilson | q-Racah

3 2Continuous dual q-Hahn | Continuous q-Hahn | Big q-Jacobi | q-Hahn | dual q-Hahn

2 1Al-SalamChihara | q-MeixnerPollaczek | Continuous q-Jacobi | Big q-Laguerre | Little q-Jacobi | q-Meixner| Quantum q-Krawtchouk | q-Krawtchouk | Ane q-Krawtchouk | Dual q-Krawtchouk

2 0/1 1Continuous big q-Hermite | Continuous q-Laguerre | Little q-Laguerre | q-Laguerre | q-Bessel | q-Charlier |Al-SalamCarlitz I | Al-SalamCarlitz II

1 0Continuous q-Hermite | StieltjesWigert | 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., Polynmes orthogonaux et applications. Proceedings ofthe Laguerre symposium held at Bar-le-Duc, October 1518, 1984., Lecture Notes in Math. 1171, Berlin, NewYork: Springer-Verlag, pp. 3662, doi:10.1007/BFb0076530, ISBN 978-3-540-16059-5, MR 838970

Askey, Richard; Wilson, James (1985), Some basic hypergeometric orthogonal polynomials that generalizeJacobi polynomials,Memoirs of the AmericanMathematical Society 54 (319): iv+55, doi:10.1090/memo/0319,ISBN 978-0-8218-2321-7, ISSN 0065-9266, MR 783216

Koekoek, Roelof; Swarttouw, Ren F. (1998), The Askey-scheme of hypergeometric orthogonal polynomialsand 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, Ren F. (2010), Hypergeometric orthogonal polynomials andtheir q-analogues, SpringerMonographs inMathematics, Berlin, NewYork: 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 Askeys scheme of hypergeometric orthog-onal polynomials, Orthogonal polynomials and their applications (Segovia, 1986), Lecture Notes in Math.1329, Berlin, New York: Springer-Verlag, pp. 4672, 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., Polynmes Orthogonaux et Applications. Proceedings of the Laguerre Symposium held at Bar-le-Duc, LectureNotes inMath. 1171, Berlin, NewYork: Springer-Verlag, pp. xxxvixxxvii, doi:10.1007/BFb0076527,ISBN 978-3-540-16059-5, MR 838967

Chapter 3

AskeyWilson polynomials

In mathematics, theAskeyWilson polynomials (or q-Wilson polynomials) are a family of orthogonal polynomialsintroduced by Askey and Wilson (1985) as q-analogs of the Wilson polynomials. They include many of the otherorthogonal polynomials in 1 variable as special or limiting cases, described in the Askey scheme. AskeyWilsonpolynomials are the special case of Macdonald polynomials (or Koornwinder polynomials) for the non-reduced aneroot system of type (C1, C1), and their 4 parameters a, b, c, d correspond to the 4 orbits of roots of this root system.They are dened by

pn(x; a; b; c; djq) = (ab; ac; ad; q)nan 43qn abcdqn1 aei aei

ab ac ad; q; q

where is a basic hypergeometric function and x = cos() and (,,,)n is the q-Pochhammer symbol. AskeyWilsonfunctions 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 generalizeJacobi polynomials,Memoirs of the AmericanMathematical 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 itsApplications 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719

Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, Ren F. (2010), Askey-Wilsonclass, in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook ofMathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248

Koornwinder, TomH. (2012), Askey-Wilson polynomial, Scholarpedia 7 (7): 7761, doi:10.4249/scholarpedia.7761

7

Chapter 4

Barnes integral

In mathematics, a Barnes integral or MellinBarnes integral is a contour integral involving a product of gammafunctions. They were introduced by Ernest William Barnes (1908, 1910). They are closely related to generalizedhypergeometric series.The integral is usually taken along a contour which is a deformation of the imaginary axis passing to the left of allpoles of factors of the form (a + s) and to the right of all poles of factors of the form (a s).

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

2F1(a; b; c; z) =(c)

(a)(b)

1

2i

Z i1i1

(a+ s)(b+ s)(s)(c+ s)

(z)s ds:

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 hypergeometricfunctions pFq in a similar way.

4.2 Barnes lemmasThe rst Barnes lemma (Barnes 1908) states

1

2i

Z i1i1

(a+ s)(b+ s)(c s)(d s)ds = (a+ c)(a+ d)(b+ c)(b+ d)(a+ b+ c+ d)

:

This is an analogue of Gausss 2F1 summation formula, and also an extension of Eulers beta integral. The integralin it is sometimes called Barness beta integral.The second Barnes lemma (Barnes 1910) states

1

2i

Z i1i1

(a+ s)(b+ s)(c+ s)(1 d s)(s)(e+ s)

ds

=(a)(b)(c)(1 d+ a)(1 d+ b)(1 d+ c)

(e a)(e b)(e c)where e = a + b + c d + 1. This is an analogue of Saalschtzs summation formula.

8

4.3. Q-BARNES INTEGRALS 9

4.3 q-Barnes integralsThere are analogues of Barnes integrals for basic hypergeometric series, and many of the other results can also beextended 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. LondonMath. Soc. s26: 141177. 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-matics 41: 136140. JFM 41.0503.01.

Gasper, George; Rahman, Mizan (2004). Basic hypergeometric series. Encyclopedia of Mathematics and itsApplications 96 (2nd ed.). Cambridge University Press. ISBN 978-0-521-83357-8. MR 2128719.

Chapter 5

Basic hypergeometric series

In mathematics, Heines basic hypergeometric series, or hypergeometric q-series, are q-analog generalizations ofgeneralized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series xn is calledhypergeometric if the ratio of successive terms xn/xn is a rational function of n. If the ratio of successive terms isa rational function of qn, then the series is called a basic hypergeometric series. The number q is called the base.The basic hypergeometric series 21(q,q;q;q,x) was rst considered by Eduard Heine (1846). It becomes thehypergeometric series F(,;;x) in the limit when the base q is 1.

