Nonterminating Basic Hypergeometric Series and the $q$-Zeilberger Algorithm

Details Nonterminating Basic Hypergeometric Series and the $q$-Zeilberger Algorithm Nonterminating Basic Hypergeometric Series and the $q$-Zeilberger Algorithm

2005-09-13

arxiv.org/abs/math/0509281v1

Mathematics

Combinatorics

30 pages

Scientific paper

We present a systematic method for proving nonterminating basic hypergeometric identities. Assume that $k$ is the summation index. By setting a parameter $x$ to $xq^n$, 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 $_8\phi_7$ series, the Rogers-Fine identity, and the limiting case of Watson's formula that implies the Rogers-Ramanujan identities.

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