Les cours du CIRM

[1] S. A. Abramov Applicability of Zeilberger’s algorithm to hypergeometric terms, Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation (2002), p. 1-7 (electronic) | DOI

[2] S. A. Abramov When does Zeilberger’s algorithm succeed?, Adv. in Appl. Math., Volume 30 (2003) no. 3, pp. 424-441 | DOI

[3] S. A. Abramov Rational solutions of linear differential and difference equations with polynomial coefficients, U.S.S.R. Comput. Math. and Math. Phys., Volume 29 (1991) no. 6, pp. 7-12 Translation from Zh. Vychisl. Mat. i Mat. Fiz. 29(11), 1611–1620 (1989)

[4] S. A. Abramov Rational solutions of linear difference and $q$-difference equations with polynomial coefficients, Program. Comput. Software, Volume 21 (1995) no. 6, pp. 273-278 Translation from Programmirovanie 6, 3–11 (1995)

[5] S. A. Abramov; H. Q. Le Applicability of Zeilberger’s algorithm to rational functions, Formal power series and algebraic combinatorics (Moscow, 2000), Springer, Berlin, 2000, pp. 91-102

[6] S. A. Abramov; H. Q. Le A criterion for the applicability of Zeilberger’s algorithm to rational functions, Discrete Math., Volume 259 (2002) no. 1-3, pp. 1-17 | DOI

[7] S. A. Abramov; M. Petkovšek On the structure of multivariate hypergeometric terms, Adv. in Appl. Math., Volume 29 (2002) no. 3, pp. 386-411 | DOI

[8] Moa Apagodu The sharpening of Wilf-Zeilberger theory, ProQuest LLC, Ann Arbor, MI, 2006 Thesis (Ph.D.)–Rutgers The State University of New Jersey - New Brunswick

[9] Richard Askey The world of $q$, CWI Quarterly, Volume 5 (1992) no. 4, pp. 251-269

[10] Moa Apagodu; Doron Zeilberger Multi-variable Zeilberger and Almkvist-Zeilberger algorithms and the sharpening of Wilf-Zeilberger theory, Adv. in Appl. Math., Volume 37 (2006) no. 2, pp. 139-152 | DOI

[11] Gert Almkvist; Doron Zeilberger The method of differentiating under the integral sign, J. Symbolic Comput., Volume 10 (1990) no. 6, pp. 571-591

[12] Alin Bostan; Frédéric Chyzak; Pierre Lairez 3 pages. In preparation, 2011

[13] Alin Bostan; Frédéric Chyzak; Mark van Hoeij; Lucien Pech Explicit formula for the generating series of diagonal 3D rook paths, Sém. Loth. Comb. (2011) (27 pp. Accepted)

[14] I. N. Bernšteĭn Modules over a ring of differential operators. An investigation of the fundamental solutions of equations with constant coefficients, Funkcional. Anal. i Priložen., Volume 5 (1971) no. 2, pp. 1-16

[15] I. N. Bernšteĭn Analytic continuation of generalized functions with respect to a parameter, Funkcional. Anal. i Priložen., Volume 6 (1972) no. 4, pp. 26-40

[16] Harald Böing; Wolfram Koepf Algorithms for $q$-hypergeometric summation in computer algebra, J. Symbolic Comput., Volume 28 (1999) no. 6, pp. 777-799 (Orthogonal polynomials and computer algebra) | DOI

[17] R. J. Blodgelt Problem E3376, Amer. Math. Monthly (1990)

[18] Manuel Bronstein; Marko Petkovšek Ore rings, linear operators and factorization, Programmirovanie (1994) no. 1, pp. 27-44

[19] Manuel Bronstein; Marko Petkovšek An introduction to pseudo-linear algebra, Theoret. Comput. Sci., Volume 157 (1996) no. 1, pp. 3-33 Algorithmic complexity of algebraic and geometric models (Creteil, 1994) | DOI

[20] Shaoshi Chen; Frédéric Chyzak; Ruyong Feng; Ziming Li On the Existence of Telescopers for Hyperexponential-Hypergeometric Sequences (2011) (21 pages. In preparation)

[21] Shaoshi Chen Some applications of differential-difference algebra to creative telescoping, École polytechnique, February (2011) (Ph. D. Thesis)