5.1 DenitionThere are two forms of basic hypergeometric series, the unilateral basic hypergeometric series , and the moregeneral bilateral basic geometric series . The unilateral basic hypergeometric series is dened as

jk

a1 a2 : : : ajb1 b2 : : : bk

; q; z

=

1Xn=0

(a1; a2; : : : ; aj ; q)n(b1; b2; : : : ; bk; q; q)n

(1)nq(n2)

1+kjzn

where

(a1; a2; : : : ; am; q)n = (a1; q)n(a2; q)n : : : (am; q)n

and where

(a; q)n =n1Yk=0

(1 aqk) = (1 a)(1 aq)(1 aq2) (1 aqn1):

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

k+1k

a1 a2 : : : ak ak+1b1 b2 : : : bk

; q; z

=

1Xn=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 = ... = abk,and very well poised if in addition a2 = a3 = qa11/2.The bilateral basic hypergeometric series, corresponding to the bilateral hypergeometric series, is dened as

j k

a1 a2 : : : ajb1 b2 : : : bk

; q; z

=

1Xn=1

(a1; a2; : : : ; aj ; q)n(b1; b2; : : : ; bk; q)n

(1)nq(n2)

kjzn:

10

5.2. SIMPLE SERIES 11

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

k k

a1 a2 : : : akb1 b2 : : : bk

; q; z

=

1Xn=1

(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

12 CHAPTER 5. BASIC HYPERGEOMETRIC SERIES

1Xn=1

qn(n+1)/2zn = (q; q)1 (1/z; q)1 (zq; q)1:

Ken Ono gives a related formal power series

A(z; q)def=

1

1 + z

1Xn=0

(z; q)n(zq; q)n z

n =1Xn=0

(1)nz2nqn2 :

5.5 Watsons contour integralAs an analogue of the Barnes integral for the hypergeometric series, Watson showed that

21(a; b; c; q; z) =12i

(a; b; q)1(q; c; q)1

Z i1i1

(qqs; cqs; q)1(aqs; bqs; q)1

(z)ssins ds

where the poles of (aqs; bqs; q)1 lie to the left of the contour and the remaining poles lie to the right. There is asimilar contour integral for rr. This contour integral gives an analytic continuation of the basic hypergeometricfunction 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): 211223, doi:10.1017/S0305004100058114, MR 600238.[2] Benaoum, H. B., "h-analogue of Newtons binomial formula, Journal of Physics A: Mathematical and General 31 (46):

L751L754, arXiv:math-ph/9812011, doi:10.1088/0305-4470/31/46/001.

5.7 References Andrews, G. E. (2010), q-Hypergeometric and Related Functions, in Olver, Frank W. J.; Lozier, Daniel M.;Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge UniversityPress, ISBN 978-0521192255, MR 2723248

W.N. Bailey, Generalized Hypergeometric Series, (1935) Cambridge Tracts in Mathematics and MathematicalPhysics, 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 and Ken Ono, A q-series identity and the Arithmetic of Hurwitz Zeta Functions, (2003)Proceedings of the American Mathematical Society 131, pp. 719724

Sylvie Corteel and Jeremy Lovejoy, Frobenius Partitions and the Combinatorics of Ramanujans 1 1 Summa-tion

Fine, Nathan J. (1988), Basic hypergeometric series and applications, Mathematical Surveys and Monographs27, 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 itsApplications 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8,MR 2128719

Heine, Eduard (1846), "ber die Reihe 1+ (q1)(q1)(q1)(q1) x+ (q1)(q+11)(q1)(q+11)(q1)(q21)(q1)(q+11) x

2+ ", Journalfr die reine und angewandte Mathematik 32: 210212

Eduard Heine, Theorie der Kugelfunctionen, (1878) 1, pp 97125. Eduard Heine, Handbuch die Kugelfunctionen. Theorie und Anwendung (1898) Springer, Berlin.

Chapter 6

Bilateral hypergeometric series

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

an/an

of two terms is a rational function of n. The denition of the generalized hypergeometric series is similar, except thatthe terms with negative n must vanish; the bilateral series will in general have innite numbers of non-zero terms forboth positive and negative n.The bilateral hypergeometric series fails to converge formost rational functions, though it can be analytically continuedto a function dened for most rational functions. There are several summation formulas giving its values for specialvalues where it does converge.

6.1 DenitionThe bilateral hypergeometric series pHp is dened by

pHp(a1; : : : ; ap; b1; : : : ; bp; z) = pHp

a1 : : : apb1 : : : bp

; z

=

1Xn=1

(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 the rising factorial or Pochhammer symbol.Usually the variable z is taken to be 1, in which case it is omitted from the notation. It is possible to dene theseries pHq with dierent p and q in a similar way, but this either fails to converge or can be reduced to the usualhypergeomtric series by changes of variables.

6.2 Convergence and analytic continuationSuppose that none of the variables a or b are integers, so that all the terms of the series are nite and non-zero. Thenthe terms with n1, so the series cannot converge unless|z|=1. When |z|=1, the series converges if

1:

13

14 CHAPTER 6. BILATERAL HYPERGEOMETRIC SERIES

The bilateral hypergeometric series can be analytically continued to a multivalued meromorphic function of severalvariables 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 positiveconverge for |z| 1 to a function satisfying an inhomogeneous linear equation with singularities at z = 0 and z=1, socan also be continued to a multivalued function with these points as branch points. The sum of these functions givesthe analytic continuation of the bilateral hypergeometric series to all values of z other than 0 and 1, and satises alinear dierential equation in z similar to the hypergeometric dierential equation.

