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Les cours du CIRM

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Frédéric Chyzak
Creative Telescoping for Parametrised Integration and Summation
Les cours du CIRM, 2 no. 1: Journées Nationales de Calcul Formel (2011), Exp. No. 2, 37 p., doi: 10.5802/ccirm.14
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Bibliographie

[1] S. A. Abramov, Applicability of Zeilberger’s algorithm to hypergeometric terms, in Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, ACM, 2002, p. 1-7 (electronic) Article |  MR 2035226 |  Zbl 1072.68642
[2] S. A. Abramov, “When does Zeilberger’s algorithm succeed?”, Adv. in Appl. Math. 30 (2003) no. 3, p. 424-441 Article |  MR 1973952 |  Zbl 1030.33011
[3] S. A. Abramov, “Rational solutions of linear differential and difference equations with polynomial coefficients”, U.S.S.R. Comput. Math. and Math. Phys. 29 (1991) no. 6, p. 7-12, Translation from Zh. Vychisl. Mat. i Mat. Fiz. 29(11), 1611–1620 (1989)  MR 1025995 |  Zbl 0719.65063
[4] S. A. Abramov, “Rational solutions of linear difference and $q$-difference equations with polynomial coefficients”, Program. Comput. Software 21 (1995) no. 6, p. 273-278, Translation from Programmirovanie 6, 3–11 (1995)  MR 1615571 |  Zbl 0910.65107
[5] S. A. Abramov & H. Q. Le, Applicability of Zeilberger’s algorithm to rational functions, Formal power series and algebraic combinatorics (Moscow, 2000), Springer, 2000, p. 91–102  MR 1798204 |  Zbl 0963.65026
[6] S. A. Abramov & H. Q. Le, “A criterion for the applicability of Zeilberger’s algorithm to rational functions”, Discrete Math. 259 (2002) no. 1-3, p. 1-17 Article |  MR 1948770 |  Zbl 1023.33017
[7] S. A. Abramov & M. Petkovšek, “On the structure of multivariate hypergeometric terms”, Adv. in Appl. Math. 29 (2002) no. 3, p. 386-411 Article |  MR 1942630 |  Zbl 1057.33017
[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  MR 2709554
[9] Richard Askey, “The world of $q$”, CWI Quarterly 5 (1992) no. 4, p. 251-269  MR 1213742 |  Zbl 0765.33015
[10] Moa Apagodu & Doron Zeilberger, “Multi-variable Zeilberger and Almkvist-Zeilberger algorithms and the sharpening of Wilf-Zeilberger theory”, Adv. in Appl. Math. 37 (2006) no. 2, p. 139-152 Article |  MR 2251432 |  Zbl 1108.05010
[11] Gert Almkvist & Doron Zeilberger, “The method of differentiating under the integral sign”, J. Symbolic Comput. 10 (1990) no. 6, p. 571-591  MR 1087980 |  Zbl 0717.33004
[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. 5 (1971) no. 2, p. 1-16  MR 290097 |  Zbl 0233.47031
[15] I. N. Bernšteĭn, “Analytic continuation of generalized functions with respect to a parameter”, Funkcional. Anal. i Priložen. 6 (1972) no. 4, p. 26-40  MR 320735 |  Zbl 0282.46038
[16] Harald Böing & Wolfram Koepf, “Algorithms for $q$-hypergeometric summation in computer algebra”, J. Symbolic Comput. 28 (1999) no. 6, p. 777-799, Orthogonal polynomials and computer algebra Article |  MR 1750546 |  Zbl 0946.65008
[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, p. 27-44  MR 1291354 |  Zbl 0828.16035
[19] Manuel Bronstein & Marko Petkovšek, “An introduction to pseudo-linear algebra”, Theoret. Comput. Sci. 157 (1996) no. 1, p. 3-33, Algorithmic complexity of algebraic and geometric models (Creteil, 1994) Article |  MR 1383396 |  Zbl 0868.34004
[20] Shaoshi Chen, Frédéric Chyzak, Ruyong Feng & Ziming Li, “On the Existence of Telescopers for Hyperexponential-Hypergeometric Sequences”, 21 pages. In preparation, 2011
[21] Shaoshi Chen, Some applications of differential-difference algebra to creative telescoping, Ph. D. Thesis, École polytechnique, 2011