[22] William Y. C. Chen; Qing-Hu Hou; Yan-Ping Mu Applicability of the $q$-analogue of Zeilberger’s algorithm, J. Symbolic Comput., Volume 39 (2005) no. 2, pp. 155-170 | DOI

[23] Frédéric Chyzak An extension of Zeilberger’s fast algorithm to general holonomic functions, Discrete Math., Volume 217 (2000) no. 1-3, pp. 115-134 Formal power series and algebraic combinatorics (Vienna, 1997)

[24] Frédéric Chyzak Fonctions holonomes en calcul formel, École polytechnique (1998) http://algo.inria.fr/chyzak/Publications/these.ps (Ph. D. Thesis)

[25] Frédéric Chyzak Gröbner bases, symbolic summation and symbolic integration, Gröbner bases and applications (Linz, 1998) (London Math. Soc. Lecture Note Ser.), Volume 251, Cambridge Univ. Press, Cambridge, 1998, pp. 32-60

[26] Frédéric Chyzak; Manuel Kauers; Bruno Salvy A non-holonomic systems approach to special function identities, ISSAC 2009—Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2009, pp. 111-118

[27] F. Chyzak; A. Quadrat; D. Robertz Effective algorithms for parametrizing linear control systems over Ore algebras, Appl. Algebra Engrg. Comm. Comput., Volume 16 (2005) no. 5, pp. 319-376

[28] William Y. C. Chen; Lisa H. Sun Extended Zeilberger’s algorithm for identities on Bernoulli and Euler polynomials, J. Number Theory, Volume 129 (2009) no. 9, pp. 2111-2132 | DOI

[29] Frédéric Chyzak; Bruno Salvy Non-commutative elimination in Ore algebras proves multivariate identities, J. Symbolic Comput., Volume 26 (1998) no. 2, pp. 187-227

[30] Gustav Doetsch Integraleigenschaften der Hermiteschen Polynome, Math. Z., Volume 32 (1930) no. 1, pp. 587-599 | DOI

[31] Shalosh B. Ekhad; Doron Zeilberger A high-school algebra, “formal calculus”, proof of the Bieberbach conjecture [after L. Weinstein], Jerusalem combinatorics ’93 (Contemp. Math.), Volume 178, Amer. Math. Soc., Providence, RI, 1994, pp. 113-115

[32] Shalosh B. Ekhad; Doron Zeilberger A WZ proof of Ramanujan’s formula for $\pi$, Geometry, Analysis, and Mechanics (J. M. Rassias, ed.), World Scientific, Singapore, 1994, pp. 107-108

[33] Shalosh B. Ekhad; Doron Zeilberger The number of solutions of $X^2=0$ in triangular matrices over $GF\left(q\right)$, Electron. J. Combin., Volume 3 (1996) no. 1, Research Paper 2, 2 pp. pages http://www.combinatorics.org/Volume_3/Abstracts/v3i1r2.html

[34] Mary Celine Fasenmyer Some generalized hypergeometric polynomials, University of Michigan (1945) (Ph. D. Thesis)

[35] Mary Celine Fasenmyer A note on pure recurrence relations, Amer. Math. Monthly, Volume 56 (1949), pp. 14-17

[36] André Galligo Some algorithmic questions on ideals of differential operators, EUROCAL ’85, Vol. 2 (Linz, 1985) (Lecture Notes in Comput. Sci.), Volume 204, Springer, Berlin, 1985, pp. 413-421

[37] Ira M. Gessel Applications of the classical umbral calculus, Algebra Universalis, Volume 49 (2003) no. 4, pp. 397-434 (Dedicated to the memory of Gian-Carlo Rota) | DOI

[38] I. M. Gel’fand; M. I. Graev; V. S. Retakh General hypergeometric systems of equations and series of hypergeometric type, Russian Math. Surveys, Volume 47 (1992) no. 4, pp. 1-88 Translation from Uspekhi Matematicheskikh Nauk 47(4(286)), 3–82 | DOI

[39] Qiang-Hui Guo; Qing-Hu Hou; Lisa H. Sun Proving hypergeometric identities by numerical verifications, J. Symbolic Comput., Volume 43 (2008), pp. 895-907

[40] M. L. Glasser; E. Montaldi Some integrals involving Bessel functions, J. Math. Anal. Appl., Volume 183 (1994) no. 3, pp. 577-590 | DOI