6.3 Summation formulas

6.3.1 Dougalls bilateral sum

2H2(a; b; c; d; 1) =

1X1

(a)n(b)n(c)n(d)n

=(d)(c)(1 a)(1 b)(c+ d a b 1)

(c a)(c b)(d a)(d b)(Dougall 1907)This is sometimes written in the equivalent form

1Xn=1

(a+ n)(b+ n)

(c+ n)(d+ n)=

2

sin(a) sin(b)(c+ d a b 1)

(c a)(d a)(c b)(d b) :

6.3.2 Baileys formula(Bailey 1959) gave the following generalization of Dougalls formula:

3H3(a; b; f + 1; d; e; f ; 1) =1X1

(a)n(b)n(f + 1)n(d)n(e)n(f)n

= (d)(e)(1 a)(1 b)(d+ e a b 2)

(d a)(d b)(e a)(e b)

where

= f1 [(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 series 3H3", The Quarterly Jour-nal of Mathematics. Oxford. Second Series 10: 9294, doi:10.1093/qmath/10.1.92, ISSN 0033-5606, MR0107727

Dougall, J. (1907), On Vandermondes Theorem and Some More General Expansions, Proc. EdinburghMath. Soc. 25: 114132, 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 with ISBN 978-0-521-09061-2)

Chapter 7

Binomial transform

In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that com-putes its forward dierences. It is closely related to the Euler transform, which is the result of applying the binomialtransform to the sequence associated with its ordinary generating function.

7.1 DenitionThe binomial transform, T, of a sequence, {an}, is the sequence {sn} dened by

sn =

nXk=0

(1)kn

k

ak:

Formally, one may write (Ta)n = sn for the transformation, where T is an innite-dimensional operator with matrixelements Tnk:

sn = (Ta)n =1Xk=0

Tnkak:

The transform is an involution, that is,

TT = 1

or, using index notation,

1Xk=0

TnkTkm = nm

where nm is the Kronecker delta. The original series can be regained by

an =nX

k=0

(1)kn

k

sk:

The binomial transform of a sequence is just the nth forward dierences of the sequence, with odd dierences carryinga negative sign, namely:

s0 = a0

15

16 CHAPTER 7. BINOMIAL TRANSFORM

s1 = (4a)0 = a1 + a0s2 = (42a)0 = (a2 + a1) + (a1 + a0) = a2 2a1 + a0...sn = (1)n(4na)0where is the forward dierence operator.Some authors dene the binomial transform with an extra sign, so that it is not self-inverse:

tn =nX

k=0

(1)nkn

k

ak

whose inverse is

an =nX

k=0

n

k

tk:

7.2 ExampleBinomial transforms can be seen in dierence tables. Consider the following:The top line 0, 1, 10, 63, 324, 1485,... (a sequence dened by (2n2 + n)3n 2) is the (noninvolutive version of the)binomial transform of the diagonal 0, 1, 8, 36, 128, 400,... (a sequence dened by n22n 1).

7.3 Shift statesThe binomial transform is the shift operator for the Bell numbers. That is,

Bn+1 =nX

k=0

n

k

Bk

where the Bn are the Bell numbers.

7.4 Ordinary generating functionThe transform connects the generating functions associated with the series. For the ordinary generating function, let

f(x) =1Xn=0

anxn

and

g(x) =1Xn=0

snxn

then

g(x) = (Tf)(x) =1

1 xf

x

x 1:

7.5. EULER TRANSFORM 17

7.5 Euler transformThe relationship between the ordinary generating functions is sometimes called the Euler transform. It commonlymakes its appearance in one of two dierent ways. In one form, it is used to accelerate the convergence of analternating series. That is, one has the identity

1Xn=0

(1)nan =1Xn=0

(1)nna0

2n+1

which is obtained by substituting x=1/2 into the last formula above. The terms on the right hand side typically becomemuch smaller, much more rapidly, thus allowing rapid numerical summation.The Euler transform can be generalized (Borisov B. and Shkodrov V., 2007):

1Xn=0

(1)nn+ p

n

an =

1Xn=0

(1)nn+ p

n

na02n+p+1

where p = 0, 1, 2,...The Euler transform is also frequently applied to the Euler hypergeometric integral 2F1 . Here, the Euler transformtakes the form:

2F1(a; b; c; z) = (1 z)b 2F1c a; b; c; z

z 1:

The binomial transform, and its variation as the Euler transform, is notable for its connection to the continued fractionrepresentation 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 functionFor the exponential generating function, let

f(x) =1Xn=0

anxn

n!

and

g(x) =1Xn=0

snxn

n!

18 CHAPTER 7. BINOMIAL TRANSFORM

then

g(x) = (Tf)(x) = exf(x):The Borel transform will convert the ordinary generating function to the exponential generating function.

7.7 Integral representationWhen the sequence can be interpolated by a complex analytic function, then the binomial transform of the sequencecan be represented by means of a NrlundRice integral on the interpolating function.

7.8 GeneralizationsProdinger gives a related, modular-like transformation: letting

un =

nXk=0

n

k

ak(c)nkbk

gives

U(x) =1

cx+ 1B

ax

cx+ 1

where U and B are the ordinary generating functions associated with the series fung and fbng , respectively.The rising k-binomial transform is sometimes dened as

nXj=0

n

j

jkaj :

The falling k-binomial transform is

nXj=0

n

j

jnkaj

Both are homomorphisms of the kernel of the Hankel transform of a series.In the case where the binomial transform is dened as

nXi=0

(1)nin

i

ai = bn:

Let this be equal to the function J(a)n = bn:If a new forward dierence table is made and the rst elements from each row of this table are taken to form a newsequence fbng , then the second binomial transform of the original sequence is,

J2(a)n =nXi=0

(2)nin

i

ai:

7.9. SEE ALSO 19

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

Jk(a)n = bn =nXi=0

(k)nin

i

ai:

Its inverse is,

Jk(b)n = an =nXi=0

knin

i

bi:

This can be generalized as,

Jk(a)n = bn = (E k)na0where E is the shift operator.Its inverse is

Jk(b)n = an = (E+ k)nb0:

7.9 See also Newton series Hankel matrix Mbius 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,

Chapter 8

Conuent hypergeometric function

In mathematics, a conuent hypergeometric function is a solution of a conuent hypergeometric equation, whichis a degenerate form of a hypergeometric dierential equation where two of the three regular singularities merge intoan irregular singularity. (The term "conuent" refers to the merging of singular points of families of dierentialequations; conuere is Latin for to ow together.) There are several common standard forms of conuent hyper-geometric functions:

Kummers (conuent hypergeometric) function M(a, b, z), introduced by Kummer (1837), is a solutionto Kummers dierential equation. There is a dierent and unrelated Kummers function bearing the samename.

