Some functions that generalize the Askey-Wilson polynomials

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.