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 nd a recurrence relation of the summation by using the q-Zeilberger algorithm. This method applies to almost all nonterminating basic hypergeometric summation formu- las in the book of Gasper and Rahman. Furthermore, by compar- ing 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.

  AMS Classification:  33D15, 33F10

  Keywords:  basic hypergeometric series, q-Zeilberger algorithm, Bailey's very-well-poised 6 6 summation formula, Sears-Carlitz transformation, Rogers- Fine identity

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