Nonterminating Basic Hypergeometric
Series and the q-Zeilberger Algorithm
Yong-Chuan Chen,Qing-Hu Hou and
Yan-Ping Mu
Center for Combinatorics and LPMC,
Nankai University, Tianjin 300071, P.R. China
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 xqn, 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 
series, the Rogers-Fine identity,
and the limiting case of Watson's formula that implies the
Rogers-Ramanujan identities.
