An extension of Zeilberger's fast algorithm to general holonomic functions

Discrete Mathematics

Volumn 217, Issue 1-3, 2000, Pages 115-134

  Chyzak, Frédéric ^a  

^a INRIA ROCQUENCOURT   (France)

Author keywords

Holonomic functions; Symbolic integration; Symbolic summation; Zeilberger's algorithm; finite functions

Indexed keywords

EID: 0345755649     PISSN: 0012365X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0012-365X(99)00259-9     Document Type: Article

References (35)

1. S.A. Abramov, Rational solutions of linear differential and difference equations with polynomial coefficients, USSR Comput. Math. Math. Phys. 29(11) (1989) 1611-1620, transl. from: Zh. vychisl. Mat. i mat. Fiz.
2. S.A. Abramov, Rational solutions of linear differential and difference equations with polynomial coefficients, USSR Comput. Math. Math. Phys. 29(11) (1989) 1611-1620, transl. from: Zh. vychisl. Mat. i mat. Fiz.
3. S.A. Abramov, Rational solutions of linear difference and q-difference equations with polynomial coefficients, in: A. Levelt (Ed.), Symbolic and Algebraic Computation, Proceedings of ISSAC'95, Montreal, Canada, ACM Press, New York, 1995, pp. 285-289.
4. S.A. Abramov, M. Bronstein, M. Petkovšek, On polynomial solutions of linear operator equations, in: A. Levelt (Ed.), Symbolic and Algebraic computation, Proceedings of ISSAC'95, Montreal, Canada, ACM Press, New York, 1995, pp. 290-296.
5. S.A. Abramov, E.V. Zima, A universal program to uncouple linear systems, in: Proceedings of CMCP'96 7, Proceedings of the International Conference on Computational Modeling and Computing in Physics, Dubna, Russia, September 16-21, 1996, pp. 16-26.
6. G. Almkvist, D. Zeilberger, The method of differentiating under the integral sign, J. Symbolic Comput. 10 (1990) 571-591.
7. M.A. Barkatou, An algorithm for computing a companion block diagonal form for a system of linear differential equations, Appl. Algebra Eng. Commun. Comput. 4 (1993) 185-195.
8. M.A. Barkatou, private communication, 1997.
9. I.N. Bernstein, Modules over a ring of differential operators, study of the fundamental solutions of equations with constant coefficients, Funct. Anal. Appl. 5(2) (1971) 1-16 (Russian); 89-101 (English translation).
10. I.N. Bernstein, The analytic continuation of generalized functions with respect to a parameter, Funct. Anal. Appl. 6(4) (1972) 26-40 (Russian); 273-285 (English translation).
11. J.E. Björk, Rings of Differential Operators, North-Holland, Amsterdam, 1979.
12. M. Bronstein, M. Petkovšek, An introduction to pseudo-linear algebra, Theoret. Comput. Sci. 157(1) (1996).
13. N.J. Calkin, A curious binomial identity, Discrete Math. 131(1-3) (1994) 335-337.
14. F. Chyzak, B. Salvy, Non-commutative elimination in Ore algebras proves multivariate holonomic identities, J. Symbolic Comput. 26(2) (1998) 187-227.
15. P.M. Cohn, Free Rings and Their Relations, London Mathematical Society Monographs, No. 2, Academic Press, New York, 1971.
16. S.C. Coutinho, A Primer of Algebraic D-modules, London Mathematical Society Student Texts, No. 33, Cambridge University Press, Cambridge, MA, 1995.
17. J.C. Faugère, P. Gianni, D. Lazard, T. Mora, Efficient computation of zero-dimensional Gröbner bases by change of ordering, J. Symbolic Comput. 16 (1993) 329-344.
18. R.W. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. USA 75(1) (1978) 40-42.
19. R.L. Graham, D.E. Knuth, O. Patashnik, Concrete Mathematics, A Foundation for Computer Science, Addison-Wesley, Reading, MA, 1989.
20. M. Karr, Summation in finite terms, J. ACM 28(2) (1981) 305-350.
21. M. Karr, Theory of summation in finite terms, J. Symbolic Comput. 1 (1985) 303-315.
22. T.H. Koornwinder, On Zeilberger's algorithm and its q-analogue, J. Computat. Appl. Math. 48 (1993) 91-111.
23. L. Lipshitz, D-finite power series, J. Algebra 122 (1989) 353-373.
24. P. Paule, A. Riese, A Mathematica q-analogue of Zeilberger's algorithm based on an algebraically motivated approach to q-hypergeometric telescoping, in: Special functions, q-series and related topics, Toronto, Ontario, 1995, Fields Inst. Commun. 14, Amer. Math. Soc., Providence, RI, 1997, pp. 179-210.
25. M. Petkovšek, H. Wilf, D. Zeilberger, A = B, A.K. Peters, Ltd., Wellesley, Massachussets, 1996, ISBN 1-56881-063-6.
26. A. Riese, A generalization of Gosper's algorithm to bibasic hypergeometric summation, The Electron. J. Combin. 3(R19) (1996) 16 pp.
27. C. Sabbah, Systèmes holonomes d'équations aux -différences, in: M. Kashiwara, T. Monteiro-Fernandes, P. Schapira (Eds.), D-Modules and Microlocal Geometry, Proceedings of the Conference D-Modules and Microlocal Geometry, Lisbon, 1990, Walter de Gruyter & Co., Berlin, 1993, pp. 125-147.
28. N. Takayama, An approach to the zero recognition problem by Buchberger algorithm, J. Symbolic Comput. 14 (1992) 265-282.
29. K. Wegschaider, Computer generated proofs of binomial multi-sum identities, diploma dissertation, RISC, Univ. Linz (1997). Available at the URL http://www.risc.uni-linz.ac.at/research/combinat/risc/
30. H.S. Wilf, D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc. 3 (1990) 147-158.
31. H.S. Wilf, D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and 'q') multisum/integral identities, Invent. Math. 108 (1992) 575-633.
32. H.S. Wilf, D. Zeilberger, Rational function certification of multisum/integral/'q' identities, Bull. Amer. Math. Soc. 27(1) (1992) 148-153.
33. D. Zeilberger, A fast algorithm for proving terminating hypergeometric identities, Discrete Math. 80 (1990) 207-211.
34. D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990) 321-368.
35. D. Zeilberger, The method of creative telescoping, J. Symbolic Comput. 11 (1991) 195-204.