Démonstration « automatique » d'identités et fonctions hypergéométriques

Cartier, Pierre

Séminaire Bourbaki : volume 1991/92, exposés 745-759, Astérisque, no. 206 (1992), Exposé no. 746, 51 p.

[1] M. Abramowitz et I. Stegun, Handbook of Mathematical functions, Dover, New-York, 1965.

[2] A. Erdelyi (éditeur), Higher transcendental functions, 3 volumes, McGraw-Hill, New York, 1953.

[3] I. Gradshteyn et I.: Ryzhik, Table of integrals, series and products, Academic Press, New York, 1980. | Zbl

[4] P. Appell et J. Kampé De Fériet, Fonctions hypergéométriques et hypersphériques, Gauthier-Villars, Paris, 1926. | JFM

[5] J. E. Rainville, Special functions, MacMillan, New York, 1960. | Zbl | MR

[6] L. Slater, Confluent hypergeometric functions, Cambridge University Press, Cambridge, 1960. | Zbl | MR

[7] E. Whittaker et G. Watson, Modern Analysis, Cambridge University Press, Cambridge, 1946.

[8] M. Aigner, Combinatorial theory, Springer, Berlin, 1979 (voir surtout le chapitre 3). | Zbl | MR

[9] G. Andrews, The theory of partitions, Addison-Wesley, Reading, 1976. | Zbl | MR

[10] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974. ¿ | Zbl | MR

[11] R. Graham, D. Knuth et O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, 1989. | Zbl | MR

[12] D. Knuth, The art of computer programming, 3 volumes, Addison-Wesley, Reading, 1968-1973. | Zbl | MR

[13] P. Macmahon, Combinatory analysis, 2 volumes, Chelsea, New York, 1960 (réimpression). | Zbl | MR

[14] J. Riordan, An introduction to combinatorial analysis, John Wiley, New York, 1958. | Zbl | MR

[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. | Zbl | MR

[16] W. Bailey, Generalized hypergeometric series, Cambridge University Press, Cambridge, 1935 (réimprimé par Stechert-Hafner, New York, 1964). | Zbl | MR

[17] H. Exton, q-hypergeometric functions and applications, Ellis Horwood/John Wiley, New York, 1983. | Zbl | MR

[18] N. Fine, Basic hypergeometric series and applications, Math. Surv. 27, Amer. Math. Soc., Providence, 1988. | Zbl | MR

[19] G. Gasper et M. Rahman, Basic hypergeometric series, Cambridge University Press, Cambridge, 1990. | Zbl | MR

[20] E. Heine, Handbuch der Kugelfunktionen. Theorie und Anwendung, 2 volumes, Springer, 1898 (=Physica Verlag, Würzburg, 1961). | Zbl | MR

[21] L. Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. | Zbl | MR

[1] D. Zeilberger, Sister Celine's technique and its generalizations, J. Math. Anal. Appl. 85 (1982), 114-145. | Zbl | MR

[2] D. Zeilberger, A Holonomic systems approach to special functions identities, J. of Computational and Applied Math. 32 (1990), 321-368. | Zbl | MR

[3] D. Zeilberger, A Fast Algorithm for proving terminating hypergeometric identities, Discrete Math. 80 (1990), 207-211. | Zbl | MR

[4] D. Zeilberger. The method of creative telescoping, J. Symbolic Computation 11 (1991). 195-204. | Zbl | MR

[5] D. Zeilberger, Closed Form (pun intended !), to appear in : "Special volume in memory of Emil Grosswald", M. Knopp, ed., Contemporary Mathematics, AMS. | Zbl | MR

[6] D. Zeilberger, Three recitations on Holonomic Systems and Hypergeometric Series, Proceedings of the Séminaire Lotharingien de combinatoire 24, IRMA, Strasbourg, à paraître. | Zbl

[7] D. Zeilberger, Plain (Lagrange interpolation) proofs of Fancy (representation theory) formulas, en préparation.

[8] S. B. Ekhad, Short proofs of two hypergeometric summation formulas of Karlsson, Proc. Amer. Math. Soc. 107 (1989), 1143-1144. | Zbl | MR

[9] S. B. Ekhad, A very short proof of Dixon's theorem, J. Comb. Theo., Series A 54 (1990), 141-142. | Zbl | MR

[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. | Zbl | MR

[11] S. B. Ekhad, Short Proof Of A "Strange" Combinatorial Identity Conjectured by Gosper, Discrete Math. à paraître. | Zbl

[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. | Zbl

[13] S. B. Ekhad and S. Tre, A purely verification proof of the first Rogers-Ramanujan identity, J. Comb. The. Ser. A 54 (1990), 309-311. | Zbl | MR

[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. | Zbl | MR

[15] H. S. Wilf and D. Zeilberger, Towards computerized proofs of identities, Bulletin of the Amer. Math. Soc. 23 (1990), 77-83. | Zbl | MR

[16] H. S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc. 3 (1990), 147-158. | Zbl | MR

[17] H. S. Wilf and D. Zeilberger, A general theory of multi-variate hypergeometric identities. en préparation.

