Askey-Wilson Polynomials Research Articles

On a conjecture involving Askey–Wilson polynomials

The following relation holds: ( ( q 1 / 2 + q − 1 / 2 2 ) 2 − 1 ) 2 ( x 2 − ( q 1 / 2 + q − 1 / 2 2 ) 2 ) ( 1 − x 2 ) D q 2 P n ( x ; 1 , q 1 / 2 , 0 , 0 | q ) = ∑ j = − 4 2 a n , j P n + j ( x ; 1 , q 1 / 2 , 0 , 0 | q ) ( a n , − 4 ≠ 0 ) , P n ( x ; a , b , c , d | q ) being the monic Askey–Wilson polynomials.

Q-fractional integral operators with two parameters

We use the Poisson kernel of the continuous q-Hermite polynomials to introduce families of integral operators. One of them is semigroups of linear operators. We describe the eigenvalues and eigenfunctions of one family of operators. The action of the semigroups of operators on the Askey–Wilson polynomials is shown to only change the parameters but preserves the degrees, hence we produce transmutation relations for the Askey–Wilson polynomials. The transmutation relations are then used to derive bilinear generating functions involving the Askey–Wilson polynomials.

Indeterminate moment problem associated with continuous dual q-Hahn polynomials

We study a limiting case of the Askey–Wilson polynomials when one of the parameters goes to infinity, namely continuous dual q-Hahn polynomials when q>1. Solutions to the associated indeterminate moment problem by general theory are found and an orthogonality relation is established.

Proof of two conjectures on Askey-Wilson polynomials

We give positive answer to two conjectures posed by M. E. H Ismail in his monograph [Classical and quantum orthogonal polynomials in one variable, Cambridge University Press, Cambridge, 2005]. These results generalize the classical problems of Sonine and Hahn.

Charting the [formula omitted]-Askey scheme. II. The [formula omitted]-Zhedanov scheme

This is the second in a series of papers which intend to explore conceptual ways of distinguishing between families in the q-Askey scheme and uniform ways of parametrizing the families. For a system of polynomials pn(x) in the q-Askey scheme satisfying Lpn=hnpn with L a second order q-difference operator the q-Zhedanov algebra is the algebra generated by operators L and X (multiplication by x). It has two relations in which essentially five coefficients occur. Vanishing of one or more of the coefficients corresponds to a subfamily or limit family of the Askey–Wilson polynomials. An arrow from one family to another means that in the latter family one more coefficient vanishes. This yields the q-Zhedanov scheme given in this paper.The q-hypergeometric expression of pn(x) can be interpreted as an expansion of pn(x) in terms of certain Newton polynomials. In our previous paper (Contemporary Math. 780) we used Verde-Star’s clean parametrization of such expansions and we obtained a q-Verde-Star scheme, where vanishing of one or more of these parameters corresponds to a subfamily or limit family. The actions of the operators L and X on the Newton polynomials can be expressed in terms of the Verde-Star parameters, and thus the coefficients for the q-Zhedanov algebra can be expressed in terms of these parameters. There are interesting differences between the q-Verde-Star scheme and the q-Zhedanov scheme, which are discussed in the paper.

Formula omitted]-fractional Askey–Wilson integrals and related semigroups of operators

We introduce three one parameter semigroups of operators and determine their spectra. Two of them are fractional integrals associated with the Askey–Wilson operator. We also study these families as families of positive linear approximation operators. Applications include connection relations and bilinear formulas for the Askey–Wilson polynomials, and solving certain dual integral equations.

Symmetry of terminating basic hypergeometric series representations of the Askey–Wilson polynomials

In this paper, we explore the symmetric nature of the terminating basic hypergeometric series representations of the Askey–Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy. In particular we identify and classify the set of 4 and 7 equivalence classes of terminating balanced ϕ34 and terminating very-well-poised W78 basic hypergeometric series which are connected with the Askey–Wilson polynomials. We study the inversion properties of these equivalence classes and also identify the connection of both sets of equivalence classes with the symmetric group S6, the symmetry group of the terminating balanced ϕ34. We then use terminating balanced ϕ34 and terminating very-well poised W78 transformations to give a broader interpretation of Watson's q-analog of Whipple's theorem and its converse.

