A Quadratic Transformation of a Basic Hypergeometric Series

1980 ◽  
Vol 11 (3) ◽  
pp. 425-427 ◽  
Author(s):  
Arun Verma
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Quadratic Transformation

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 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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