Extensions of Bailey′s transform and applications

Joshi, C. M.; Vyas, Yashoverdhan

doi:10.1155/ijmms.2005.1909

Cited by 10 publications

(5 citation statements)

References 16 publications

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“…Following results will be needed in our work: W. N. Bailey [2,3] established a series transform known as Bailey's transform and gave a mechanism to derive ordinary and q-hypergeometric identities and Rogers-Ramanujan type identities. Its extensions and other types of Bailey's transform have also appeared in [5,6].…”

Section: Introductionmentioning

confidence: 99%

Extensions of the classical theorems for very well-poised hypergeometric functions

Srivástava

Vyas

Fatawat

2017

RACSAM

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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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Section: Introductionmentioning

confidence: 99%

Extensions of the classical theorems for very well-poised hypergeometric functions

Srivástava

Vyas

Fatawat

2017

RACSAM

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“…W. N. Bailey [2,3] developed a series transform known as Bailey's transform and gave a mechanism to derive ordinary and q-hypergeometric identities and Rogers-Ramanujan type identities. Its extensions and other types of Bailey's transform and their applications in deriving many new summation and transformation formulae can be found in [6,7]. It states that if…”

Section: Preliminariesmentioning

confidence: 99%

Summations and transformations for very well-poised hypergeometric functions 2q+5F2q+4(1) and 2q+7F2q+6(1) with arbitrary integral parameter differences

Vyas¹,

Fatawat²

2022

MMN

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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.

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“…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.…”

Section: Introduction Motivation and Preliminariesmentioning

confidence: 99%

Discrete Analogues of the Erdélyi Type Integrals for Hypergeometric Functions

Vyas

Bhatnagar

Fatawat

et al. 2022

Journal of Mathematics

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Gasper followed the fractional calculus proof of an Erdélyi integral to derive its discrete analogue in the form of a hypergeometric expansion. To give an alternative proof, we derive it by following a procedure analogous to a triple series manipulation-based proof of the Erdélyi integral, due to “Joshi and Vyas”. Motivated from this alternative way of proof, we establish the discrete analogues corresponding to many of the Erdélyi type integrals due to “Joshi and Vyas” and “Luo and Raina” in the form of new hypergeometric expansion formulas. Moreover, the applications of investigated discrete analogues in deriving some expansion formulas involving orthogonal polynomials of the Askey-scheme and a new generalization of Whipple’s transformation for a balanced 4F3 in the form of an m + 4 F m + 3 transformation, are also discussed.

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Extensions of Bailey′s transform and applications

Cited by 10 publications

References 16 publications

Extensions of the classical theorems for very well-poised hypergeometric functions

Extensions of the classical theorems for very well-poised hypergeometric functions

Summations and transformations for very well-poised hypergeometric functions 2q+5F2q+4(1) and 2q+7F2q+6(1) with arbitrary integral parameter differences

Discrete Analogues of the Erdélyi Type Integrals for Hypergeometric Functions

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