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The present paper aims to derive summation and transformation formulae for the generalized very well-poised hypergeometric functions 2q+5F2q+4(1) and 2q+7F2q+6(1) having arbitrary integral parameter differences. These results are derived with the help of Bailey's transform and the extension of Saalschutz summation theorem for the series r+3Fr+2(1), where r pairs of parameters differ by positive integers. The particularizations of these generalized identities give classical summation theorems due to Dougall, transformation formula due to Whipple and, other related results. Furthermore, the application of 2q+5F2q+4(1) summation to the limiting case, when q→1, of one of the Andrews' q-identities gives a Srivastava-Daoust type multiple hypergeometric series. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we derive two most general possible q-hypergeometric expansion formulae for 12 Φ 11(q) and r Φ s (z), respectively. The results are unique in the sense that no such results are available in the literature beyond 4 Φ 3(q) so far and have the advantage that these results are also applicable to the top-level 10 Φ 9(q) biorthogonal rational functions. [Copyright &y& Elsevier]

Abstract: Recently we discovered several new Erdélyi type integrals. In the present paper, it is shown how the q-extensions of all those integrals involving and representing certain q-hypergeometric functions can be developed. The well-known special cases and applications of these q-integrals are also pointed out. [Copyright &y& Elsevier]

It is shown how series manipulation technique and certain classical summation theorems for hypergeometric series can be used to prove Erde´lyi's integral representations for 2F1(z), originally proved using fractional calculus. The method not only leads to generalizations but also leads to new integrals of Erde´lyi type for certain q+1Fq(z) and corresponding Pochhammer contour integrals. The technique outlined here, compared to the method of fractional calculus, seems to be more effective as it not only provides transparent elementary proofs of Erde´lyi's integrals but even leads to various generalizations. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]