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.

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The well-known Bailey's transform is extended. Using the extended transform, we derive hitherto undiscovered ordinary and q-hypergeometric identities and discuss their particular cases of importance, namely, two new q-sums for Saalschützian 4 Φ 3 , new double series Rogers-Ramanujan-type identities of modulo 81, discrete extension of the q-analogs of two quadratic transformations of 2 F 1 , and two new quadratic-cubic transformations of 3 F 2 .

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.