[41] R. William Gosper Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. U.S.A., Volume 75 (1978) no. 1, pp. 40-42

[42] Joachim Hornegger Hypergeometrische Summation und polynomiale Rekursion, Universität Erlangen–Nürnberg (1992) (Diplomarbeit)

[43] Qing-Hu Hou $k$-free recurrences of double hypergeometric terms, Adv. in Appl. Math., Volume 32 (2004) no. 3, pp. 468-484 | DOI

[44] N. Jacobson Pseudo-linear transformations, Ann. of Math. (2), Volume 38 (1937) no. 2, pp. 484-507 | DOI

[45] Masaki Kashiwara On the holonomic systems of linear differential equations. II, Invent. Math., Volume 49 (1978) no. 2, pp. 121-135 | DOI

[46] Manuel Kauers Summation algorithms for Stirling number identities, J. Symbolic Comput., Volume 42 (2007) no. 10, pp. 948-970 | DOI

[47] Tom H. Koornwinder On Zeilberger’s algorithm and its $q$-analogue, Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), Volume 48 (1993) no. 1-2, pp. 91-111 | DOI

[48] Christoph Koutschan A fast approach to creative telescoping, Math. Comput. Sci., Volume 4 (2010) no. 2-3, pp. 259-266 | DOI

[49] A. Kandri-Rody; V. Weispfenning Noncommutative Gröbner bases in algebras of solvable type, J. Symbolic Comput., Volume 9 (1990) no. 1, pp. 1-26

[50] Kha Le On the $q$-analogue of Zeilberger’s algorithm for rational functions, Programmirovanie (2001) no. 1, pp. 49-58 | DOI

[51] L. Lipshitz The diagonal of a $D$-finite power series is $D$-finite, J. Algebra, Volume 113 (1988) no. 2, pp. 373-378

[52] L. Lipshitz $D$-finite power series, J. Algebra, Volume 122 (1989) no. 2, pp. 353-373

[53] John E. Majewicz WZ-style certification and Sister Celine’s technique for Abel-type sums, J. Differ. Equations Appl., Volume 2 (1996) no. 1, pp. 55-65 | DOI

[54] John E. Majewicz WZ certification of Abel-type identities and Askey’s positivity conjecture, ProQuest LLC, Ann Arbor, MI, 1997 Thesis (Ph.D.)–Temple University

[55] Mohamud Mohammed; Doron Zeilberger Sharp upper bounds for the orders of the recurrences output by the Zeilberger and $q$-Zeilberger algorithms, J. Symbolic Comput., Volume 39 (2005) no. 2, pp. 201-207 | DOI

[56] Hiromasa Nakayama; Kenta Nishiyama An algorithm of computing inhomogeneous differential equations for definite integrals, Proceedings of the Third international congress conference on Mathematical software (ICMS’10) (2010), pp. 221-232 http://dl.acm.org/citation.cfm?id=1888390.1888442

[57] Toshinori Oaku Algorithms for integrals of holonomic functions over domains defined by polynomial inequalities, http://arxiv.org/abs/1108.4853v1, 2011 (29 pages)

[58] Toshinori Oaku An algorithm of computing $b$-functions, Duke Math. J., Volume 87 (1997) no. 1, pp. 115-132 | DOI

[59] Oystein Ore Sur les fonctions hypergéométriques de plusieurs variables, Comptes Rendus des Séances de l’Académie des Sciences, Volume 189 (1929), pp. 1238-1241

[60] Oystein Ore Sur la forme des fonctions hypergéométriques de plusieurs variables, Journal de Mathématiques Pures et Appliquées, Volume 9 (1930) no. 9, pp. 311-326

[61] Oystein Ore Linear equations in non-commutative fields, Ann. of Math. (2), Volume 32 (1931) no. 3, pp. 463-477 | DOI

[62] Oystein Ore Theory of non-commutative polynomials, Ann. of Math. (2), Volume 34 (1933) no. 3, pp. 480-508 | DOI

[63] Toshinori Oaku; Nobuki Takayama Algorithms for $D$-modules—restriction, tensor product, localization, and local cohomology groups, J. Pure Appl. Algebra, Volume 156 (2001) no. 2-3, pp. 267-308 | DOI

[64] Toshinori Oaku; Nobuki Takayama Integrals of modules and their applications, Sūrikaisekikenkyūsho Kōkyūroku (1998) no. 1038, pp. 163-169 Research on the theory and applications of computer algebra (Japanese) (Kyoto, 1997)

