DOI:10.1017/S0013091506001313 Printed in the United Kingdom NON-TERMINATING BASIC HYPERGEOMETRIC SERIES AND THE q-ZEILBERGER ALGORITHM

Abstract We present a systematic method for proving non-terminating basic hypergeometric identi-ties. 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 non-terminating basic hypergeometric summation formulae in the work of Gasper and Rahman. Furthermore, by comparing the recursions and the limit values, we may verify many classical transformation formu-lae, including the Sears–Carlitz transformation, transformations of the very well-poised 8φ7 series, the Rogers–Fine identity and the limiting case of Watson’s formula that implies the Rogers–Ramanujan identities