On Another Characterization of Askey-Wilson Polynomials

In this paper we show that the only sequences of orthogonal polynomials $$(P_n)_{n\ge 0}$$ satisfying $$\begin{aligned} \phi (x){\mathcal {D}}_q P_{n}(x)=a_n{\mathcal {S}}_q P_{n+1}(x) +b_n{\mathcal {S}}_q P_n(x) +c_n{\mathcal {S}}_q P_{n-1}(x), \end{aligned}$$ ( $$c_n\ne 0$$ ) where $$\phi $$ is a well chosen polynomial of degree at most two, $${\mathcal {D}}_q$$ is the Askey-Wilson operator and $${\mathcal {S}}_q$$ the averaging operator, are the multiple of Askey-Wilson polynomials, or specific or limiting cases of them.

Monic Bivariate Polynomials on Quadratic and q-Quadratic Lattices

Monic families of bivariate Askey–Wilson polynomials and q-Racah polynomials are explicitly given. Monic families of bivariate orthogonal polynomials, both in quadratic and q-quadratic lattices, are also explicitly given using appropriate limit relations.

Moments of orthogonal polynomials and exponential generating functions

Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ( $$q=1$$ ) classical orthogonal polynomials, and study those cases in which the exponential generating function has a nice form. In the opposite direction, we show that the generalized Dumont–Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials. Finally, we show that one of our methods can be applied to deal with the moments of Askey–Wilson polynomials.

A Littlewood–Richardson rule for Koornwinder polynomials

Koornwinder polynomials are q-orthogonal polynomials equipped with extra five parameters and the $$B C_n$$ -type Weyl group symmetry, which were introduced by Koornwinder (Contemp Math 138:189–204, 1992) as multivariate analogue of Askey–Wilson polynomials. They are now understood as the Macdonald polynomials associated with the affine root system of type $$(C^\vee _n,C_n)$$ via the Macdonald–Cherednik theory of double affine Hecke algebras. In this paper, we give explicit formulas of Littlewood–Richardson coefficients for Koornwinder polynomials, i.e., the structure constants of the product as invariant polynomials. Our formulas are natural $$(C^\vee _n,C_n)$$ -analogue of Yip’s alcove-walk formulas (Math Z 272:1259–1290, 2012) which were given in the case of reduced affine root systems.

Remarks on Askey–Wilson polynomials and Meixner polynomials of the second kind

The purpose of this note is to characterize all the sequences of orthogonal polynomials \((P_n)_{n\ge 0}\) such that $$\begin{aligned} \frac{\triangle }{\mathbf{\triangle } x(s-1/2)}P_{n+1}(x(s-1/2))=c_n(\triangle +2\,\mathrm {I})P_n(x(s-1/2)), \end{aligned}$$where \(\,\mathrm {I}\) is the identity operator, x defines a class of lattices with, generally, nonuniform step-size, and \(\triangle f(s)=f(s+1)-f(s)\). The proposed method can be applied to similar and to more general problems involving the mentioned operators, in order to obtain new characterization theorems for some specific families of classical orthogonal polynomials on lattices.

Computation of Al-Salam Carlitz and Askey-Wilson Moments using Motzkin Paths

In this paper we study the moments of polynomials from the Askey scheme, and we focus on Askey-Wilson polynomials. More precisely, we give a combinatorial proof for the case where $d=0$. Their values have already been computed by Kim and Stanton in 2015, however, the proof is not completely combinatorial, which means that an explicit bijection has not been exhibited yet. In this work, we use a new combinatorial approach for the simpler case of Al-Salam-Carlitz, using a sign reversing involution that directly operates on Motzkin path. We then generalize this method to Askey-Wilson polynomials with $d=0$ only, providing the first fully combinatorial proof for that case.

