q-Beta Integrals and Multivariate Basic Hypergeometric Series Associated to Root Systems of Type Am

2000 ◽  
Vol 4 (3) ◽  
pp. 347-373 ◽  
Author(s):  
Robert A. Gustafson ◽  
Medhat A. Rakha
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Root Systems ◽  
Beta Integrals

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.

scholarly journals Approach of q-Derivative Operators to Terminating q-Series Formulae

2018 ◽  
Vol 26 (2) ◽  
pp. 99-111
Author(s):  
Xiaoyuan Wang ◽  
Wenchang Chu
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Operator Approach ◽  
Derivative Operator ◽  
Summation Formulae

AbstractThe q-derivative operator approach is illustrated by reviewing several typical summation formulae of terminating basic hypergeometric series.

scholarly journals Inversion of Bilateral Basic Hypergeometric Series

10.37236/1703 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Michael Schlosser
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Matrix Inversion ◽  
Infinite Matrices ◽  
Inversion Result ◽  
Matrix Inverse ◽  
Bilateral Basic Hypergeometric Series ◽  
Summation Theorem ◽  
Lower Triangular

We present a new matrix inverse with applications in the theory of bilateral basic hypergeometric series. Our matrix inversion result is directly extracted from an instance of Bailey's very-well-poised ${}_6\psi_6$ summation theorem, and involves two infinite matrices which are not lower-triangular. We combine our bilateral matrix inverse with known basic hypergeometric summation theorems to derive, via inverse relations, several new identities for bilateral basic hypergeometric series.

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