G. Almkvist and D. Zeilberger, The method of differentiating under the integral sign, Journal of Symbolic Computation, vol.10, issue.6, pp.571-591, 1990.
DOI : 10.1016/S0747-7171(08)80159-9

R. Askey, The world of q, CWI Quarterly, vol.5, issue.4, pp.251-270, 1992.

T. Becker and V. Weispfenning, Gröbner bases, volume 141 of Graduate Texts in Mathematics, A computational approach to commutative algebra, 1993.

I. N. Bernstein, Structure of representations generated by vectors of highest weight, Functional Analysis and Its Applications, vol.77, issue.3, pp.1-16, 1971.
DOI : 10.1007/BF01075841

I. N. Bernstein, The analytic continuation of generalized functions with respect to a parameter, Functional Analysis and ApplicationsRussian, vol.6, issue.4, pp.26-40, 1972.

B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, 1965.

B. Buchberger, Ein algorithmisches Kriterium f??r die L??sbarkeit eines algebraischen Gleichungssystems, Aequationes Mathematicae, vol.95, issue.3, pp.374-383, 1970.
DOI : 10.1007/BF01844169

B. Buchberger, Gröbner bases: an algorithmic method in polynomial ideal theory, Recent trends in multidimensional system theory, pp.184-232, 1985.

P. Cartier, Démonstration " automatique " d'identités et fonctions hypergéométriques, Astérisque, vol.206, pp.41-91, 1992.

F. Chyzak, Holonomic systems and automatic proofs of identities, Research Report, vol.2371, 1994.
URL : https://hal.archives-ouvertes.fr/inria-00074305

F. Chyzak, An extension of Zeilberger's fast algorithm to general holonomic functions, Formal Power Series and Algebraic Combinatorics, 9th Conference, pp.172-183, 1997.
DOI : 10.1016/S0012-365X(99)00259-9

P. M. Cohn, Free Rings and Their Relations, 1971.

L. Comtet, Calcul pratique des coefficients de Taylor d'une fonction algébrique, Enseignement Mathématique, vol.10, pp.267-270, 1964.

D. Cox, J. Little, and D. O-'shea, Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, 1992.

S. B. Ekhad and J. E. Majewicz, A short WZ-style proof of Abel's identity The Foata Festschrift, The Electronic Journal of Combinatorics, vol.3, issue.2, 1996.

A. Erdélyi, Higher Transcendental Functions, pp.1-2, 1981.

M. C. Fasenmyer, Some generalized hypergeometric polynomials, Bulletin of the American Mathematical Society, vol.53, issue.8, 1945.
DOI : 10.1090/S0002-9904-1947-08893-5

M. C. Fasenmyer, Some generalized hypergeometric polynomials, Bulletin of the American Mathematical Society, vol.53, issue.8, pp.806-812, 1947.
DOI : 10.1090/S0002-9904-1947-08893-5

M. C. Fasenmyer, A Note on Pure Recurrence Relations, The American Mathematical Monthly, vol.56, issue.1, pp.14-17, 1949.
DOI : 10.2307/2305810

J. Faugère, P. Gianni, D. Lazard, and T. Mora, Efficient Computation of Zero-dimensional Gr??bner Bases by Change of Ordering, Journal of Symbolic Computation, vol.16, issue.4, pp.329-344, 1993.
DOI : 10.1006/jsco.1993.1051

A. Galligo, Some algorithmic questions on ideals of differential operators, Proceedings EUROCAL'85, pp.413-421, 1985.
DOI : 10.1007/3-540-15984-3_301

G. Gasper and M. Rahman, Basic Hypergeometric Series, of Encyclopedia of Mathematics and its Applications, 1990.
DOI : 10.1017/cbo9780511526251

A. Giovini, T. Mora, G. Niesi, L. Robbiano, and C. Traverso, ???One sugar cube, please??? or selection strategies in the Buchberger algorithm, Proceedings of the 1991 international symposium on Symbolic and algebraic computation , ISSAC '91, pp.49-54, 1991.
DOI : 10.1145/120694.120701

A. Kandri-rody and V. Weispfenning, Non-commutative Gr??bner bases in algebras of solvable type, Journal of Symbolic Computation, vol.9, issue.1, pp.1-26, 1990.
DOI : 10.1016/S0747-7171(08)80003-X

M. Kashiwara, On the holonomic systems of linear differential equations, II, Inventiones Mathematicae, vol.38, issue.2, pp.121-135, 1978.
DOI : 10.1007/BF01403082

T. H. Koornwinder, On Zeilberger's algorithm and its q-analogue, Journal of Computational and Applied Mathematics, vol.48, issue.1-2, pp.91-111, 1993.
DOI : 10.1016/0377-0427(93)90317-5

H. Kredel, Solvable Polynomial Rings, Reihe Mathematik. Verlag Shaker, 1993.

L. Lipshitz, The diagonal of a D-finite power series is D-finite, Journal of Algebra, vol.113, issue.2, pp.373-378, 1988.
DOI : 10.1016/0021-8693(88)90166-4

