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Nonsymmetric Askey–Wilson Shift Operators
Studies in Applied Mathematics, Volume 155, Issue 3, September 2025.ABSTRACT We classify the shift operators for the symmetric Askey–Wilson polynomials and construct shift operators for the nonsymmetric Askey–Wilson polynomials using two decompositions of nonsymmetric Askey–Wilson polynomials in terms of symmetric ones. These shift operators are difference–reflection operators, and we discuss the conditions under which
Max van Horssen, Philip Schlösser
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Continuous −1$-1$ hypergeometric orthogonal polynomials
Studies in Applied Mathematics, Volume 153, Issue 3, October 2024.Abstract The study of −1$-1$ orthogonal polynomials viewed as q→−1$q\rightarrow -1$ limits of the q$q$‐orthogonal polynomials is pursued. This paper presents the continuous polynomials part of the −1$-1$ analog of the q$q$‐Askey scheme. A compendium of the properties of all the continuous −1$-1$ hypergeometric polynomials and their connections is ...
Jonathan Pelletier +2 more
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Connection and linearization coefficients of the Askey–Wilson polynomials
Journal of Symbolic Computation, 2013 The linearization problem is the problem of finding the coefficients C"k(m,n) in the expansion of the product P"n(x)Q"m(x) of two polynomial systems in terms of a third sequence of polynomials R"k(x),P"n(x)Q"m(x)=@?k=0n+mC"k(m,n)R"k(x). The polynomials P"n, Q"m and R"k may belong to three different polynomial families.
Mama Foupouagnigni +2 more
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Double Affine Hecke Algebras of Rank 1 and the Z_3-Symmetric Askey-Wilson Relations
Symmetry, Integrability and Geometry: Methods and Applications, 2010 We consider the double affine Hecke algebra H=H(k_0,k_1,k_0^v,k_1^v;q) associated with the root system (C_1^v,C_1). We display three elements x, y, z in H that satisfy essentially the Z_3-symmetric Askey-Wilson relations.
Paul Terwilliger, Tatsuro Ito
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On Some Limit Cases of Askey–Wilson Polynomials
Journal of Approximation Theory, 1998 The authors derive the classical orthogonality relations and norm evaluations for the \(q\)-Racah and \(q\)-Jacobi polynomials by taking limits in the orthogonality relations and norm evaluations for the Askey-Wilson polynomials [\textit{R. Askey} and \textit{J. Wilson}, Mem. Am. Math. Soc. 54, No. 319, 1-55 (1985; Zbl 0572.33012)].
Stokman, J.V., Koornwinder, T.H.
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Equilibrium positions, shape invariance and Askey–Wilson polynomials [PDF]
Journal of Mathematical Physics, 2005 We show that the equilibrium positions of the Ruijsenaars–Schneider–van Diejen systems with the trigonometric potential are given by the zeros of the Askey–Wilson polynomials with five parameters. The corresponding single particle quantum version, which is a typical example of “discrete” quantum mechanical systems with a q-shift type kinetic term, is ...
Odake, Satoru, Sasaki, Ryu
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A Relativistic Conical Function and its Whittaker Limits
Symmetry, Integrability and Geometry: Methods and Applications, 2011 In previous work we introduced and studied a function R(a+,a−,c;v,vˆ) that is a generalization of the hypergeometric function _2F_1 and the Askey-Wilson polynomials.
Simon Ruijsenaars
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Turán Inequalities for Symmetric Askey-Wilson Polynomials
Rocky Mountain Journal of Mathematics, 2000 The authors study a renormalized \(A-W\) polynomial \(V_n(x)\). Using the Szász technique they establish the inequalities \[ 0\leq V_n^2(x)-V_{n+1} (x)V_{n-1} (x)\leq K, \] where \(K\) is independent of \(x\). The two inequalities hold under certain conditions upon parameters and variable.
Abreu, L.D., Bustoz, J.
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Bethe Ansatz Solutions to Quasi Exactly Solvable Difference Equations
Symmetry, Integrability and Geometry: Methods and Applications, 2009 Bethe ansatz formulation is presented for several explicit examples of quasi exactly solvable difference equations of one degree of freedom which are introduced recently by one of the present authors.
Ryu Sasaki, Wen-Li Yang, Yao-Zhong Zhang
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Bispectral extensions of the Askey–Wilson polynomials
Journal of Functional Analysis, 2014 Following the pioneering work of Duistermaat and Grünbaum, we call a family $\{p_n(x)\}_{n=0}^{\infty}$ of polynomials bispectral, if the polynomials are simultaneously eigenfunctions of two commutative algebras of operators: one consisting of difference operators acting on the degree index $n$, and another one of operators acting on the variable $x ...
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