Tricomis (conuent hypergeometric) function U(a, b, z) introduced by Francesco Tricomi (1947), some-times denoted by (a; b; z), is another solution to Kummers equation.

Whittaker functions (for Edmund Taylor Whittaker) are solutions toWhittakers equation. Coulomb wave functions are solutions to the Coulomb wave equation. The Kummer functions, Whittakerfunctions, and Coulomb wave functions are essentially the same, and dier from each other only by elementaryfunctions and change of variables.

8.1 Kummers equationKummers equation may be written as:

zd2w

dz2+ (b z)dw

dz aw = 0;

with a regular singular point at z = 0 and an irregular singular point at z = 1 . It has two (usually) linearlyindependent solutions M(a, b, z) and U(a, b, z).Kummers function (of the rst kind)M is a generalized hypergeometric series introduced in (Kummer 1837), givenby:

M(a; b; z) =1Xn=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

8.1. KUMMERS EQUATION 21

is the rising factorial. Another common notation for this solution is (a, b, z). Considered as a function of a, b, or zwith the other two held constant, this denes an entire function of a or z, except when b = 0, 1, 2, ... As a functionof b it is analytic except 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 Kummers function (if it is dened) is a (generalized) Laguerre polynomial.Just as the conuent dierential equation is a limit of the hypergeometric dierential equation as the singular pointat 1 is moved towards the singular point at , the conuent hypergeometric function can be given as a limit of thehypergeometric function

M(a; c; z) = limb!1 2

F1(a; b; c; z/b)

and many of the properties of the conuent hypergeometric function are limiting cases of properties of the hyperge-ometric function.Since Kummers equation is second order there must be another, independent, solution. For this we can usually usethe Tricomi conuent hypergeometric function U(a, b, z) introduced by Francesco Tricomi (1947), and sometimesdenoted by (a; b; z). The function U is dened in terms of Kummers function M by

U(a; b; z) =(1 b)

(a b+ 1)M(a; b; z) +(b 1)(a)

z1bM(a b+ 1; 2 b; z):

This is undened for integer b, but can be extended to integer b by continuity. Unlike Kummers function which isan entire function of z, U(z) usually has a singularity at zero. But see #Special cases for some examples where it isan entire function (polynomial).Note that if

(b 1)(a)

= 0;

which can occur if a is a non-positive integer, then U(a, b, z) andM(a, b, z) are not independent and another solutionis needed. Also when b is a non-positive integer we need another solution because thenM(a, b, z) is not dened. Forinstance, if a = b = 0, Kummers function is undened, but two independent solutions are w(z) = U(0; 0; z) = 1and 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 z1

ubeudu

When b = 1 this second solution is the exponential integral Ei(z).See #Special cases for solutions to some other cases.

8.1.1 Other equationsConuent Hypergeometric Functions can be used to solve the Extended Conuent Hypergeometric Equation whosegeneral form is given as:

z d2wdz2 + (b z)dwdz (

PMm=0 amz

m)w [1]

{NB that for M=0 (or when the summation involves just one term), it reduces to the conventional Conuent Hyper-geometric Equation}Thus Conuent Hypergeometric Functions can be used to solve most second-order ordinary dierential equationswhose variable coecients are all linear functions of z; because they can be transformed to the Extended ConuentHypergeometric Equation. Consider the equation:

22 CHAPTER 8. CONFLUENT HYPERGEOMETRIC FUNCTION

(A+Bz)d2w

dz2+ (C +Dz)

dw

dz+ (E + Fz)w = 0

First we move the regular singular point to 0 by using the substitution of A + Bz z which converts the equation to:

zd2w

dz2+ (C +Dz)

dw

dz+ (E + Fz)w = 0

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

z 7! 1pD2 4F z

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

zd2w

dz2+

C +

DpD2 4F z

dw

dz+

Ep

D2 4F +F

D2 4F zw = 0

whose solution is

exp1 +

DpD2 4F

z

2

w(z);

where w(z) is a solution to Kummers equation with

a =

1 +

DpD2 4F

C

2 Ep

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 12Dzw(z);

where w(z) is a conuent hypergeometric limit function satisfying

zw00(z) + Cw0(z) +E 12CD

w(z) = 0:

As noted lower down, even the Bessel equation can be solved using conuent hypergeometric functions.

8.2 Integral representationsIf Re b > Re a > 0, M(a, b, z) can be represented as an integral

M(a; b; z) =(b)

(a)(b a)Z 10

ezuua1(1 u)ba1 du:

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

8.3. ASYMPTOTIC BEHAVIOR 23

U(a; b; z) =1

(a)

Z 10

eztta1(1 + t)ba1 dt; (Re a > 0)

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

M(a; b; z) =1

2i

(b)

(a)

Z i1i1

(s)(a+ s)(b+ s)

(z)sds

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

8.3 Asymptotic behaviorIf a solution to Kummers equation is asymptotic to a power of z as z , then the power must be a. This is infact the case for Tricomis solution U(a, b, z). Its asymptotic behavior as z can be deduced from the integralrepresentations. If z = x R, then making a change of variables in the integral followed by expanding the binomialseries and integrating it formally term by term gives rise to an asymptotic series expansion, valid as x :[2]

U(a; b; x) xa 2F0a; a b+ 1; ; 1

x

;

where 2F0(; ; ;1/x) is a generalized hypergeometric series (with 1 as leading term), which generally convergesnowhere but exists as a formal power series in 1/x. This asymptotic expansion is also valid for complex z instead ofreal x, with j arg zj < 32:The asymptotic behavior of Kummers solution for large |z| is:

M(a; b; z) (b)ezzab

(a)+

(z)a(b a)

The powers of z are taken using 32 < arg z 12 .[3] The rst term is only needed when (b a) is innite (thatis, when b a is a non-positive integer) or when the real part of z is non-negative, whereas the second term is onlyneeded when (a) is innite (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 Kummers equation asymptotic to ezzab 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)abU(b a; b;z) .

