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On Convergence of q-Series Involving ?r+1r Basic Hypergeometric Series

q-Series, which are also called basic hypergeometric series, play a very important role in many fields, such as affine root systems, Lie algebras and groups, number theory, orthogonal polynomials and physics. Convergence of a q-series is an important problem in the study of q-series. There are some results about it in 1–3 . For example, Ito used inequality technique to give a sufficient condition for convergence of a special q-series called Jackson integral. In this paper, by using inequality technique, we derive the following two theorems on convergence of q-series involving r 1φr basic hypergeometric series, which can be used for convergence of special Thomae q-integral.