[22] William Y. C. Chen, Qing-Hu Hou & Yan-Ping Mu, “Applicability of the $q$-analogue of Zeilberger’s algorithm”, J. Symbolic Comput. 39 (2005) no. 2, p. 155-170 Article |  MR 2169798 |  Zbl 1126.33008
[23] Frédéric Chyzak, “An extension of Zeilberger’s fast algorithm to general holonomic functions”, Discrete Math. 217 (2000) no. 1-3, p. 115-134, Formal power series and algebraic combinatorics (Vienna, 1997)  MR 1766263 |  Zbl 0968.33011
[24] Frédéric Chyzak, Fonctions holonomes en calcul formel, Ph. D. Thesis, École polytechnique, INRIA, TU 0531. 227 pages., 1998
[25] Frédéric Chyzak, Gröbner bases, symbolic summation and symbolic integration, Gröbner bases and applications (Linz, 1998), London Math. Soc. Lecture Note Ser. 251, Cambridge Univ. Press, 1998, p. 32–60  MR 1699813 |  Zbl 0898.68040
[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, 2009, p. 111–118  MR 2742698
[27] F. Chyzak, A. Quadrat & D. Robertz, “Effective algorithms for parametrizing linear control systems over Ore algebras”, Appl. Algebra Engrg. Comm. Comput. 16 (2005) no. 5, p. 319-376  MR 2233761 |  Zbl 1109.93018
[28] William Y. C. Chen & Lisa H. Sun, “Extended Zeilberger’s algorithm for identities on Bernoulli and Euler polynomials”, J. Number Theory 129 (2009) no. 9, p. 2111-2132 Article |  MR 2528056 |  Zbl 1183.11011
[29] Frédéric Chyzak & Bruno Salvy, “Non-commutative elimination in Ore algebras proves multivariate identities”, J. Symbolic Comput. 26 (1998) no. 2, p. 187-227  MR 1635242 |  Zbl 0944.05006
[30] Gustav Doetsch, “Integraleigenschaften der Hermiteschen Polynome”, Math. Z. 32 (1930) no. 1, p. 587-599 Article |  MR 1545186 |  JFM 56.0989.03
[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. 178, Amer. Math. Soc., 1994, p. 113–115  MR 1310578 |  Zbl 0894.30013
[32] Shalosh B. Ekhad & Doron Zeilberger, A WZ proof of Ramanujan’s formula for $\pi $, in J. M. Rassias, éd., Geometry, Analysis, and Mechanics, World Scientific, 1994, p. 107–108  MR 1323194 |  Zbl 0849.33003
[33] Shalosh B. Ekhad & Doron Zeilberger, “The number of solutions of ${X}^2 = 0$ in triangular matrices over ${GF}(q)$”, Electron. J. Combin. 3 (1996) no. 1  MR 1364064 |  Zbl 0851.15010
[34] Mary Celine Fasenmyer, Some generalized hypergeometric polynomials, Ph. D. Thesis, University of Michigan, 1945
[35] Mary Celine Fasenmyer, “A note on pure recurrence relations”, Amer. Math. Monthly 56 (1949), p. 14-17  MR 30044 |  Zbl 0032.41002
[36] André Galligo, Some algorithmic questions on ideals of differential operators, EUROCAL ’85, Vol. 2 (Linz, 1985), Lecture Notes in Comput. Sci. 204, Springer, 1985, p. 413–421  MR 826576 |  Zbl 0634.16001
[37] Ira M. Gessel, “Applications of the classical umbral calculus”, Algebra Universalis 49 (2003) no. 4, p. 397-434, Dedicated to the memory of Gian-Carlo Rota Article |  MR 2022347 |  Zbl 1092.05005
[38] I. M. Gel’fand, M. I. Graev & V. S. Retakh, “General hypergeometric systems of equations and series of hypergeometric type”, Russian Math. Surveys 47 (1992) no. 4, p. 1-88, Translation from Uspekhi Matematicheskikh Nauk 47(4(286)), 3–82 Article |  MR 1208882 |  Zbl 0798.33010
[39] Qiang-Hui Guo, Qing-Hu Hou & Lisa H. Sun, “Proving hypergeometric identities by numerical verifications”, J. Symbolic Comput. 43 (2008), p. 895-907  MR 2472539 |  Zbl 1173.33304
[40] M. L. Glasser & E. Montaldi, “Some integrals involving Bessel functions”, J. Math. Anal. Appl. 183 (1994) no. 3, p. 577-590 Article |  MR 1274858 |  Zbl 0809.33001
[41] R. William Gosper, “Decision procedure for indefinite hypergeometric summation”, Proc. Nat. Acad. Sci. U.S.A. 75 (1978) no. 1, p. 40-42  MR 485674 |  Zbl 0384.40001
