scholarly journals Askey–Wilson polynomials and a double $q$-series transformation formula with twelve parameters

2019 ◽  
Vol 147 (6) ◽  
pp. 2349-2363
Author(s):  
Zhi-Guo Liu
Keyword(s):  
Transformation Formula ◽  
Series Transformation ◽  
Wilson Polynomials

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 A series transformation formula and related polynomials

2005 ◽  
Vol 2005 (23) ◽  
pp. 3849-3866 ◽  
Author(s):  
Khristo N. Boyadzhiev
Keyword(s):  
Power Series ◽  
Gamma Function ◽  
Asymptotic Series ◽  
Transformation Formula ◽  
Bell Polynomials ◽  
New Class ◽  
Series Transformation ◽  
Lerch Transcendent ◽  
Series Of Functions ◽  
Case Based

We present a formula that turns power series into series of functions. This formula serves two purposes: first, it helps to evaluate some power series in a closed form; second, it transforms certain power series into asymptotic series. For example, we find the asymptotic expansions forλ>0of the incomplete gamma functionγ(λ,x)and of the Lerch transcendentΦ(x,s,λ). In one particular case, our formula reduces to a series transformation formula which appears in the works of Ramanujan and is related to the exponential (or Bell) polynomials. Another particular case, based on the geometric series, gives rise to a new class of polynomials called geometric polynomials.

scholarly journals The Euler Series Transformation and the Binomial Identities of Ljunggren, Munarini and Simons

Integers ◽  
2010 ◽  
Vol 10 (3) ◽  
Author(s):  
Khristo N. Boyadzhiev
Keyword(s):  
Transformation Formula ◽  
Series Transformation ◽  
Euler Series ◽  
Binomial Identities

AbstractWe point out that the curious identity of Simons follows immediately from Euler's series transformation formula and also from an identity due to Ljunggren. We also mention its relation to Legendre's polynomials. At the end we use the generalized Euler series transformation to obtain two recent binomial identities of Munarini.

scholarly journals Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Massimo Gisonni ◽  
Tamara Grava ◽  
Giulio Ruzza
Keyword(s):  
Generating Functions ◽  
Combinatorial Interpretation ◽  
Hurwitz Numbers ◽  
Jacobi Ensemble ◽  
Matrix Ensembles ◽  
Wilson Polynomials ◽  
Unitary Ensemble ◽  
Topological Expansion

AbstractWe express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix ensembles. We also provide effective formulæ for generating functions of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson polynomials, generalizing the known relations between one point correlators and Wilson polynomials.

A transformation formula in the theory of partitions

1944 ◽  
Vol 11 (4) ◽  
pp. 873-887 ◽  
Author(s):  
Lowell Schoenfeld
Keyword(s):  
Transformation Formula

scholarly journals Befriending Askey–Wilson polynomials

2014 ◽  
Vol 17 (03) ◽  
pp. 1450015 ◽  
Author(s):  
Paweł J. Szabłowski
Keyword(s):  
Characteristic Function ◽  
Hermite Polynomials ◽  
Probabilistic Interpretation ◽  
Conjugate Pair ◽  
Linear Combinations ◽  
Expansion Formula ◽  
Wilson Polynomials ◽  
The Way

We recall five families of polynomials constituting a part of the so-called Askey–Wilson scheme. We do this to expose properties of the Askey–Wilson (AW) polynomials that constitute the last, most complicated element of this scheme. In doing so we express AW density as a product of the density that makes q-Hermite polynomials orthogonal times a product of four characteristic function of q-Hermite polynomials (2.9) just pawing the way to a generalization of AW integral. Our main results concentrate mostly on the complex parameters case forming conjugate pairs. We present new fascinating symmetries between the variables and some newly defined (by the appropriate conjugate pair) parameters. In particular in (3.12) we generalize substantially famous Poisson–Mehler expansion formula (3.16) in which q-Hermite polynomials are replaced by Al-Salam–Chihara polynomials. Further we express Askey–Wilson polynomials as linear combinations of Al-Salam–Chihara (ASC) polynomials. As a by-product we get useful identities involving ASC polynomials. Finally by certain re-scaling of variables and parameters we reach AW polynomials and AW densities that have clear probabilistic interpretation.

A transformation formula for Grothendieck residues and some of its applications

1989 ◽  
Vol 29 (3) ◽  
pp. 495-499
Author(s):  
A. M. Kytmanov
Keyword(s):  
Transformation Formula

scholarly journals Expansions in the Askey–Wilson polynomials

2015 ◽  
Vol 424 (1) ◽  
pp. 664-674 ◽  
Author(s):  
Mourad E.H. Ismail ◽  
Dennis Stanton
Keyword(s):  
Wilson Polynomials

scholarly journals The factorization method for the Askey–Wilson polynomials

1999 ◽  
Vol 107 (2) ◽  
pp. 219-232 ◽  
Author(s):  
Gaspard Bangerezako
Keyword(s):  
Factorization Method ◽  
Wilson Polynomials

scholarly journals A cubic transformation formula for Appell–Lauricella hypergeometric functions over finite fields

2018 ◽  
Vol 4 (2) ◽  
Author(s):  
Sharon Frechette ◽  
Holly Swisher ◽  
Fang-Ting Tu
Keyword(s):  
Finite Fields ◽  
Hypergeometric Functions ◽  
Transformation Formula
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