The classical summation and transformation theorems for very well-poised hypergeometric functions, namely, 5 F 4 (1) summation, Dougall's 7 F 6 (1) summation, Whipple's 7 F 6 (1) to 4 F 3 (1) transformation and Bailey's 9 F 8 (1) to 9 F 8 (1) transformation are extended. These extensions are derived by applying the well-known Bailey's transform method along with the classical very well-poised summation and transformation theorems for very well-poised hypergeometric functions and the Rakha and Rathie's extension of the Saalschütz's theorem. To show importance and applications of the discovered extensions, a number of special cases are pointed out, which leads not only to the extensions of other classical theorems for very well-poised and well-poised hypergeometric functions but also generate new hypergeometric summations and transformations.

This paper provides three classes of q-summation formulas in the form of general contiguous extensions of the first q-Kummer summation theorem. Their derivations are presented by using three methods, which are along the lines of the three types of well-known proofs of the q-Kummer summation theorem with a key role of the q-binomial theorem. In addition to the q-binomial theorem, the first proof makes use of Thomae’s q-integral representation and the second proof needs Heine’s transformation. Whereas the third proof utilizes only the q-binomial theorem. Subsequently, the applications of these summation formulas in obtaining the general contiguous extensions of the second and the third q-Kummer summation theorems are also presented. Furthermore, the investigated results are specialized to give many of the known as well as presumably new q-summation theorems, which are contiguous to the three q-Kummer summation theorems. This work is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including Number Theory, Theory of Partitions and Combinatorial Analysis as well as in the study of Combinatorial Generating Functions. Just as it is known in the theory of the Gauss, Kummer (or confluent), Clausen and the generalized hypergeometric functions, the parameters in the corresponding basic or quantum (or q-) hypergeometric functions are symmetric in the sense that they remain invariant when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. A case has therefore been made for the symmetry possessed not only by hypergeometric functions and basic or quantum (or q-) hypergeometric functions, which are studied in this paper, but also by the symmetric quantum calculus itself.

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ver the years, the study of Bailey transform, Bailey lemma, Bailey pair, their variants and their applications are the major subjects of interest. Of course, it is due to the efficiency of the Bailey transform and lemma in producing many ordinary and q-hypergeometric identities, multiple series summation and transformation formulas, and the Rogers-Ramanujan type identities. Andrews investigated a WP-Bailey lemma and the pairs with the help of Bailey transform and used it to derive well-known summations and multiple series transformations. In this research paper, we investigate an Andrews’ type WP-Bailey lemma and the pairs with the help of First Bailey Type Transform due to Joshi and Vyas. The investigated Andrews’ type WP-Bailey lemma is then applied to obtain terminating multiple q-hypergeometric identities and construct the WP-Bailey type chains and a binary tree. The paper is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including number theory, theory of partitions and combinatorial analysis as well as in the study of combinatorial generating functions.

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