Padé approximations to the logarithm III: Alternative methods and additional results

Ramanujan Journal

Volumn 12, Issue 3, 2006, Pages 299-314

  Driver, Kathy ^a   Prodinger, Helmut ^a   Schneider, Carsten ^b   Weideman, J A C ^c  

^a UNIVERSITY OF THE WITWATERSRAND   (South Africa)

^b JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

^c STELLENBOSCH UNIVERSITY   (South Africa)

Author keywords

Combinatorial identities; Computer algebra; Creative telescoping; Pad approximation

Indexed keywords

EID: 33751501378     PISSN: 13824090     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11139-006-0144-5     Document Type: Article

References (25)

1. Abramov, S.A., Petkovšek, M.: D'Alembertian solutions of linear differential and difference equations. In: von zur Gathen, J. (ed.) Proc. ISSAC'94 pp. 169-174. ACM Press, Baltimore, (1994)
2. Andrews, G.E.: Applications of basic hypergeometric functions. SIAM Rev. 16, 441-484 (1974)
3. Andrews, G.E., Uchimura, K.: Identities in combinatorics. IV. Differentiation and harmonic numbers. Utilitas Math. 28, 265-269 (1985)
4. Bronstein, M.: On solutions of linear ordinary difference equations in their coefficient field. J. Symbolic Comput. 29(6), 841-877 (2000)
5. Driver, K., Prodinger, H., Schneider, C., Weideman, J.A.C.: Padé approximations to the logarithm II: Identities, recurrences, and symbolic computation. Ramanujan J. 11, 139-158 (2006)
6. Egorychev, G.P.: Integral Representation and the Computation of Combinatorial Sums. American Mathematical Society (1984)
7. Gosper, R.W.: Decision procedures for indefinite hypergeometric summation. Proc. Nat. Acad. Sci. USA 75, 40-42 (1978)
8. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science. 2nd edition Addison-Wesley Publishing Company, Amsterdam, (1994)
9. Hendriks, P.A. Singer, M.F.: Solving difference equations in finite terms. J. Symbolic Comput. 27(3), 239-259 (1999)
10. Karr, M.: Summation in finite terms. J. ACM 28, 305-350 (1981)
11. Karr, M.: Theory of summation in finite terms. J. Symbolic Comput. 1, 303-315 (1985)
12. Paule, P., Schneider, C.: Computer proofs of a new family of harmonic number identities. Adv. Appl. Math. 31(2), 359-378 (2002)
13. Paule, P., Schorn, M.: A mathematica version of Zeilberger's algorithm for proving binomial coefficient identities. J. Symbolic Comput. 20(5-6), 673-698 (1995)
14. Petkovšek, M., Wilf, H.S., Zeilberger, D.: A = B. A. K. Peters Wellesley, MA, (1996)
15. Risch, R.: The problem of integration in finite terms. Trans. Amer. Math. Soc. 139, 167-189 (1969)
16. Risch, R.: The solution to the problem of integration in finite terms. Bull. Amer. Math. Soc. 76, 605-608 (1970)
17. Roman, S.M., Rota, G.-C.: The umbral calculus. Advances Math. 27(2), 95-188 (1978)
18. Schneider, C.: Symbolic summation in difference fields. Technical Report 01-17, RISC-Linz, J. Kepler University. PhD Thesis (2001)
19. Schneider, C.: A collection of denominator bounds to solve parameterized linear difference equations in ∏∑-Extensions. In: Petcu, D., Negra, V., Zaharie, D., Jebelean T., (eds.) Proc. SYNASC04, 6th Internat. Symposium on Symbolic and Numeric Algorithms for Scientific Computation. Timisoara (Romania), pp. 269-282. Mirton Publishing (2004)
20. Schneider, C.: The summation package Sigma: Underlying principles and a rhombus tiling application. Discrete Math. Theor. Comput. Sci. 6(2), 365-386 (2004)
21. Schneider, C.: Degree bounds to find polynomial solutions of parameterized linear difference equations in ∏∑-fields. Appl. Algebra Engrg. Comm. Comput. 16(1), 1-32 (2005)
22. Schneider, C.: Solving parameterized linear difference equations in terms of indefinite nested sums and products. J. Differ. Equations Appl. 11(9), 799-821 (2005)
23. Weideman, J.A.C.: http://dip.sun.ac.za/~weideman/FADE
24. Weideman, J.A.C.: Fade approximations to the logarithm. I. Derivation via differential equations. Quaest. Math. 28(3), 375-390 (2005)
25. Zeilberger, D.: A fast algorithm for proving terminating hypergeometric identities. Discrete Math. 80(2), 207-211 (1990)