[42] Joachim Hornegger, Hypergeometrische Summation und polynomiale Rekursion, Diplomarbeit, Universität Erlangen–Nürnberg, 1992
[43] Qing-Hu Hou, “$k$-free recurrences of double hypergeometric terms”, Adv. in Appl. Math. 32 (2004) no. 3, p. 468-484 Article |  MR 2041960 |  Zbl 1057.33018
[44] N. Jacobson, “Pseudo-linear transformations”, Ann. of Math. (2) 38 (1937) no. 2, p. 484-507 Article |  MR 1503347 |  Zbl 0017.15001
[45] Masaki Kashiwara, “On the holonomic systems of linear differential equations. II”, Invent. Math. 49 (1978) no. 2, p. 121-135 Article |  MR 511186 |  Zbl 0401.32005
[46] Manuel Kauers, “Summation algorithms for Stirling number identities”, J. Symbolic Comput. 42 (2007) no. 10, p. 948-970 Article |  MR 2361673 |  Zbl 1142.11008
[47] Tom H. Koornwinder, On Zeilberger’s algorithm and its $q$-analogue, in Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), 1993, p. 91-111 Article |  MR 1246853 |  Zbl 0797.65011
[48] Christoph Koutschan, “A fast approach to creative telescoping”, Math. Comput. Sci. 4 (2010) no. 2-3, p. 259-266 Article |  MR 2775992 |  Zbl 1218.68205
[49] A. Kandri-Rody & V. Weispfenning, “Noncommutative Gröbner bases in algebras of solvable type”, J. Symbolic Comput. 9 (1990) no. 1, p. 1-26  MR 1044911 |  Zbl 0715.16010
[50] Kha Le, “On the $q$-analogue of Zeilberger’s algorithm for rational functions”, Programmirovanie (2001) no. 1, p. 49-58 Article |  MR 1867720 |  Zbl 0985.33014
[51] L. Lipshitz, “The diagonal of a $D$-finite power series is $D$-finite”, J. Algebra 113 (1988) no. 2, p. 373-378  MR 929767 |  Zbl 0657.13024
[52] L. Lipshitz, “$D$-finite power series”, J. Algebra 122 (1989) no. 2, p. 353-373  MR 999079 |  Zbl 0695.12018
[53] John E. Majewicz, “WZ-style certification and Sister Celine’s technique for Abel-type sums”, J. Differ. Equations Appl. 2 (1996) no. 1, p. 55-65 Article |  MR 1375596 |  Zbl 0863.05009
[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  MR 2696103
[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. 39 (2005) no. 2, p. 201-207 Article |  MR 2169800 |  Zbl 1121.33023
[56] Hiromasa Nakayama & Kenta Nishiyama, An algorithm of computing inhomogeneous differential equations for definite integrals, in Proceedings of the Third international congress conference on Mathematical software, ICMS’10, Springer-Verlag, 2010, p. 221-232  Zbl pre05785573
[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. 87 (1997) no. 1, p. 115-132 Article |  MR 1440065 |  Zbl 0893.32009
[59] Oystein Ore, “Sur les fonctions hypergéométriques de plusieurs variables”, Comptes Rendus des Séances de l’Académie des Sciences 189 (1929), p. 1238-1241  JFM 55.0220.03
[60] Oystein Ore, “Sur la forme des fonctions hypergéométriques de plusieurs variables”, Journal de Mathématiques Pures et Appliquées 9 (1930) no. 9, p. 311-326 Article |  JFM 56.0313.13
[61] Oystein Ore, “Linear equations in non-commutative fields”, Ann. of Math. (2) 32 (1931) no. 3, p. 463-477 Article |  MR 1503010 |  Zbl 0001.26601
[62] Oystein Ore, “Theory of non-commutative polynomials”, Ann. of Math. (2) 34 (1933) no. 3, p. 480-508 Article |  MR 1503119 |  Zbl 0007.15101
[63] Toshinori Oaku & Nobuki Takayama, “Algorithms for $D$-modules—restriction, tensor product, localization, and local cohomology groups”, J. Pure Appl. Algebra 156 (2001) no. 2-3, p. 267-308 Article |  MR 1808827 |  Zbl 0983.13008
[64] Toshinori Oaku & Nobuki Takayama, “Integrals of modules and their applications”, Sūrikaisekikenkyūsho Kōkyūroku (1998) no. 1038, p. 163-169, Research on the theory and applications of computer algebra (Japanese) (Kyoto, 1997)  MR 1672325 |  Zbl 0944.14501
