Cauchy Augmentation for Basic Hypergeometric Series

William Y. C. Chen  and  Amy M. Fu

  Abstract:  We present a technique of deriving basic hypergeometric identities from their specializations with a fewer number of parameters by using the classical Cauchy identity on the expansion of the power of x in terms of the q-binomial coefficients. We call method the Cauchy augmentation. Despite its simple appearance, the Cauchy identity plays a marvelous role for parameter augmentation. For example, from the Euler identity one can reach the q-Gauss summation formula by using the Cauchy augmentation twice. This idea also applies to Jackson's to transformation formula. Moreover, we obtain a transformation formula analogous to Jackson's formula.

  AMS Classification:  05A30,33D15.

  Keywords: Cauchy augmentation, Euler identity, Cauchy identity, Gauss identity, basic hypergeometric series, Jackson's transformation formula.

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