8.4 RelationsThere are many relations between Kummer functions for various arguments and their derivatives. This section givesa 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 functionM(a, b, z) can be written as a linear combination of any two of its contiguous functions, with rational coecients interms of a, b, and z. This gives (42)=6 relations, given by identifying any two lines on the right hand side of

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 Kummers transformationKummers functions are also related by Kummers transformations:

M(a; b; z) = ezM(b a; b;z)U(a; b; z) = z1bU (1 + a b; 2 b; z)

8.5 Multiplication theoremThe following multiplication theorems hold true:

U(a; b; z) = e(1t)zXi=0

(t 1)izii!

U(a; b+ i; zt)

= e(1t)ztb1Xi=0

1 1t

ii!

U(a i; b i; zt):

8.6 Connection with Laguerre polynomials and similar representationsIn terms of Laguerre polynomials, Kummers functions have several expansions, for example

Ma; b; xyx1

= (1 x)a Pn a(n)b(n) L(b1)n (y)xn (Erdelyi 1953, 6.12)

8.7 Special casesFunctions that can be expressed as special cases of the conuent hypergeometric function include:

Some elementary functions (the left-hand side is not dened when b is a non-positive integer, but the right-handside 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) = ezR1z

uaeudu (a polynomial if a is a non-positive integer)U(1;b;z)(b1) +

M(1;b;z)(b) = z

1bez

8.7. SPECIAL CASES 25

U(a; a+ 1; z) = za

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.

Batemans function Bessel functions and many related functions such as Airy functions, Kelvin functions, Hankel functions. Forexample, the special case b = 2a the function reduces to a Bessel function:

1F1(a; 2a; x) = ex2 0F1

; a+ 12 ;

x2

16

= e

x2

x4

12a

a+ 12

Ia 12

x2

:

This identity is sometimes also referred to as Kummers second transformation. Similarly

U(a; 2a; x) =ex2px12aK

a 12

x2

;

When a is a non-positive integer, this equals 2aax2

where is a Bessel polynomial.

The error function can be expressed as

erf(x) = 2p

Z x0

et2

dt =2xp

1F112 ;

32 ;x2

:

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) PoissonCharlier function Toronto functions Whittaker functions M,(z), W,(z) are solutions of Whittakers equation that can be expressed in terms ofKummer functions M and U by

M;(z) = e z2 z+

12M

+ 12 ; 1 + 2; z

W;(z) = e

z2 z+12U

+ 12 ; 1 + 2; z

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

EhN ; 2pi = 22 p2 1+p2 p

1F1

p2 ; 12 ;

2

22

EhN; 2

pi=22 p2 U p2 ; 12 ; 222

In the second formula the functions second branch cut can be chosen by multiplying with (1)p .

26 CHAPTER 8. CONFLUENT HYPERGEOMETRIC FUNCTION

8.8 Application to continued fractionsBy applying a limiting argument to Gausss continued fraction it can be shown that

M(a+ 1; b+ 1; z)

M(a; b; z)=

1

1b a

b(b+ 1)z

1 +

a+ 1

(b+ 1)(b+ 2)z

1b 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 thatdoes not include a pole.

8.9 Notes[1] Campos, LMBC (2001). On Some Solutions of the Extended Conuent Hypergeometric Dierential 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. Theyswitch the sign of the exponent in exp(ia) in the right half-plane but this is unimportant because the term is negligiblethere 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 Functionswith Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 504, ISBN 978-0486612720, MR0167642.

Chistova, E.A. (2001), c/c024700, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

Daalhuis, Adri B. Olde (2010), Conuent hypergeometric function, in Olver, Frank W. J.; Lozier, DanielM.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge UniversityPress, ISBN 978-0521192255, MR 2723248

Erdlyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz & Tricomi, Francesco G. (1953). Higher transcen-dental functions. Vol. I. New YorkTorontoLondon: McGrawHill Book Company, Inc. MR 0058756.

Kummer, Ernst Eduard (1837). De integralibus quibusdam denitis et seriebus innitis. Journal fr diereine und angewandte Mathematik (in Latin) 17: 228242. doi:10.1515/crll.1837.17.228. ISSN 0075-4102.

Slater, Lucy Joan (1960). Conuent hypergeometric functions. Cambridge, UK: Cambridge University Press.MR 0107026.

Tricomi, Francesco G. (1947). Sulle funzioni ipergeometriche conuenti. Annali di Matematica Pura edApplicata. Serie Quarta (in Italian) 26: 141175. doi:10.1007/bf02415375. ISSN 0003-4622. MR 0029451.

Tricomi, Francesco G. (1954). Funzioni ipergeometriche conuenti. Consiglio Nazionale Delle RicercheMono-grae Matematiche (in Italian) 1. Rome: Edizioni cremonese. ISBN 978-88-7083-449-9. MR 0076936.

8.11. EXTERNAL LINKS 27

8.11 External links Conuent 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

Chapter 9

Dixons identity

In mathematics, Dixons identity (or Dixons theorem or Dixons formula) is any of several dierent but closelyrelated identities proved by A. C. Dixon, some involving nite sums of products of three binomial coecients, andsome evaluating a hypergeometric sum. These identities famously follow from the MacMahon Master theorem, andcan now be routinely proved by computer algorithms (Ekhad 1990).

9.1 StatementsThe original identity, from (Dixon 1891), is

aXk=a

(1)k

2a

k + a

3=

(3a)!

(a!)3:

A generalization, also sometimes called Dixons identity, is

aXk=a

(1)ka+ 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 terminatingwell-poised hypergeometric series

b+ c

b a

c+ a

c a3F2(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 Dixons theorem evaluating a well-poised 3F2generalized hypergeometric series at 1, from (Dixon 1902):

3F2(a; b; c; 1 + a b; 1 + a c; 1) = (1 + a/2)(1 + a/2 b c)(1 + a b)(1 + a c)(1 + a)(1 + a b c)(1 + a/2 b)(1 + a/2 c) :

This holds for Re(1 + 12a b c) > 0. As c tends to it reduces to Kummers formula for the hypergeometricfunction 2F1 at 1. Dixons theorem can be deduced from the evaluation of the Selberg integral.