On the Askey–Wilson polynomials and a $q$-beta integral

On the Askey–Wilson polynomials and a $q$-beta integral

Degenerate Sklyanin algebras, Askey–Wilson polynomials and Heun operators

The q-difference equation, the shift and the contiguity relations of the Askey–Wilson polynomials are cast in the framework of the three and four-dimensional degenerate Sklyanin algebras and . It is shown that the q-para Racah polynomials corresponding to a non-conventional truncation of the Askey–Wilson polynomials form a basis for a finite-dimensional representation of . The first order Heun operators defined by a degree raising condition on polynomials are shown to form a five-dimensional vector space that encompasses . The most general quadratic expression in the five basis operators and such that it raises degrees by no more than one is identified with the Heun–Askey–Wilson operator.

Multiple Askey–Wilson polynomials and related basic hypergeometric multiple orthogonal polynomials

We first show how one can obtain Al-Salam--Chihara polynomials, continuous dual $q$-Hahn polynomials, and Askey--Wilson polynomials from the little $q$-Laguerre and the little $q$-Jacobi polynomials by using special transformations. This procedure is then extended to obtain multiple Askey--Wilson, multiple continuous dual $q$-Hahn, and multiple Al-Salam--Chihara polynomials from the multiple little $q$-Laguerre and the multiple little $q$-Jacobi polynomials.

Terminating Basic Hypergeometric Representations and Transformations for the Askey-Wilson Polynomials.

In this survey paper, we exhaustively explore the terminating basic hypergeometric representations of the Askey–Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy.

Elements of the q-Askey Scheme in the Algebra of Symmetric Functions

The classical q-hypergeometric orthogonal polynomials are assembled into a hierarchy called the q-Askey scheme. At the top of the hierarchy, there are two closely related families, the Askey-Wilson and q-Racah polynomials. As it is well known, their construction admits a generalization leading to remarkable orthogonal symmetric polynomials in several variables. We construct an analogue of the multivariable q-Racah polynomials in the algebra of symmetric functions. Next, we show that our q-Racah symmetric functions can be degenerated into the big q-Jacobi symmetric functions, introduced in a recent paper by the second author. The latter symmetric functions admit further degenerations leading to new symmetric functions, which are analogues of q-Meixner and Al-Salam--Carlitz polynomials. Each of the four families of symmetric functions (q-Racah, big q-Jacobi, q-Meixner, and Al-Salam--Carlitz) forms an orthogonal system of functions with respect to certain measure living on a space of infinite point configurations. The orthogonality measures of the four families are of independent interest. We show that they are linked by limit transitions which are consistent with the degenerations of the corresponding symmetric functions.

A Quantum Algebra Approach to Multivariate Askey–Wilson Polynomials

AbstractWe study matrix elements of a change of basis between two different bases of representations of the quantum algebra ${\mathcal{U}}_q(\mathfrak{s}\mathfrak{u}(1,1))$. The two bases, which are multivariate versions of Al-Salam–Chihara polynomials, are eigenfunctions of iterated coproducts of twisted primitive elements. The matrix elements are identified with Gasper and Rahman’s multivariate Askey–Wilson polynomials, and from this interpretation we derive their orthogonality relations. Furthermore, the matrix elements are shown to be eigenfunctions of the twisted primitive elements after a change of representation, which gives a quantum algebraic derivation of the fact that the multivariate Askey–Wilson polynomials are solutions of a multivariate bispectral $q$-difference problem.

On Matrix Product Ansatz for Asymmetric Simple Exclusion Process with Open Boundary in the Singular Case

We study a substitute for the matrix product ansatz for asymmetric simple exclusion process with open boundary in the “singular case” $$\alpha \beta =q^N\gamma \delta ,$$ when the standard form of the matrix product ansatz of Derrida et al. (J Phys A 26(7):1493–1517, 1993) does not apply. In our approach, the matrix product ansatz is replaced with a pair of linear functionals on an abstract algebra. One of the functionals, $$\varphi _1,$$ is defined on the entire algebra, and determines stationary probabilities for large systems on $$L\ge N+1$$ sites. The other functional, $$\varphi _0,$$ is defined only on a finite-dimensional linear subspace of the algebra, and determines stationary probabilities for small systems on $$L< N+1$$ sites. Functional $$\varphi _0$$ vanishes on non-constant Askey–Wilson polynomials and in non-singular case becomes an orthogonality functional for the Askey–Wilson polynomials.