[18] H. S. Wilf, 54 computer-generated proofs of binomial coefficient identities, à paraître.

[19] G. Almkvist et D. Zeilberger, The method of differentiating under the integral sign, Journ. Symb. Computation, 10 (1990), 571- 591. | Zbl | MR

[20] G. Almkvist et D. Zeilberger, A MAPLE program that finds, and proves, recurrences and differential equations satisfied by hyperexponential definite integrals, SIGSAM Bulletin 25 (1991),...

[21] D. Zeilberger, A MAPLE program for proving hypergeometric series, SIGSAM Bulletin 25 (1991),...

[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 CCCR 5 (2) (1971), 1-16. | Zbl | MR

[2] J. Bernstein, The analytic continuation of generalized functions with respect to a parameter, Funct. Anal. and Appl. 6 (1972), 273- 285. | Zbl | MR

[3] J.-E. Björk, Rings of Differential Operators, North Holland, Amsterdam, 1979. | Zbl | MR

[4] A. Borel et al., Algebraic D-modules, Perspectives in Math. 2, Academic Press, Boston, 1987. | Zbl | MR

[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. | Zbl | MR

[6] F. Ehlers, The Weyl Algebra, Chapitre V de [4], 173-205.

[7] A. Galligo, Some algorithmic questions on ideals of differential operators, Lect. Note in Comp. Sci. 204 (1985), 413-421. | Zbl | MR

[8] R. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. USA 75 (1978), 40-42. | Zbl | MR

[9] M. Kashiwara, B-functions and holonomic systems, Invent. Math. 38 (1976), 33-53. | Zbl | MR

[10] M. Kashiwara, Vanishing cycles sheaves and holonomic systems of differential equations, Springer Lect. Notes in Math. 1016 (1983), 134-142. | Zbl | MR

[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). | Zbl | MR | Numdam

[12] R. Risch, The solution of the problem in integrating in finite terms, Bull. Amer. Math. Soc. 76 (1970), 605-608. | Zbl | MR

[13] N. Takayama, Gröbner basis and the problem of contiguous relations, Japan Journ. Appl. Math. 6 (1989), 147-160. | Zbl | MR

[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.

[15] N. Takayama, An approach to the zero recognition problem by Buchberger algorithm, Journ. Symb. Computation, | Zbl

[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. | Zbl | MR

[2] K. Aomoto, q-analogue of de Rham cohomology associated with Jackson integrals, Proc. Japan Acad. 66 (1990), 161-164. | Zbl | MR

[3] K. Aomoto, Finiteness of a cohomology associated with certain Jackson integrals, Tohoku J. Math. 43 (1991), 75-101. | Zbl | MR

[4] C. Sabbah, Systèmes holonomes d'équations aux q-différences, Pré-publication Centre de Math. Ecole Polytechnique, Palaiseau, 1991.

[1] P. Cartier et D. Foata, Problèmes combinatoires de commutation et réarrangements, Lect. Notes Math. vol. 85, Springer 1969. | Zbl | MR

[2] R. Apéry, Irrationalité de ζ(2) et ζ(3), Astérisque61 (1979), 11-13. | Zbl | Numdam

[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 : | Zbl

[4] I. G. Macdonald, Affine root systems and Dedekind $\eta$-function, Invent. Math. 15 (1972), 91-143. | Zbl

[5] I. G. Macdonald, Some conjectures for root systems, SIAM Journ. Math. Anal. 13 (1982), 988-1007. | Zbl | MR

[6] D. Zeilberger, Unified approach to MacDonald's root system conjectures, SIAM Journ. Math. Anal. 19 (1988), 987-1013. | Zbl | MR

[7] F. Garvan et G. Bonnet, MacDonald's constant term conjectures for exceptional root systems, Bull. Amer. Math. Soc. 24 (1991), 343-347. | Zbl | MR

[8] A. Helversen-Pasotto, L'identité de Barnes pour les corps finis, C.R. Acad. Sci. Paris, Série A, 286 (1978), 297-300. | Zbl | MR

[9] J. Greene et D. Stanton, A character sum evaluation and Gaussian hypergeometric series, Journ. Number Theory, 23 (1986), 136-148. | Zbl | MR