9.2 q-analoguesA q-analogue of Dixons formula for the basic hypergeometric series in terms of the q-Pochhammer symbol is givenby

28

9.3. REFERENCES 29

43

a qa1/2 b c

a1/2 aq/b aq/c; q; qa1/2/bc

=

(aq; aq/bc; qa1/2/b; qa1/2/c; q)1(aq/b; aq/c; qa1/2; qa1/2/bc; q)1

where |qa1/2/bc| < 1.

9.3 References Dixon, A.C. (1891), On the sum of the cubes of the coecients in a certain expansion by the binomialtheorem, Messenger of Mathematics 20: 7980, JFM 22.0258.01

Dixon, A.C. (1902), Summation of a certain series, Proc. LondonMath. Soc. 35 (1): 284291, doi:10.1112/plms/s1-35.1.284, JFM 34.0490.02

Ekhad, Shalosh B. (1990), A very short proof of Dixons theorem, Journal of Combinatorial Theory. SeriesA 54 (1): 141142, doi:10.1016/0097-3165(90)90014-N, ISSN 1096-0899, MR 1051787, Zbl 0707.05007

Gessel, Ira; Stanton, Dennis (1985), Short proofs of Saalschtzs and Dixons theorems, Journal of Combi-natorial Theory. Series A 38 (1): 8790, doi:10.1016/0097-3165(85)90026-3, ISSN 1096-0899, MR 773560,Zbl 0559.05008

Ward, James (1991), 100 years of Dixons identity, Irish Mathematical Society Bulletin (27): 4654, ISSN0791-5578, MR 1185413, Zbl 0795.01009

Wilf, Herbert S. (1994),Generatingfunctionology (2nd ed.), Boston,MA:Academic Press, ISBN0-12-751956-4, Zbl 0831.05001

Chapter 10

Dougalls formula

Dougalls formula may refer to one of two formulas for hypergeometric series, both named after John Dougall:

Dougalls formula for the sum of a 7F6 hypergeometric series Dougalls formula for the sum of a bilateral hypergeometric series

30

Chapter 11

Elliptic hypergeometric series

In mathematics, an elliptic hypergeometric series is a series cn such that the ratio cn/cn is an elliptic function ofn, analogous to generalized hypergeometric series where the ratio is a rational function of n, and basic hypergeometricseries 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 DenitionsThe q-Pochhammer symbol is dened by

(a; q)n =

n1Yk=0

(1 aqk) = (1 a)(1 aq)(1 aq2) (1 aqn1):

(a1; a2; : : : ; am; q)n = (a1; q)n(a2; q)n : : : (am; q)n:

The modied Jacobi theta function with argument x and nome p is dened by

(x; p) = (x; p/x; p)1

(x1; :::; xm; p) = (x1; p):::(xm; p)

The elliptic shifted factorial is dened by

(a; q; p)n = (a; p)(aq; p):::(aqn1; p)

(a1; :::; am; q; p)n = (a1; q; p)n (am; q; p)nThe theta hypergeometric series rEr is dened by

r+1Er(a1; :::ar+1; b1; :::; br; q; p; z) =1Xn=0

(a1; :::; ar+1; q; p)n(q; b1; :::; br; q; p)n

zn

The very well poised theta hypergeometric series rVr is dened by

r+1Vr(a1; a6; a7; :::ar+1; q; p; z) =1Xn=0

(a1q2n; p)

(a1; p)

(a1; a6; a7; :::; ar+1; q; p)n(q; a1q/a6; a1q/a7; :::; a1q/ar+1; q; p)n

(qz)n

31

32 CHAPTER 11. ELLIPTIC HYPERGEOMETRIC SERIES

The bilateral theta hypergeometric series rGr is dened by

rGr(a1; :::ar; b1; :::; br; q; p; z) =1X

n=1

(a1; :::; ar; q; p)n(b1; :::; br; q; p)n

zn

11.2 Denitions of additive elliptic hypergeometric seriesThe elliptic numbers are dened by

[a;; ] =1(a; e

i )

1(; ei )

where the Jacobi theta function is dened by

1(x; q) =1X

n=1(1)nq(n+1/2)2e(2n+1)ix

The additive elliptic shifted factorials are dened by

[a;; ]n = [a;; ][a+ 1;; ]:::[a+ n 1;; ] [a1; :::; am;; ] = [a1;; ]:::[am;; ]

The additive theta hypergeometric series rer is dened by

r+1er(a1; :::ar+1; b1; :::; br;; ; z) =1Xn=0

[a1; :::; ar+1;; ]n[1; b1; :::; br;; ]n

zn

The additive very well poised theta hypergeometric series rvr is dened by

r+1vr(a1; a6; :::ar+1;; ; z) =1Xn=0

[a1 + 2n;; ]

[a1;; ]

[a1; a6; :::; ar+1;; ]n[1; 1 + a1 a6; :::; 1 + a1 ar+1;; ]n z

n

11.3 References Frenkel, Igor B.; Turaev, Vladimir G. (1997), Elliptic solutions of the Yang-Baxter equation and modularhypergeometric functions, The Arnold-Gelfand mathematical seminars, Boston, MA: Birkhuser Boston, pp.171204, ISBN 978-0-8176-3883-2, MR 1429892

Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and itsApplications 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 mathe-matical physics (St. Petersburg, 2001), NATO Sci. Ser. II Math. Phys. Chem. 77, Dordrecht: Kluwer Acad.Publ., pp. 307327, arXiv:math/0303204, MR 2000728

Spiridonov, V. P. (2003), Theta hypergeometric integrals, Rossiskaya Akademiya Nauk. Algebra i Analiz15 (6): 161215, 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, Rossiskaya AkademiyaNauk. MoskovskoeMatematicheskoe Obshchestvo. UspekhiMatematicheskikhNauk 63 (3): 372, doi:10.1070/RM2008v063n03ABEH004533,MR 2479997

Warnaar, S. Ole (2002), Summation and transformation formulas for elliptic hypergeometric series, Con-structive Approximation. an International Journal for Approximations and Expansions 18 (4): 479502, 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 dened by a MellinBarnes integral

