Démonstration «automatique» d'identités et fonctions hypergéométriques

 Access to full text

 Full (PDF)

1. [1] M. Abramowitz et I. Stegun, Handbook of Mathematical functions, Dover, New-York, 1965. 
2. [2] A. Erdelyi (éditeur), Higher transcendental functions, 3 volumes, McGraw-Hill, New York, 1953. 
3. [3] I. Gradshteyn et I.: Ryzhik, Table of integrals, series and products, Academic Press, New York, 1980. Zbl0918.65002
4. [4] P. Appell et J. Kampé De Fériet, Fonctions hypergéométriques et hypersphériques, Gauthier-Villars, Paris, 1926. Zbl52.0361.13JFM52.0361.13
5. [5] J.E. Rainville, Special functions, MacMillan, New York, 1960. Zbl0092.06503MR107725
6. [6] L. Slater, Confluent hypergeometric functions, Cambridge University Press, Cambridge, 1960. Zbl0086.27502MR107026
7. [7] E. Whittaker et G. Watson, Modern Analysis, Cambridge University Press, Cambridge, 1946. Zbl0108.26903
8. [8] M. Aigner, Combinatorial theory, Springer, Berlin, 1979 (voir surtout le chapitre 3). Zbl0858.05001MR542445
9. [9] G. Andrews, The theory of partitions, Addison-Wesley, Reading, 1976. Zbl0371.10001MR557013
10. [10] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974. ¿ Zbl0283.05001MR460128
11. [11] R. Graham, D. Knuth et O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, 1989. Zbl0668.00003MR1397498
12. [12] D. Knuth, The art of computer programming, 3 volumes, Addison-Wesley, Reading, 1968-1973. Zbl0191.18001MR378456
13. [13] P. Macmahon, Combinatory analysis, 2 volumes, Chelsea, New York, 1960 (réimpression). Zbl0101.25102MR141605
14. [14] J. Riordan, An introduction to combinatorial analysis, John Wiley, New York, 1958. Zbl0078.00805MR96594
15. [15] G. Andrews, q-series : Their development and application in analysis, number theory, combinatorics, physics and computer algebra,CBMS Regional Conference Lecture Series, 66, Amer. Math. Soc., Providence, 1986. Zbl0594.33001MR858826
16. [16] W. Bailey, Generalized hypergeometric series, Cambridge University Press, Cambridge, 1935 (réimprimé par Stechert-Hafner, New York, 1964). Zbl0011.02303MR185155
17. [17] H. Exton, q-hypergeometric functions and applications, Ellis Horwood/John Wiley, New York, 1983. Zbl0514.33001MR708496
18. [18] N. Fine, Basic hypergeometric series and applications, Math. Surv.27, Amer. Math. Soc., Providence, 1988. Zbl0647.05004MR956465
19. [19] G. Gasper et M. Rahman, Basic hypergeometric series, Cambridge University Press, Cambridge, 1990. Zbl0695.33001MR1052153
20. [20] E. Heine, Handbuch der Kugelfunktionen. Theorie und Anwendung, 2 volumes, Springer, 1898 (=Physica Verlag, Würzburg, 1961). Zbl0103.29304MR204726
21. [21] L. Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. Zbl0135.28101MR201688
22. [1] D. Zeilberger, Sister Celine's technique and its generalizations, J. Math. Anal. Appl.85 (1982), 114-145. Zbl0485.05003MR647562
23. [2] D. Zeilberger, A Holonomic systems approach to special functions identities, J. of Computational and Applied Math.32 (1990), 321-368. Zbl0738.33001MR1090884
24. [3] D. Zeilberger, A Fast Algorithm for proving terminating hypergeometric identities, Discrete Math.80 (1990), 207-211. Zbl0701.05001MR1048463
25. [4] D. Zeilberger. The method of creative telescoping, J. Symbolic Computation11 (1991). 195-204. Zbl0738.33002MR1103727
26. [5] D. Zeilberger, Closed Form (pun intended !), to appear in : "Special volume in memory of Emil Grosswald", M. Knopp, ed., Contemporary Mathematics, AMS. Zbl0808.05010MR1210544
27. [6] D. Zeilberger, Three recitations on Holonomic Systems and Hypergeometric Series, Proceedings of the Séminaire Lotharingien de combinatoire24, IRMA, Strasbourg, à paraître. Zbl0981.05514
28. [7] D. Zeilberger, Plain (Lagrange interpolation) proofs of Fancy (representation theory) formulas, en préparation. 