[65] Toshinori Oaku; Nobuki Takayama; Uli Walther A localization algorithm for $D$-modules, J. Symbolic Comput., Volume 29 (2000) no. 4-5, pp. 721-728 Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998) | DOI

[66] Peter Paule Greatest factorial factorization and symbolic summation, J. Symbolic Comput., Volume 20 (1995) no. 3, pp. 235-268 | DOI

[67] Peter Paule; Axel Riese A Mathematica $q$-analogue of Zeilberger’s algorithm based on an algebraically motivated approach to $q$-hypergeometric telescoping, Special functions, $q$-series and related topics (Toronto, ON, 1995) (Fields Inst. Commun.), Volume 14, Amer. Math. Soc., Providence, RI, 1997, pp. 179-210

[68] Helmut Prodinger Descendants in heap ordered trees, or, A triumph of computer algebra, Electron. J. Combin., Volume 3 (1996) no. 1, Research Paper 29, 9 pp. pages http://www.combinatorics.org/Volume_3/Abstracts/v3i1r29.html

[69] Peter Paule; Markus Schorn A Mathematica version of Zeilberger’s algorithm for proving binomial coefficient identities, J. Symbolic Comput., Volume 20 (1995) no. 5-6, pp. 673-698 Symbolic computation in combinatorics $\Delta {}_1$ (Ithaca, NY, 1993) | DOI

[70] R. Piessens; P. Verbaeten Numerical solution of the Abel integral equation, Nordisk Tidskr. Informationsbehandling (BIT), Volume 13 (1973), pp. 451-457

[71] Earl D. Rainville Special functions, The Macmillan Co., New York, 1960

[72] Axel Riese Fine-tuning Zeilberger’s algorithm. The methods of automatic filtering and creative substituting, Symbolic computation, number theory, special functions, physics and combinatorics (Gainesville, FL, 1999) (Dev. Math.), Volume 4, Kluwer Acad. Publ., Dordrecht, 2001, pp. 243-254

[73] Axel Riese qMultiSum—a package for proving $q$-hypergeometric multiple summation identities, J. Symbolic Comput., Volume 35 (2003) no. 3, pp. 349-376 | DOI

[74] Axel Riese A generalization of Gosper’s algorithm to bibasic hypergeometric summation, Electron. J. Combin., Volume 3 (1996) no. 1, Research Paper 19, 16 pp. pages http://www.combinatorics.org/Volume_3/Abstracts/v3i1r19.html

[75] Mikio Sato Theory of prehomogeneous vector spaces (algebraic part)—the English translation of Sato’s lecture from Shintani’s note, Nagoya Math. J., Volume 120 (1990), pp. 1-34 http://projecteuclid.org/getRecord?id=euclid.nmj/1118782193 (Notes by Takuro Shintani, Translated from the Japanese by Masakazu Muro)

[76] Mutsumi Saito; Bernd Sturmfels; Nobuki Takayama Gröbner deformations of hypergeometric differential equations, Algorithms and Computation in Mathematics, 6, Springer-Verlag, Berlin, 2000

[77] Bernd Sturmfels; Nobuki Takayama Gröbner bases and hypergeometric functions, Gröbner bases and applications (Linz, 1998) (London Math. Soc. Lecture Note Ser.), Volume 251, Cambridge Univ. Press, Cambridge, 1998, pp. 246-258

[78] R. P. Stanley Differentiably finite power series, European J. Combin., Volume 1 (1980) no. 2, pp. 175-188

[79] Richard P. Stanley Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999

[80] Volker Strehl Binomial identities—combinatorial and algorithmic aspects, Discrete Math., Volume 136 (1994) no. 1-3, pp. 309-346 (Trends in discrete mathematics) | DOI

[81] James Joseph Sylvester A method of determining by mere inspection the derivatives from two equations of any degree, Philosophical Magazine, Volume XVI (1840), pp. 132-135 Available from his Collected Mathematical Pepers (http://www.archive.org/details/collectedmathema00sylv).

