The summation package Sigma: Underlying principles and a rhombus tiling application

Discrete Mathematics and Theoretical Computer Science

Volumn 6, Issue 2, 2004, Pages 365-386

  Schneider, Carsten ^a  

^a JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

Author keywords

Rhombus tilings; Symbolic summation

Indexed keywords

EID: 15844368051     PISSN: 13658050     EISSN: 13658050     Source Type: Journal    
DOI: None     Document Type: Article

References (36)

1. S. A. Abramov and M. Petkovšek. D'Alembertian solutions of linear differential and difference equations. In J. von zur Gathen, editor, Proc. ISSAC'94, pages 169-174. ACM Press, Baltimore, 1994.
2. S. A. Abramov, P. Paule, and M. Petkovšek. q-Hypergeometric solutions of q-difference equations. Discrete Math., 180(1-3):3-22, 1998.
3. M. Bronstein. On solutions of linear ordinary difference equations in their coefficient field. J. Symbolic Comput., 29(6):841-877, June 2000.
4. W. Chu and L. De Donno. Hypergeometric series and harmonic number identities. Preprint, 2004.
5. F. Chyzak. An extension of Zeilberger's fast algorithm to general holonomic functions. Discrete Math., 217:115-134, 2000.
6. R. M. Cohn. Difference Algebra. Interscience Publishers, John Wiley & Sons, 1965.
7. F. Chyzak and B. Salvy. Non-commutative elimination in ore algebras proves multivariate identities. J. Symbolic Comput., 26(2):187-227, 1998.
8. K. Driver, H. Prodinger, C. Schneider, and A. Weideman. Padé approximations to the logarithm II: Identities, recurrences, and symbolic computation. To appear in Ramanujan Journal, 2004.
9. K. Driver, H. Prodinger, C. Schneider, and A. Weideman. Padé approximations to the logarithm III: Alternative methods and additional results. To Appear in Ramanujan Journal, 2004.
10. M. Fulmek and C. Krattenthaler. The number of rhombus tilings of a symmetric hexagon which contains a fixed rhombus on the symmetric axis, II. European J. Combin., 21(5):601-640, 2000.
11. R. W. Gosper. Decision procedures for indefinite hypergeometric summation. Proc. Nat. Acad. Sci. U.S.A., 75:40-42, 1978.
12. P. A. Hendriks and M. F. Singer. Solving difference equations in finite terms. J. Symbolic Comput., 27(3):239-259, 1999.
13. M. Karr. Summation in finite terms. J. ACM, 28:305-350, 1981.
14. M. Karr. Theory of summation in finite terms. J. Symbolic Comput., 1:303-315, 1985.
15. C. Krattenthaler and T. Rivoal. Hypergéométrie et fonction zêta de Riemann. Preprint, 2004.
16. C. Mallinger. Algorithmic manipulations and transformations of univariate holonomic functions and sequences. Master's thesis, RISC, J. Kepler University, Linz, August 1996.
17. M. Petkovšek. Hypergeometric solutions of linear recurrences with polynomial coefficients. J. Symbolic Comput., 14(2-3):243-264, 1992.
18. P. Paule and A. Riese. A Mathematica q-analogue of Zeilberger's algorithm based on an algebraically motivated aproach to q-hypergeometric telescoping. In M. Ismail and M. Rahman, editors, Special Functions, q-Series and Related Topics, volume 14, pages 179-210. Fields Institute Toronto, AMS, 1997.
19. P. Paule and M. Schorn. A Mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. J. Symbolic Comput., 20(5-6):673-698, 1995.
20. [PS95b] P. Paule and V. Strehl. Symbolic summation - some recent developments. In J. Fleischer et al., editor, Computer Algebra in Science and Engineering - Algorithms, Systems, and Applications, pages 138-162. World Scientific, Singapore, 1995.
21. P. Paule and C. Schneider. Computer proofs of a new family of harmonic number identities. Adv. in Appl. Math., 31(2):359-378, 2003.
22. M. Petkovšek, H. S. Wilf, and D. Zeilberger. A = B. A. K. Peters, Wellesley, MA, 1996.
23. A. Riese. qMultisum - A package for proving q-hypergeometric multiple summation identities. J. Symbolic Comput., 35:349-377, 2003.
24. [Ris69] R. Risch. The problem of integration in finite terms. Trans. Amer. Math. Soc., 139:167-189, 1969.
25. [Ris70] R. Risch. The solution to the problem of integration in finite terms. Bull. Amer. Math. Soc., 76:605-608, 1970.
26. C. Schneider. An implementation of Karr's summation algorithm in Mathematica. Sém. Lothar. Combin., S43b:1-10, 2000.
27. C. Schneider. Symbolic summation in difference fields. Technical Report 01-17, RISC-Linz, J. Kepler University, November 2001. PhD Thesis.
28. C. Schneider. Degree bounds to find polynomial solutions of parameterized linear difference equations in ΠΣ-fields. SFB-Report 02-21, J. Kepler University, Linz, November 2002.
29. C. Schneider. Solving parameterized linear difference equations in ΠΣ-fields. SFB-Report 02-19, J. Kepler University, Linz, November 2002.
30. C. Schneider. A collection of denominator bounds to solve parameterized linear difference equations in ΠΣ-extensions. To appear in SYNASC 2004, 2004.
31. C. Schneider. Product representations in ΠΣ-fields. To appear in Annals of Combinatorics, 2004.
32. C. Schneider. Symbolic summation with single-nested sum extensions. Proc. ISSAC'04, pages 282-289, 2004.
33. B. Salvy and P. Zimmermann. Gfun: A package for the manipulation of generating and holonomic functions in one variable. ACM Trans. Math. Software, 20:163-177, 1994.
34. M. van Hoeij. Rational solutions of linear difference equations. In O. Gloor, editor, Proc. ISSAC'98, pages 120-123. ACM Press, 1998.
35. K. Wegschaider. Computer generated proofs of binomial multi-sum identities. Diploma thesis, RISC Linz, Johannes Kepler University, May 1997.
36. D. Zeilberger. A fast algorithm for proving terminating hypergeometric identities. Discrete Math., 80(2):207-211, 1990.