Multivariable Askey-Wilson Polynomials Research Articles

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.

A bispectral q-hypergeometric basis for a class of quantum integrable models

For the class of quantum integrable models generated from the q−Onsager algebra, a basis of bispectral multivariable q−orthogonal polynomials is exhibited. In the first part, it is shown that the multivariable Askey-Wilson polynomials with N variables and N + 3 parameters introduced by Gasper and Rahman [Dev. Math. 13, 209 (2005)] generate a family of infinite dimensional modules for the q−Onsager algebra, whose fundamental generators are realized in terms of the multivariable q−difference and difference operators proposed by Iliev [Trans. Am. Math. Soc. 363, 1577 (2011)]. Raising and lowering operators extending those of Sahi [SIGMA 3, 002 (2007)] are also constructed. In the second part, finite dimensional modules are constructed and studied for a certain class of parameters and if the N variables belong to a discrete support. In this case, the bispectral property finds a natural interpretation within the framework of tridiagonal pairs. In the third part, eigenfunctions of the q−Dolan-Grady hierarchy are considered in the polynomial basis. In particular, invariant subspaces are identified for certain conditions generalizing Nepomechie’s relations. In the fourth part, the analysis is extended to the special case q = 1. This framework provides a q−hypergeometric formulation of quantum integrable models such as the open XXZ spin chain with generic integrable boundary conditions (q ≠ 1).

Integrable Boundary Interactions for Ruijsenaars’ Difference Toda Chain

We endow Ruijsenaars’ open difference Toda chain with a one-sided boundary interaction of Askey–Wilson type and diagonalize the quantum Hamiltonian by means of deformed hyperoctahedral q-Whittaker functions that arise as a t = 0 degeneration of the Macdonald–Koornwinder multivariate Askey–Wilson polynomials. This immediately entails the quantum integrability, the bispectral dual system, and the n-particle scattering operator for the chain in question.

Multivariable Askey–Wilson function and bispectrality

For every positive integer d, we define a meromorphic function F d (n;z), where n,z∈ℂ d , which is a natural extension of the multivariable Askey–Wilson polynomials of Gasper and Rahman (Theory and Applications of Special Functions, Dev. Math., vol. 13, pp. 209–219, Springer, New York, 2005). It is defined as a product of very-well-poised 8 φ 7 series and we show that it is a common eigenfunction of two commutative algebras ${\mathcal{A}}_{z}$ and ${\mathcal{A}}_{n}$ of difference operators acting on z and n, with eigenvalues depending on n and z, respectively. In particular, this leads to certain identities connecting products of very-well-poised 8 φ 7 series.

Bispectral commuting difference operators for multivariable Askey-Wilson polynomials

We construct a commutative algebra A z \mathcal A_{z} , generated by d d algebraically independent q q -difference operators acting on variables z 1 , z 2 , … , z d z_1,z_2,\dots ,z_d , which is diagonalized by the multivariable Askey-Wilson polynomials P n ( z ) P_n(z) considered by Gasper and Rahman (2005). Iterating Sears’ 4 ϕ 3 {}_4\phi _3 transformation formula, we show that the polynomials P n ( z ) P_n(z) possess a certain duality between z z and n n . Analytic continuation allows us to obtain another commutative algebra A n \mathcal A_{n} , generated by d d algebraically independent difference operators acting on the discrete variables n 1 , n 2 , … , n d n_1,n_2,\dots ,n_d , which is also diagonalized by P n ( z ) P_n(z) . This leads to a multivariable q q -Askey-scheme of bispectral orthogonal polynomials which parallels the theory of symmetric functions.

An asymptotic formula for the Koornwinder polynomials

A formula for the large-degree asymptotics of Koornwinder's multivariate Askey–Wilson polynomials is presented. In the special case of a single variable, this asymptotic formula agrees with the known leading asymptotics of the Askey–Wilson polynomials determined by Ismail and Wilson.

An asymptotic formula for the Koornwinder polynomials

A formula for the large-degree asymptotics of Koornwinder's multivariate Askey-Wilson polynomials is presented. In the special case of a single variable, this asymptotic formula agrees with the known leading asymptotics of the Askey-Wilson polynomials determined by Ismail and Wilson.

On $BC$ type basic hypergeometric orthogonal polynomials

The five parameter family of multivariable Askey-Wilson polynomials is studied with four parameters generically complex. The multivariable Askey-Wilson polynomials form an orthogonal system with respect to an explicit (in general complex) measure. A partially discrete orthogonality measure is obtained by shifting the contour to the torus while picking up residues. A parameter domain is given for which the partially discrete orthogonality measure is positive. The orthogonality relations and norm evaluations for multivariable q-Racah polynomials and multivariable big and little q-Jacobi polynomials are proved by taking suitable limits in the orthogonality relations for the multivariable Askey-Wilson polynomials. In particular new proofs of several well known q-analogues of the Selberg integral are obtained.