29. [8] S.B. Ekhad, Short proofs of two hypergeometric summation formulas of Karlsson, Proc. Amer. Math. Soc.107 (1989), 1143-1144. Zbl0688.33001MR1019759
30. [9] S.B. Ekhad, A very short proof of Dixon's theorem, J. Comb. Theo., Series A54 (1990), 141-142. Zbl0707.05007MR1051787
31. [10] S.B. Ekhad, A one-line proof o f the Habsieger-Zeilberger G2 constant term identity, J. Comput. Appl. Math.34 (1991), 133-134. Zbl0737.33010MR1095202
32. [11] S.B. Ekhad, Short Proof Of A "Strange" Combinatorial Identity Conjectured by Gosper, Discrete Math. à paraître. Zbl0741.05004
33. [12] S.B. Ekhad, A Short, Elementary and Easy. WZ proof of the Askey-Gasper inequality that was used by de Branges in his proof of the Bieberbach conjecture, à paraître. Zbl0780.33004
34. [13] S.B. Ekhad and S. Tre, A purely verification proof of the first Rogers-Ramanujan identity, J. Comb. The. Ser. A54 (1990), 309-311. Zbl0702.05007MR1060003
35. [14] S.B. Ekhad and D. Zeilberger, A 21st century proof of Dougall's hypergeometric sum identity, J. Math. Anal. Appl.147 (1990), 610- 611. Zbl0714.33002MR1050232
36. [15] H.S. Wilf and D. Zeilberger, Towards computerized proofs of identities, Bulletin of the Amer. Math. Soc.23 (1990), 77-83. Zbl0718.05010MR1019401
37. [16] H.S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc.3 (1990), 147-158. Zbl0695.05004MR1007910
38. [17] H.S. Wilf and D. Zeilberger, A general theory of multi-variate hypergeometric identities. en préparation. 
39. [18] H.S. Wilf, 54 computer-generated proofs of binomial coefficient identities, à paraître. 
40. [19] G. Almkvist et D. Zeilberger, The method of differentiating under the integral sign, Journ. Symb. Computation, 10 (1990), 571- 591. Zbl0717.33004MR1087980
41. [20] G. Almkvist et D. Zeilberger, A MAPLE program that finds, and proves, recurrences and differential equations satisfied by hyperexponential definite integrals, SIGSAM Bulletin25 (1991),... 
42. [21] D. Zeilberger, A MAPLE program for proving hypergeometric series, SIGSAM Bulletin25 (1991),... 
43. [1] J. Bernstein, Modules over the ring of differential operators. A study of the fundamental solutions of equations with constant coefficients, Funk. Analisis, Akademia Nauk CCCR5 (2) (1971), 1-16. Zbl0233.47031MR290097
44. [2] J. Bernstein, The analytic continuation of generalized functions with respect to a parameter, Funct. Anal. and Appl.6 (1972), 273- 285. Zbl0282.46038MR320735
45. [3] J.-E. Björk, Rings of Differential Operators, North Holland, Amsterdam, 1979. Zbl0499.13009MR549189
46. [4] A. Borel et al., Algebraic D-modules, Perspectives in Math.2, Academic Press, Boston, 1987. Zbl0642.32001MR882000
47. [5] B. Buchberger, An algorithmic method in polynomial ideal theory, N.K. Bose ed. Recents trends in multidimensional systems theory, D. Reidel Publishing Corp., 1985. Zbl0587.13009MR835951
48. [6] F. Ehlers, The Weyl Algebra, Chapitre V de [4], 173-205. 