[82] Nobuki Takayama Gröbner basis and the problem of contiguous relations, Japan J. Appl. Math., Volume 6 (1989) no. 1, pp. 147-160 | DOI

[83] Nobuki Takayama An algorithm of constructing the integral of a module — an infinite dimensional analog of Gröbner basis, Symbolic and Algebraic Computation (1990), pp. 206-211 Proceedings of ISSAC’90 (Kyoto, Japan)

[84] Nobuki Takayama Gröbner basis, integration and transcendental functions, Symbolic and Algebraic Computation (1990), pp. 152-156 Proceedings of ISSAC’90 (Kyoto, Japan)

[85] Nobuki Takayama An approach to the zero recognition problem by Buchberger algorithm, J. Symbolic Comput., Volume 14 (1992) no. 2-3, pp. 265-282

[86] Akalu Tefera Improved algorithms and implementations in the multi-WZ theory, ProQuest LLC, Ann Arbor, MI, 2000 Thesis (Ph.D.)–Temple University

[87] Akalu Tefera MultInt, a MAPLE package for multiple integration by the WZ method, J. Symbolic Comput., Volume 34 (2002) no. 5, pp. 329-353 | DOI

[88] Harrison Tsai Weyl closure of a linear differential operator, J. Symbolic Comput., Volume 29 (2000) no. 4-5, pp. 747-775 Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998) | DOI

[89] Harrison Tsai Algorithms for associated primes, Weyl closure, and local cohomology of $D$-modules, Local cohomology and its applications (Guanajuato, 1999) (Lecture Notes in Pure and Appl. Math.), Volume 226, Dekker, New York, 2002, pp. 169-194

[90] Alfred van der Poorten A proof that Euler missed$...$Apéry’s proof of the irrationality of $\zeta \left(3\right)$, Math. Intelligencer, Volume 1 (1979) no. 4, pp. 195-203 (An informal report) | DOI

[91] Petrus Verbaeten The automatic construction of pure recurrence relations, Proceedings of Eurosam 74, Sigsam Bulletin (1974), pp. 96-98 https://lirias.kuleuven.be/handle/123456789/132545

[92] Pierre Verbaeten Rekursiebetrekkingen voor lineaire hypergeometrische funkties, Department of Computer Science, K. U. Leuven (1976) (Ph. D. Thesis)

[93] Kurt Wegschaider Computer generated proofs of binomial multi-sum identities, RISC, J. Kepler University, May (1997) (Diplomarbeit)

[94] Herbert S. Wilf; Doron Zeilberger An algorithmic proof theory for hypergeometric (ordinary and “$q$”) multisum/integral identities, Invent. Math., Volume 108 (1992) no. 3, pp. 575-633

[95] Herbert S. Wilf; Doron Zeilberger Rational function certification of multisum/integral/“$q$” identities, Bull. Amer. Math. Soc. (N.S.), Volume 27 (1992) no. 1, pp. 148-153

[96] Lily Yen Contributions to the proof theory of hypergeometric identities, University of Pennsylvania (1993) (Ph. D. Thesis)

[97] Lily Yen A two-line algorithm for proving terminating hypergeometric identities, J. Math. Anal. Appl., Volume 198 (1996) no. 3, pp. 856-878 | DOI

[98] Lily Yen A two-line algorithm for proving $q$-hypergeometric identities, J. Math. Anal. Appl., Volume 213 (1997) no. 1, pp. 1-14 | DOI

[99] Doron Zeilberger The algebra of linear partial difference operators and its applications, SIAM J. Math. Anal., Volume 11 (1980) no. 6, pp. 919-932

[100] Doron Zeilberger Sister Celine’s technique and its generalizations, J. Math. Anal. Appl., Volume 85 (1982) no. 1, pp. 114-145

[101] Doron Zeilberger A fast algorithm for proving terminating hypergeometric identities, Discrete Math., Volume 80 (1990) no. 2, pp. 207-211

[102] Doron Zeilberger A holonomic systems approach to special functions identities, J. Comput. Appl. Math., Volume 32 (1990) no. 3, pp. 321-368

[103] Doron Zeilberger The method of creative telescoping, J. Symbolic Comput., Volume 11 (1991) no. 3, pp. 195-204

[104] Doron Zeilberger Towards a WZ evaluation of the Mehta integral, SIAM J. Math. Anal., Volume 25 (1994) no. 2, pp. 812-814 | DOI

[105] Bao-Yin Zhang A new elementary algorithm for proving $q$-hypergeometric identities, J. Symbolic Comput., Volume 35 (2003) no. 3, pp. 293-303 | DOI