Positivity of the Poisson Kernel for the Continuous q-Jacobi Polynomials and Some Quadratic Transformation Formulas for Basic Hypergeometric Series

1986 ◽  
Vol 17 (4) ◽  
pp. 970-999 ◽  
Author(s):  
George Gasper ◽  
Mizan Rahman
Keyword(s):  
Hypergeometric Series ◽  
Poisson Kernel ◽  
Jacobi Polynomials ◽  
Basic Hypergeometric Series ◽  
Quadratic Transformation ◽  
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 Multiple big q-Jacobi polynomials

2020 ◽  
Vol 10 (02) ◽  
pp. 2050013
Author(s):  
Fethi Bouzeffour ◽  
Mubariz Garayev
Keyword(s):  
Difference Equation ◽  
Hypergeometric Series ◽  
Recurrence Relations ◽  
Jacobi Polynomials ◽  
Basic Hypergeometric Series ◽  
Type Ii ◽  
Third Order ◽  
Raising And Lowering Operators ◽  
Explicit Formulae ◽  
Lowering Operators

Here, we investigate type II multiple big [Formula: see text]-Jacobi orthogonal polynomials. We provide their explicit formulae in terms of basic hypergeometric series, raising and lowering operators, Rodrigues formulae, third-order [Formula: see text]-difference equation, and we obtain recurrence relations.

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