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.

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.

scholarly journals From Kajihara’s transformation formula to deformed Macdonald–Ruijsenaars and Noumi–Sano operators

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Martin Hallnäs ◽  
Edwin Langmann ◽  
Masatoshi Noumi ◽  
Hjalmar Rosengren
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Root Systems ◽  
Higher Order ◽  
Explicit Description ◽  
Transformation Formula ◽  
Inverse Limits ◽  
First Order ◽  
Order Case ◽  
Multiple Basic Hypergeometric Series

AbstractKajihara obtained in 2004 a remarkable transformation formula connecting multiple basic hypergeometric series associated with A-type root systems of different ranks. By specialisations of his formula, we deduce kernel identities for deformed Macdonald–Ruijsenaars (MR) and Noumi–Sano (NS) operators. The deformed MR operators were introduced by Sergeev and Veselov in the first order case and by Feigin and Silantyev in the higher order cases. As applications of our kernel identities, we prove that all of these operators pairwise commute and are simultaneously diagonalised by the super-Macdonald polynomials. We also provide an explicit description of the algebra generated by the deformed MR and/or NS operators by a Harish-Chandra type isomorphism and show that the deformed MR (NS) operators can be viewed as restrictions of inverse limits of ordinary MR (NS) operators.

Sign in / Sign up

Export Citation Format

Share Document