On Certain Transformation Formulas Involving Basic Hypergeometric Series

2019 ◽  
Vol 90 (4) ◽  
pp. 623-627
Author(s):  
Satya Prakash Singh ◽  
Vijay Yadav
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Transformation Formulas

scholarly journals Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series

2020 ◽  
Author(s):  
Victor J. W. Guo ◽  
Michael J. Schlosser
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Double Series ◽  
Summation Formula ◽  
Transformation Formula ◽  
Quadratic Transformation ◽  
Ultraspherical Polynomials ◽  
Higher Power ◽  
Series Transformation ◽  
Transformation Formulas

AbstractSeveral new q-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Zudilin. More concretely, the results in this paper include q-analogues of supercongruences (referring to p-adic identities remaining valid for some higher power of p) established by Long, by Long and Ramakrishna, and several other q-supercongruences. The six basic hypergeometric transformation formulas which are made use of are Watson’s transformation, a quadratic transformation of Rahman, a cubic transformation of Gasper and Rahman, a quartic transformation of Gasper and Rahman, a double series transformation of Ismail, Rahman and Suslov, and a new transformation formula for a nonterminating very-well-poised $${}_{12}\phi _{11}$$ 12 ϕ 11 series. Also, the nonterminating q-Dixon summation formula is used. A special case of the new $${}_{12}\phi _{11}$$ 12 ϕ 11 transformation formula is further utilized to obtain a generalization of Rogers’ linearization formula for the continuous q-ultraspherical polynomials.

Some q-transformation formulas and Hecke type identities

2019 ◽  
Vol 15 (07) ◽  
pp. 1349-1367 ◽  
Author(s):  
Chun Wang ◽  
Shane Chern
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Transformation Formulas

In this paper, we establish certain transformations on basic hypergeometric series. Some applications of these transformation formulas to Hecke type identities will be discussed. We also study other [Formula: see text]-series transformations that may lead to certain Rogers–Ramanujan type identities.

scholarly journals Transformation Formulas for Bilinear Sums of Basic Hypergeometric Series

2016 ◽  
Vol 59 (01) ◽  
pp. 136-143
Author(s):  
Yasushi Kajihara
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Type A ◽  
Transformation Formula ◽  
Bilinear Transformation ◽  
Multivariate Generalization ◽  
The One ◽  
Transformation Formulas

Abstract A master formula of transformation formulas for bilinear sums of basic hypergeometric series is proposed. It is obtained from the author’s previous results on a transformation formula for Milne’s multivariate generalization of basic hypergeometric series of type A with diòerent dimensions and it can be considered as a generalization of theWhipple–Sears transformation formula for terminating balanced µh series. As an application of the master formula, the one-variable cases of some transformation formulas for bilinear sums of basic hypergeometric series are given as examples. The bilinear transformation formulas seemto be new in the literature, even in the one-variable case.

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