H m;np;q

z

(a1; A1) (a2; A2) : : : (ap; Ap)(b1; B1) (b2; B2) : : : (bq; Bq)=

1

2i

ZL

(Qm

j=1 (bj +Bjs))(Qn

j=1 (1 aj Ajs))(Qq

j=m+1 (1 bj Bjs))(Qp

j=n+1 (aj +Ajs))zs ds

where L is a certain contour separating the poles of the two factors in the numerator. Another generalization of FoxH-function is given by Innayat Hussain AA (1987). For a further generalization of this function, useful in Physicsand 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...pand k = 1...q (Srivastava 1984, p. 50):

H m;np;q

z

(a1; C) (a2; C) : : : (ap; C)(b1; C) (b2; C) : : : (bq; C)=

1

CGm;np;q

a1; : : : ; apb1; : : : ; bq

z1/C :12.1 References

Fox, Charles (1961), The G and H functions as symmetrical Fourier kernels, Transactions of the AmericanMathematical Society 98: 395429, 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: 41094117.

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: 41194128.

Mathai, A. M.; Saxena, RamKishore (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: Springer-Verlag, ISBN 978-1-4419-0915-2, MR 2562766

Rathie, Arjun K. (1997), A new generalization of generalized hypergeometric function, Le Matematiche LII:297310.

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

FoxWright function

In mathematics, the FoxWright function (also known as FoxWright Psi function or justWright function, notto 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):

pq

(a1; A1) (a2; A2) : : : (ap; Ap)(b1; B1) (b2; B2) : : : (bq; Bq)

; z

=

1Xn=0

(a1 +A1n) (ap +Apn)(b1 +B1n) (bq +Bqn)

zn

n!:

Its normalisation

pq

(a1; A1) (a2; A2) : : : (ap; Ap)(b1; B1) (b2; B2) : : : (bq; Bq)

; z

=

(b1) (bq)(a1) (ap)

1Xn=0

(a1 +A1n) (ap +Apn)(b1 +B1n) (bq +Bqn)

zn

n!

becomes pFq(z) for A...p = B...q = 1.The FoxWright function is a special case of the Fox H-function (Srivastava & Manocha 1984, p. 50):

pq

(a1; A1) (a2; A2) : : : (ap; Ap)(b1; B1) (b2; B2) : : : (bq; Bq)

; z

= H1;pp;q+1

z

(1 a1; A1) (1 a2; A2) : : : (1 ap; Ap)(0; 1) (1 b1; B1) (1 b2; B2) : : : (1 bq; Bq):

13.1 References Wright, E. M. (1935). The asymptotic expansion of the generalized hypergeometric function. Proc. LondonMath. Soc. 10 (4): 286293. 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 FoxWright Psi Functions for Certain RationalParameters. Computers Math. Applic. 30 (11): 7382.

35

Chapter 14

Frobenius solution to the hypergeometricequation

In the following we solve the second-order dierential equation called the hypergeometric dierential equation usingFrobenius method, named after Ferdinand Georg Frobenius. This is a method that uses the series solution for adierential equation, where we assume the solution takes the form of a series. This is usually the method we use forcomplicated ordinary dierential equations.The solution of the hypergeometric dierential equation is very important. For instance, Legendres dierentialequation can be shown to be a special case of the hypergeometric dierential equation. Hence, by solving the hyper-geometric dierential equation, one may directly compare its solutions to get the solutions of Legendres dierentialequation, after making the necessary substitutions. For more details, please check the hypergeometric dierentialequation.We shall prove that this equation has three singularities, namely at x = 0, x = 1 and around innity. However, as thesewill turn out to be regular singular points, we will be able to assume a solution on the form of a series. Since this is asecond-order dierential 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 bedened (depending on the value of the parameters of the equation). This is why we shall study the dierent cases forthe parameters and modify our assumed solution accordingly.

14.1 The equationSolve the hypergeometric equation around all singularities:

x(1 x)y00 + f (1 + + )xg y0 y = 0

14.2 Solution around x = 0Let

P0(x) = ;P1(x) = (1 + + )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. Lets start with x = 0. To see if it is regular, we study the following limits:

limx!a

(x a)P1(x)P2(x)

= limx!0

(x 0)( (1 + + )x)x(1 x) = limx!0

x( (1 + + )x)x(1 x) =

limx!a

(x a)2P0(x)P2(x)

= limx!0

(x 0)2()x(1 x) = limx!0

x2()x(1 x) = 0

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

y =1Xr=0

arxr+c

with a0 0. Hence,

y0 =1Xr=0

ar(r + c)xr+c1

y00 =1Xr=0

ar(r + c)(r + c 1)xr+c2:

Substituting these into the hypergeometric equation, we get

x1Xr=0

ar(r+c)(r+c1)xr+c2x21Xr=0

ar(r+c)(r+c1)xr+c2+1Xr=0

ar(r+c)xr+c1(1++)x

1Xr=0

ar(r+c)xr+c1

1Xr=0

arxr+c = 0

That is,

1Xr=0

ar(r+c)(r+c1)xr+c11Xr=0

ar(r+c)(r+c1)xr+c+1Xr=0

ar(r+c)xr+c1(1++)

1Xr=0

ar(r+c)xr+c

1Xr=0

arxr+c = 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:

1Xr=0

ar(r + c)(r + c 1)xr+c1 1Xr=1

ar1(r + c 1)(r + c 2)xr+c1 + 1Xr=0

ar(r + c)xr+c1

(1 + + )1Xr=1

ar1(r + c 1)xr+c1 1Xr=1

ar1xr+c1 = 0

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

a0(c(c 1) + c)xc1 +1Xr=1

ar(r + c)(r + c 1)xr+c1 1Xr=1

ar1(r + c 1)(r + c 2)xr+c1

+ 1Xr=1

ar(r + c)xr+c1 (1 + + )