L. Lipshitz, D-finite power series, Journal of Algebra, vol.122, issue.2, pp.353-373, 1989.
DOI : 10.1016/0021-8693(89)90222-6

URL : http://doi.org/10.1016/0021-8693(89)90222-6

T. Mora, An introduction to commutative and noncommutative Gr??bner bases, Theoretical Computer Science, vol.134, issue.1, pp.131-173, 1994.
DOI : 10.1016/0304-3975(94)90283-6

O. Ore, Theory of Non-Commutative Polynomials, The Annals of Mathematics, vol.34, issue.3, pp.480-508, 1933.
DOI : 10.2307/1968173

S. Parnes and S. B. Ekhad, A WZ-style proof of Jacobi polynomials' generating function, Discrete Mathematics, vol.110, issue.1-3, pp.263-264, 1992.
DOI : 10.1016/0012-365X(92)90715-R

P. Paule, Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type, The Electronic Journal of Combinatorics, vol.1, issue.R10, pp.1-9, 1994.

P. Paule and A. 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, Fields Proceedings of the Workshop, pp.12-23, 1995.

P. Paule and M. Schorn, A Mathematica Version of Zeilberger's Algorithm for Proving Binomial Coefficient Identities, Journal of Symbolic Computation, vol.20, issue.5-6, pp.673-698, 1995.
DOI : 10.1006/jsco.1995.1071

A. Riese, A generalization of Gosper's algorithm to bibasic hypergeometric summation, The Electronic Journal of Combinatorics, vol.3, issue.R19, pp.1-16, 1996.

C. Sabbah, M. Kashiwara, T. Monteiro-fernandes, and P. Schapira, Systèmes holonomes d'´ equations aux q-différences, D-Modules and Microlocal Geometry Proceedings of the Conference D-Modules and Microlocal Geometry, pp.125-147, 1990.

B. Salvy and P. Zimmermann, GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Transactions on Mathematical Software, vol.20, issue.2, pp.163-177, 1994.
DOI : 10.1145/178365.178368

URL : https://hal.archives-ouvertes.fr/hal-00917741

R. P. Stanley, Differentiably Finite Power Series, European Journal of Combinatorics, vol.1, issue.2, pp.175-188, 1980.
DOI : 10.1016/S0195-6698(80)80051-5

V. Strehl, Binomial identities ??? combinatorial and algorithmic aspects, Discrete Mathematics, vol.136, issue.1-3, pp.309-346, 1994.
DOI : 10.1016/0012-365X(94)00118-3

URL : http://doi.org/10.1016/0012-365x(94)00118-3

N. Takayama, Gr??bner basis and the problem of contiguous relations, Japan Journal of Applied Mathematics, vol.34, issue.9, pp.147-160, 1989.
DOI : 10.1007/BF03167920

N. Takayama, An algorithm of constructing the integral of a module ? an infinite dimensional analog of Gröbner basis, Symbolic and Algebraic Computation Proceedings of ISSAC'90, pp.206-211, 1990.

N. Takayama, Gröbner basis, integration and transcendental functions, Symbolic and Algebraic Computation Proceedings ISSAC'90, pp.152-156, 1990.

N. Takayama, An approach to the zero recognition problem by Buchberger algorithm, Journal of Symbolic Computation, vol.14, issue.2-3, pp.265-282, 1992.
DOI : 10.1016/0747-7171(92)90039-7

N. Takayama, An Algorithm for Finding Recurrence Relations of Binomial Sums and its Complexity, Journal of Symbolic Computation, vol.20, issue.5-6, pp.637-651, 1996.
DOI : 10.1006/jsco.1995.1069

V. Weispfenning, Finite Gröbner bases in non-Noetherian skew polynomial rings, Symbolic and Algebraic Computation Proceedings ISSAC'92, pp.329-334, 1992.

H. S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and ???q???) multisum/integral identities, Inventiones Mathematicae, vol.54, issue.1, pp.575-633, 1992.
DOI : 10.1007/BF02100618

H. S. Wilf and D. Zeilberger, Rational Function Certification\\ Of Multisum/integral/``$q$'' Identities, Bulletin of the American Mathematical Society, vol.27, issue.1, pp.148-153, 1992.
DOI : 10.1090/S0273-0979-1992-00297-5

D. Zeilberger, A fast algorithm for proving terminating hypergeometric identities, Discrete Mathematics, vol.80, issue.2, pp.207-211, 1990.
DOI : 10.1016/0012-365X(90)90120-7

D. Zeilberger, A holonomic systems approach to special functions identities, Journal of Computational and Applied Mathematics, vol.32, issue.3, pp.321-368, 1990.
DOI : 10.1016/0377-0427(90)90042-X

D. Zeilberger, A maple program for proving hypergeometric identities, ACM SIGSAM Bulletin, vol.25, issue.3, pp.4-13, 1991.
DOI : 10.1145/122514.122515

D. Zeilberger, The method of creative telescoping, Journal of Symbolic Computation, vol.11, issue.3, pp.195-204, 1991.
DOI : 10.1016/S0747-7171(08)80044-2