Non-Terminating Basic Hypergeometric Series and the q-Zeilberger Algorithm

DOI10.1017/S0013091506001313zbMATH Open1183.33029arXivmath/0509281MaRDI QIDQ3533804

Yan-Ping Mu, William Y. C. Chen, Qing-Hu Hou

Publication date: 24 October 2008

Published in: Proceedings of the Edinburgh Mathematical Society (Search for Journal in Brave)

Abstract: We present a systematic method for proving nonterminating basic hypergeometric identities. Assume that $k$ is the summation index. By setting a parameter $x$ to $x{q}^n$, we may find a recurrence relation of the summation by using the $q$-Zeilberger algorithm. This method applies to almost all nonterminating basic hypergeometric summation formulas in the book of Gasper and Rahman. Furthermore, by comparing the recursions and the limit values, we may verify many classical transformation formulas, including the Sears-Carlitz transformation, transformations of the very-well-poised ${}_{8}ph{i}_7$ series, the Rogers-Fine identity, and the limiting case of Watson's formula that implies the Rogers-Ramanujan identities.

Full work available at URL: https://arxiv.org/abs/math/0509281

zbMATH Keywords

basic hypergeometric series\(q\)-Zeilberger algorithmRogers-Fine identityBailey's very well-poised \(_6 \psi_6\) summation formulaSears-Carlitz transformation

Mathematics Subject Classification ID

Basic hypergeometric functions in one variable, ({}_rphi_s) (33D15) Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.) (33F10)

Related Items (7)

The method of combinatorial telescoping ⋮ Applicability of the \(q\)-analogue of Zeilberger's algorithm ⋮ Nonterminating well–poised hypergeometric series ⋮ The Abel Lemma and the $q$-Gosper Algorithm ⋮ Parameter augmentation and the \(q\)-Gosper algorithm ⋮ The \(q\)-WZ method for infinite series ⋮ Title not available (Why is that?)

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