1Xr=1

ar1(r + c 1)xr+c1 1Xr=1

ar1xr+c1 = 0

Now, from the linear independence of all powers of x, that is, of the functions 1, x, x2, etc., the coecients of xkvanish for all k. Hence, from the rst term, we have

a0(c(c 1) + c) = 0

38 CHAPTER 14. FROBENIUS SOLUTION TO THE HYPERGEOMETRIC EQUATION

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

c(c 1 + ) = 0:Hence,

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

((r + c)(r + c 1) + (r + c))ar + ((r + c 1)(r + c 2) (1 + + )(r + c 1) )ar1 = 0Hence,

ar =(r + c 1)(r + c 2) + (1 + + )(r + c 1) +

(r + c)(r + c 1) + (r + c) ar1

=(r + c 1)(r + c+ + 1) +

(r + c)(r + c+ 1) ar1

But

(r + c 1)(r + c+ + 1) + = (r + c 1)(r + c+ 1) + (r + c 1) + = (r + c 1)(r + c+ 1) + (r + c+ 1)

Hence, we get the recurrence relation

ar =(r + c+ 1)(r + c+ 1)

(r + c)(r + c+ 1) ar1; for r 1:

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

a1 =(c+ )(c+ )

(c+ 1)(c+ )a0

a2 =(c+ + 1)(c+ + 1)

(c+ 2)(c+ + 1)a1 =

(c+ + 1)(c+ )(c+ )(c+ + 1)

(c+ 2)(c+ 1)(c+ )(c+ + 1)a0 =

(c+ )2(c+ )2(c+ 1)2(c+ )2

a0

a3 =(c+ + 2)(c+ + 2)

(c+ 3)(c+ + 2)a2 =

(c+ )2(c+ + 2)(c+ )2(c+ + 2)

(c+ 1)2(c+ 3)(c+ )2(c+ + 2)a0 =

(c+ )3(c+ )3(c+ 1)3(c+ )3

a0

As we can see,

ar =(c+ )r(c+ )r(c+ 1)r(c+ )r

a0; for r 0

Hence, our assumed solution takes the form

y = a0

1Xr=0

(c+ )r(c+ )r(c+ 1)r(c+ )r

xr+c:

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

14.3. ANALYSIS OF THE SOLUTION IN TERMS OF THE DIFFERENCE 1 OF THE TWO ROOTS 39

14.3 Analysis of the solution in terms of the dierence 1 of the tworoots

14.3.1 not an integerThen y1 = y|c and y2 = y|c . Since

y = a0

1Xr=0

(c+ )r(c+ )r(c+ 1)r(c+ )r

xr+c;

we have

y1 = a0

1Xr=0

()r()r(1)r()r

xr = a0 2F1(; ; ;x)

y2 = a0

1Xr=0

(+ 1 )r( + 1 )r(1 + 1)r(1 + )r x

r+1

= a0x1

1Xr=0

(+ 1 )r( + 1 )r(1)r(2 )r x

r

= a0x1

2F1( + 1; + 1; 2 ;x)

Hence, y = A0y1 +B0y2: Let A a0 = a and B a0 = B. Then

y = A2F1(; ; ;x) +Bx1

2F1( + 1; + 1; 2 ;x)

14.3.2 = 1Then y1 = y|c . Since = 1, we have

y = a0

1Xr=0

(c+ )r(c+ )r(c+ 1)2r

xr+c:

Hence,

y1 = a0

1Xr=0

()r()r(1)r(1)r

xr = a02F1(; ; 1;x)

y2 =@y

@c

c=0

:

To calculate this derivative, let

Mr =(c+ )r(c+ )r

(c+ 1)2r:

Then

ln(Mr) = ln(c+ )r(c+ )r

(c+ 1)2r

= ln(c+ )r + ln(c+ )r 2 ln(c+ 1)r

40 CHAPTER 14. FROBENIUS SOLUTION TO THE HYPERGEOMETRIC EQUATION

But

ln(c+ )r = ln ((c+ )(c+ + 1) (c+ + r 1)) =r1Xk=0

ln(c+ + k):

Hence,

ln(Mr) =r1Xk=0

ln(c+ + k) +r1Xk=0

ln(c+ + k) 2r1Xk=0

ln(c+ 1 + k)

=r1Xk=0

(ln(c+ + k) + ln(c+ + k) 2 ln(c+ 1 + k))

Dierentiating both sides of the equation with respect to c, we get:

1

Mr

@Mr@c

=

r1Xk=0

1

c+ + k+

1

c+ + k 2c+ 1 + k

:

Hence,

@Mr@c

=(c+ )r(c+ )r

(c+ 1)2r

r1Xk=0

1

c+ + k+

1

c+ + k 2c+ 1 + k

:

Now,

y = a0xc1Xr=0

(c+ )r(c+ )r(c+ 1)2r

xr = a0xc1Xr=0

Mrxr:

Hence,

@y

@c= a0x

c ln(x)1Xr=0

(c+ )r(c+ )r(c+ 1)2r

xr + a0xc1Xr=0

(c+ )r(c+ )r

(c+ 1)2r

(r1Xk=0

1

c+ + k+

1

c+ + k 2c+ 1 + k

)!xr

= a0xc1Xr=0

(c+ )r(c+ )r(c+ 1)r)2

lnx+

r1Xk=0

1

c+ + k+

1

c+ + k 2c+ 1 + k

!xr:

For c = 0, we get

y2 = a0

1Xr=0

()r()r(1)2r

lnx+

r1Xk=0

1

+ k+

1

+ k 2

1 + k

!xr:

Hence, y = Cy1 + Dy2. Let Ca0 = C and Da0 = D. Then

y = C2F1(; ; 1;x) +D1Xr=0

()r()r(1)2r

ln(x) +

r1Xk=0

1

+ k+

1

+ k 2

1 + k

!xr

14.3. ANALYSIS OF THE SOLUTION IN TERMS OF THE DIFFERENCE 1 OF THE TWO ROOTS 41

14.3.3 an integer and 1 0

From the recurrence relation

ar =(r + c+ 1)(r + c+ 1)

(r + c)(r + c+ 1) ar1;

we see that when c = 0 (the smaller root), a . Hence, we must make the substitution a0 = b0 (c - ci), where ciis the root for which our solution is inn