q-analogue of a transformation of Whipple (Q788870)

Of late, a great resurgence of interest in the theory of q-analogues of hypergeometric series has taken place. Here, the authors begin by first establishing a quite general formula where a particular bi-basic series on the bases q and \(q^ 2\) is expressed as a series involving the single base \(q^ 2\) with argument \(q^ 2\) of the form \({}_ 5\Phi_ 4\). By suitable specialisation of this result, q-analogues of two formulae of \textit{F. J. W. Whipple} [Proc. Lond. Math. Soc., II. Ser. 26, 257-272 (1927)] connecting nearly-poised \({}_ 4F_ 3\) and Saalschützian \({}_ 5F_ 4\) series are deduced. The rest of this most interesting paper is devoted to obtaining various new summation theorems and transformations for both basic hypergeometric series and their ordinary counterparts, again utilising the same main result. The formulae given in this paper seem generally to be rather too lengthy for explicit inclusion in a short review.