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The q-Markov-WZ Method
Mohamud Mohammed
Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA
Annals of Combinatorics 9 (2) p.205-221 June, 2005
AMS Subject Classification: 05A10, 05A19, 33F10, 33C20, 33-04
Andrei Markov's 1890 beautiful ad-hoc method of transforming a series of hypergeometric type into a rapidly-converging series was upgraded recently to a full-fiedged method by Mohammed and Zeilberger, but only for the ordinary case. In this article, the
q-case is developed and it is shown how Markov's ad-hoc method, when coupled with q-WZ theory and q-Gosper's algorithm, leads to a new class of identities and very fast convergence-acceleration series that can be applied to any infinite series of
q-hypergeometric type.
Keywords: WZ theory, series convergence, q-hypergeometric, q-Gosper's algorithm, q-Zeilberger's algorithm


1. G. Almkvist and D. Zeilberger, The method of differentiating under the integral sign, J. Symbolic Comput. 10 (1990) 571--591. [available on-line from the second author's website.]

2. T. Amdeberhan and D. Zeilberger, Hypergeometric series acceleration via the WZ method, Electron. J. Combin. 4 (2) [Wilf Festschrifft volume] (1997) R3 (4 pages). [available on-line from the authors' websites.]

3. T. Amdeberhan and D. Zeilberger, q-Apéry irrationality proofs by q-WZ pairs, Adv. Appl. Math. 20 (1998) 275--283. [available on-line from the authors' websites.]

4. R. Apéry, Interpolation de fractions continues et irrationalité de certaine constantes, Bull. Sect. des Sci. #3 (1981) 37--53.

5. S. Fischler, Irationalité de valeurs de zêta [d'après Apéry, Rivoal, ... ], Séminaire Bourbaki- Novembre 2002, Exposé numéro 910, to appear in Astérique. Available On-line from arXiv.org.

6. T.H. Koornwinder, On Zeilberger's algorithm and its q-analog, J. Comput. Appl. Math. 48 (1993) 91--111.

7. M. Kondratieva and S. Sadov, Summation techniques forgotten for a century: Markov (1890)- Wilf-Zeilberger (1990), ICM-2002, Abstract of Short Communications. Higher Education Press, Beijing (2002) p. 404.

8. M. Kondratieva and S. Sadov, Markov's transformation of series and the WZ method, preprint, Available from http://xxx.arXiv.org/math.CA/0405592.

9. A.A. Markoff, Mémoire sur la transformation des séries peu convergentes en séries tres convergentes, Mémoires de L'Académie Impériale des Sciences de St.-Pétersbourg, VIIe Série, Tome XXXVII, No 9.

10. M. Mohammed and D. Zeilberger, The Markov WZ Method, Electron. J. Combin. 11 (2004) #R53.

11. M. Petkovsek, H.S. Wilf, and D. Zeilberger, A=B, AK Peters, Wellesley, 1996. [available on-line from the authors' websites.]

12. A.V. Sills, Finite rogers-ramanujan type identities, Electron. J. Combin. 10 (2003) #R13.

13. W. Koepf, Hypergeometric summation, In: An algorithmic Approach to Summation and Special Function Identity, Advanced lectures in mathemetics, Vieweg, 1998.

14. H.S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities, Invent. Math. 108 (1992) 575--633. [available on-line from the authors' websites.]

15. D. Zeilberger, A fast algorithm for proving terminating hypergeometric identities, Discrete Math. 80 (1990) 207--211. [available on-line from the author's website.]

16. D. Zeilberger, Closed form (pun intended!), In: Special Volume in Memory of Emil Grosswald, M. Knopp and M. Sheingorn, Eds., Contemporary Mathematics 143, AMS, Providence (1993) pp. 579--607. [available on-line from the author's website.]

17. D. Zeilberger, Computerized deconstruction, Adv. Appl. Math. 30 (2003) 633--654. [available on-line from the author's website.]