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

TL;DR

This paper introduces a systematic method using the $q$-Zeilberger algorithm to prove and verify a wide range of nonterminating basic hypergeometric identities and transformations, including classical and famous identities.

Contribution

The paper develops a general approach leveraging the $q$-Zeilberger algorithm to prove and verify nonterminating basic hypergeometric identities, applicable to many classical formulas.

Findings

Successfully proves numerous classical hypergeometric identities.

Verifies transformation formulas such as Sears-Carlitz and Watson's limit.

Applies to almost all nonterminating basic hypergeometric summation formulas in Gasper and Rahman's book.

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 $xq^n$, 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.