Inversion Techniques And Q-Series Identities

This dissertation studies the applications of the inversion techniques and its equivalent form in finding and proving the hypergeometric series identities. The content is as follows:1. Chu[26] derived the q-duplicate inverse series relations. Unfortunately, he has not found an appropriate identity which fits in the q-Duplicate inverse series relations.We find that the very-well-poised 6 5 [40, (11.20)] series and Watson's transformation formula [-40. (III.17)] can fit our scheme and give some terminating summation formulas, including those due to Andrews [(II.17)][40]. by using the q-Duplicate inverse series relations.2. Firstly, we constructed q-triplicatc inverse series relations and use them to derive some new summation formulas, including the generalization of G. E. Andrews' identity [40. (11.17)].Secondly, we get some terminating identities by using Slater's identities 79]. Finally, we establish the q-multiplicate inverse series relations.3. W'e establish the generalized Bailey's transformation. Bailey's Lemma, Bailey pairs and some of G. E. Andrews's important results. As applications, several identities of Rogers-Ramanujan type and transformation formulas of basic hypergeometric series are demonstrated, especially, we give the concrete applications of (a, q, 2)-Bailey pairs and (a, q, 3)-Bailey pairs.4. We show some new transformation formulas by using Bressoud's matrix inverse. As applications of Gasper's matrix inverse, we generalize Jackson's bibasic summation formula ([40, (II.22)]) and get the closed form of general 2λ+6W2λ+5 and 2λ+6V2λ+5. A transformation formula containing four independent bases is found and applied to derive a few summation formulas for basic hypevgeometric series. The ordinary hypergeometric limits of these formulas are also obtained.5. By using F. H. Jackson's identity [55]. we derive a generalization of Subbarao and Verma's summation formula (5.1.6) which generalized Chu's summation formula (5.1.5) involving four arbitrary sequences and other general bilateral summations. Subsequently, we exhibit the dual formulas for the above mentioned identities. And we derive a few transformation formulas with eight arbitrary sequences. Finally, another generalization of Gasper's bibasic summation formula is also obtained.