49. [7] A. Galligo, Some algorithmic questions on ideals of differential operators, Lect. Note in Comp. Sci.204 (1985), 413-421. Zbl0634.16001MR826576
50. [8] R. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. USA75 (1978), 40-42. Zbl0384.40001MR485674
51. [9] M. Kashiwara, B-functions and holonomic systems, Invent. Math.38 (1976), 33-53. Zbl0354.35082MR430304
52. [10] M. Kashiwara, Vanishing cycles sheaves and holonomic systems of differential equations, SpringerLect. Notes in Math.1016 (1983), 134-142. Zbl0566.32022MR726425
53. [11] B. Malgrange, L'involutivité des caractéristiques des systèmes différentiels et microdifférentiels, Sém. Bourbaki, exposé 522, 1977-1978 (Lect. Notes in Math.710, 277-289). Zbl0423.46033MR554227
54. [12] R. Risch, The solution of the problem in integrating in finite terms, Bull. Amer. Math. Soc.76 (1970), 605-608. Zbl0196.06801MR269635
55. [13] N. Takayama, Gröbner basis and the problem of contiguous relations, Japan Journ. Appl. Math.6 (1989), 147-160. Zbl0691.68032MR981518
56. [14] N. Takayama, An algorithm for constructing the integral o f a module - an infinite dimensional analog of Gröbner basis, Proceedings of ISSAC'90, A.C.M. Press. 
57. [15] N. Takayama, An approach to the zero recognition problem by Buchberger algorithm, Journ. Symb. Computation, Zbl0763.65007
58. [1] K. Aomoto, A note on holonomic q-difference systems, Algebraic Analysis (in honor of M. Sato), M. Kashiwara and T. Kawai eds., Academic Press, (1988), 25-28. Zbl0674.33006MR992444
59. [2] K. Aomoto, q-analogue of de Rham cohomology associated with Jackson integrals, Proc. Japan Acad.66 (1990), 161-164. Zbl0718.33011MR1078398
60. [3] K. Aomoto, Finiteness of a cohomology associated with certain Jackson integrals, Tohoku J. Math.43 (1991), 75-101. Zbl0769.33016MR1088716
61. [4] C. Sabbah, Systèmes holonomes d'équations aux q-différences, Pré-publication Centre de Math. Ecole Polytechnique, Palaiseau, 1991. 
62. [1] P. Cartier et D. Foata, Problèmes combinatoires de commutation et réarrangements, Lect. Notes Math. vol. 85, Springer1969. Zbl0186.30101MR239978
63. [2] R. Apéry, Irrationalité de ζ(2) et ζ(3), Astérisque61 (1979), 11-13. Zbl0401.10049
64. [3] A. van des Poorten, A proof that Euler missed... Apery's proof of the irrationality of ζ(3), Math. Intelligencer, 1 (1979), 195-203. Sur les conjectures de Macdonald : Zbl0409.10028
65. [4] I.G. Macdonald, Affine root systems and Dedekind η-function, Invent. Math.15 (1972), 91-143. Zbl0244.17005
66. [5] I.G. Macdonald, Some conjectures for root systems, SIAM Journ. Math. Anal.13 (1982), 988-1007. Zbl0498.17006MR674768
67. [6] D. Zeilberger, Unified approach to Macdonald's root system conjectures, SIAM Journ. Math. Anal.19 (1988), 987-1013. Zbl0658.05005MR946656
68. [7] F. Garvan et G. Bonnet, Macdonald's constant term conjectures for exceptional root systems, Bull. Amer. Math. Soc.24 (1991), 343-347. Zbl0737.33011MR1078471
69. [8] A. Helversen-Pasotto, L'identité de Barnes pour les corps finis, C.R. Acad. Sci. Paris, Série A, 286 (1978), 297-300. Zbl0373.12009MR476707
70. [9] J. Greene et D. Stanton, A character sum evaluation and Gaussian hypergeometric series, Journ. Number Theory, 23 (1986), 136-148. Zbl0588.10038MR840021

1. Dominique Foata, Commutation et réarrangements, hier et aujourd'hui
2. Pierre Cartier, Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents
3. Stéphane Fischler, Irrationalité de valeurs de zêta