[65] Toshinori Oaku, Nobuki Takayama & Uli Walther, “A localization algorithm for $D$-modules”, J. Symbolic Comput. 29 (2000) no. 4-5, p. 721-728, Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998) Article |  MR 1769663 |  Zbl 1012.13010
[66] Peter Paule, “Greatest factorial factorization and symbolic summation”, J. Symbolic Comput. 20 (1995) no. 3, p. 235-268 Article |  MR 1378099 |  Zbl 0854.68047
[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. 14, Amer. Math. Soc., 1997, p. 179–210  Zbl 0869.33010
[68] Helmut Prodinger, “Descendants in heap ordered trees, or, A triumph of computer algebra”, Electron. J. Combin. 3 (1996) no. 1  MR 1410884 |  Zbl 0885.05004
[69] Peter Paule & Markus Schorn, “A Mathematica version of Zeilberger’s algorithm for proving binomial coefficient identities”, J. Symbolic Comput. 20 (1995) no. 5-6, p. 673-698, Symbolic computation in combinatorics Δ 1 (Ithaca, NY, 1993) Article |  MR 1395420 |  Zbl 0851.68052
[70] R. Piessens & P. Verbaeten, “Numerical solution of the Abel integral equation”, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), p. 451-457  MR 329293 |  Zbl 0266.65081
[71] Earl D. Rainville, Special functions, The Macmillan Co., New York, 1960  MR 107725 |  Zbl 0092.06503
[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. 4, Kluwer Acad. Publ., 2001, p. 243–254  MR 1880090 |  Zbl 1037.33018
[73] Axel Riese, “qMultiSum—a package for proving $q$-hypergeometric multiple summation identities”, J. Symbolic Comput. 35 (2003) no. 3, p. 349-376 Article |  MR 1962799 |  Zbl 1020.33007
[74] Axel Riese, “A generalization of Gosper’s algorithm to bibasic hypergeometric summation”, Electron. J. Combin. 3 (1996) no. 1  MR 1394550 |  Zbl 0885.33012
[75] Mikio Sato, “Theory of prehomogeneous vector spaces (algebraic part)—the English translation of Sato’s lecture from Shintani’s note”, Nagoya Math. J. 120 (1990), p. 1-34, Notes by Takuro Shintani, Translated from the Japanese by Masakazu Muro Article |  MR 1086566 |  Zbl 0715.22014
[76] Mutsumi Saito, Bernd Sturmfels & Nobuki Takayama, Gröbner deformations of hypergeometric differential equations, Algorithms and Computation in Mathematics 6, Springer-Verlag, Berlin, 2000  MR 1734566 |  Zbl 0946.13021
[77] Bernd Sturmfels & Nobuki Takayama, Gröbner bases and hypergeometric functions, Gröbner bases and applications (Linz, 1998), London Math. Soc. Lecture Note Ser. 251, Cambridge Univ. Press, 1998, p. 246–258  MR 1708882 |  Zbl 0918.33004
[78] R. P. Stanley, “Differentiably finite power series”, European J. Combin. 1 (1980) no. 2, p. 175-188  MR 587530 |  Zbl 0445.05012
[79] Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, Cambridge, 1999  MR 1676282 |  Zbl 0928.05001
[80] Volker Strehl, “Binomial identities—combinatorial and algorithmic aspects”, Discrete Math. 136 (1994) no. 1-3, p. 309-346, Trends in discrete mathematics Article |  MR 1313292 |  Zbl 0823.33003
[81] James Joseph Sylvester, “A method of determining by mere inspection the derivatives from two equations of any degree”, Philosophical Magazine XVI (1840), p. 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. 6 (1989) no. 1, p. 147-160 Article |  MR 981518 |  Zbl 0691.68032
[83] Nobuki Takayama, An algorithm of constructing the integral of a module — an infinite dimensional analog of Gröbner basis, in Symbolic and Algebraic Computation, ACM and Addison-Wesley, Proceedings of ISSAC’90 (Kyoto, Japan), 1990, p. 206-211
[84] Nobuki Takayama, Gröbner basis, integration and transcendental functions, in Symbolic and Algebraic Computation, ACM and Addison-Wesley, Proceedings of ISSAC’90 (Kyoto, Japan), 1990, p. 152-156
