Abstract:
We determine all biinfinite tridiagonal matrices for which some family of eigenfunctions are also eigenfunctions of a second order q-difference operator. The solution is described in terms of an arbitrary solution of a q-analogue of Gauss hypergeometric equation depending on five free parameters and extends the four dimensional family of solutions given by the Askey-Wilson polynomials. There is some evidence that this bispectral problem, for an arbitrary order q-difference operator, is intimately related with some q-deformation of the Toda lattice hierarchy and its Virasoro symmetries. When tridiagonal matrices are replaced by the Schroedinger operator, and q= 1, this statement holds with Toda replaced by KdV. In this context, this paper determines the analogs of the Bessel and Airy potentials.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received: 7 May 1996/Accepted: 30 August 1996
Rights and permissions
About this article
Cite this article
Grünbaum, F., Haine, L. Some Functions that Generalize the Askey–Wilson Polynomials . Comm Math Phys 184, 173–202 (1997). https://doi.org/10.1007/s002200050057
Issue Date:
DOI: https://doi.org/10.1007/s002200050057