q-Extensions of certain Erdélyi type integrals

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.

A note on the $l$-fold Bailey Lemma and Mixed Mock Modular forms

“…The type of Bailey pairs we have constructed in Theorem 1 allow us relate a 2l-fold q-series with a l-fold q-series. The case l = 1 of Theorem 1 gives a 2-fold Bailey pair of the Joshi-Vyas type [11].…”

Discrete Analogues of the Erdélyi Type Integrals for Hypergeometric Functions

“…where R(β 1 ) > R(α 2 ) > 0, is known as Euler's integral ( [4], p. 47, eorem 16). An extension of (8) was developed by Bateman [5] which is as follows:…”

“…As a matter of fact, the further generalizations of these 4 Φ 3 expansions were developed by Joshi and Vyas [11], in the form of 12 Φ 11 (q) and r Φ s (z) expansions, along with the applicability to biorthogonal rational functions ( [11], p. 221, Section 3). Many researchers, for example, [7,8,[11][12][13][14][15][16][17][18][19] have studied and investigated several kinds of integrals that involve and represent the hypergeometric functions, on account of the numerous applications of these integrals, for example, as shown in [13].…”

“…Furthermore, they obtained many other varieties of Erdélyi's integrals, for example ( [15], p. 132, Equations (4.1), (4.4), (4.5), (4.7) to (4.9) and (5.2)). In the sequel, they [16] also discussed q-extensions of the Erdélyi type integrals, by using series rearrangement. Certain classical series rearrangements were also used by [20,21] to derive extensions of the Bailey transform, which in turn, were applied to establish remarkable ordinary and q-hypergeometric identities.…”