[85] Nobuki Takayama, “An approach to the zero recognition problem by Buchberger algorithm”, J. Symbolic Comput. 14 (1992) no. 2-3, p. 265-282  MR 1187235 |  Zbl 0763.65007
[86] Akalu Tefera, Improved algorithms and implementations in the multi-WZ theory, ProQuest LLC, Ann Arbor, MI, 2000, Thesis (Ph.D.)–Temple University  MR 2701445
[87] Akalu Tefera, “MultInt, a MAPLE package for multiple integration by the WZ method”, J. Symbolic Comput. 34 (2002) no. 5, p. 329-353 Article |  MR 1937465 |  Zbl 1015.33013
[88] Harrison Tsai, “Weyl closure of a linear differential operator”, J. Symbolic Comput. 29 (2000) no. 4-5, p. 747-775, Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998) Article |  MR 1769665 |  Zbl 1008.16026
[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. 226, Dekker, 2002, p. 169–194  MR 1888199 |  Zbl 0992.68245
[90] Alfred van der Poorten, “A proof that Euler missed$\ldots $Apéry’s proof of the irrationality of $\zeta (3)$”, Math. Intelligencer 1 (1979) no. 4, p. 195-203, An informal report Article |  MR 547748 |  Zbl 0409.10028
[91] Petrus Verbaeten, The automatic construction of pure recurrence relations, in Proceedings of Eurosam 74, Sigsam Bulletin, 1974, p. 96-98
[92] Pierre Verbaeten, Rekursiebetrekkingen voor lineaire hypergeometrische funkties, Ph. D. Thesis, Department of Computer Science, K. U. Leuven, 239 pp., 1976
[93] Kurt Wegschaider, Computer generated proofs of binomial multi-sum identities, Diplomarbeit, RISC, J. Kepler University, 1997
[94] Herbert S. Wilf & Doron Zeilberger, “An algorithmic proof theory for hypergeometric (ordinary and “$q$”) multisum/integral identities”, Invent. Math. 108 (1992) no. 3, p. 575-633  MR 1163239 |  Zbl 0739.05007
[95] Herbert S. Wilf & Doron Zeilberger, “Rational function certification of multisum/integral/“$q$” identities”, Bull. Amer. Math. Soc. (N.S.) 27 (1992) no. 1, p. 148-153  MR 1145580 |  Zbl 0759.05007
[96] Lily Yen, Contributions to the proof theory of hypergeometric identities, Ph. D. Thesis, University of Pennsylvania, 1993  MR 2689630
[97] Lily Yen, “A two-line algorithm for proving terminating hypergeometric identities”, J. Math. Anal. Appl. 198 (1996) no. 3, p. 856-878 Article |  MR 1377830 |  Zbl 0857.33002
[98] Lily Yen, “A two-line algorithm for proving $q$-hypergeometric identities”, J. Math. Anal. Appl. 213 (1997) no. 1, p. 1-14 Article |  MR 1469359 |  Zbl 0903.33008
[99] Doron Zeilberger, “The algebra of linear partial difference operators and its applications”, SIAM J. Math. Anal. 11 (1980) no. 6, p. 919-932  MR 595820 |  Zbl 0458.39002
[100] Doron Zeilberger, “Sister Celine’s technique and its generalizations”, J. Math. Anal. Appl. 85 (1982) no. 1, p. 114-145  MR 647562 |  Zbl 0485.05003
[101] Doron Zeilberger, “A fast algorithm for proving terminating hypergeometric identities”, Discrete Math. 80 (1990) no. 2, p. 207-211  MR 1048463 |  Zbl 0701.05001
[102] Doron Zeilberger, “A holonomic systems approach to special functions identities”, J. Comput. Appl. Math. 32 (1990) no. 3, p. 321-368  MR 1090884 |  Zbl 0738.33001
[103] Doron Zeilberger, “The method of creative telescoping”, J. Symbolic Comput. 11 (1991) no. 3, p. 195-204  MR 1103727 |  Zbl 0738.33002
[104] Doron Zeilberger, “Towards a WZ evaluation of the Mehta integral”, SIAM J. Math. Anal. 25 (1994) no. 2, p. 812-814 Article |  MR 1266590 |  Zbl 0934.33002
[105] Bao-Yin Zhang, “A new elementary algorithm for proving $q$-hypergeometric identities”, J. Symbolic Comput. 35 (2003) no. 3, p. 293-303 Article |  MR 1962797 |  Zbl 1020.33006
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