Generalized basic hypergeometric series with unconnected bases

In a series of recent papers Verma and Upadhyay (7,8,9) developed the theory of basic hypergeometric series with two bases q and q ½ . These investigations were made in an attempt to discover a summation formula for a bilateral basic hypergeometric series 2 Ψ 2 analogous to that for a 2 H 2 (cf. Bailey (2,3)) and in finding relations between certain q -infinite products. In one of their papers they mentioned that it did not seem possible to develop the corresponding general theory for basic series with two unconnected bases q and q 1 . A recent paper by Andrews (1) indicates that transformations between basic hypergeometric series with two unconnected bases can be very interesting and useful in the study of ‘mock’ theta functions and their extensions. Besides this interest, such a theory also enables one to extend the entire existing transformation theory of the generalized basic hypergeometric series.