Multivariable Askey–Wilson function and bispectrality

Abstract

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.