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-differe nce operator. The solution is described in terms of an arbitrary solut ion 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 thi s bispectral problem, for an arbitrary order q-difference operator, is intimately related with some q-deformation of the Toda lattice hierar chy and its Virasoro symmetries. When tridiagonal matrices are replace d by the Schroedinger operator, and q = 1, this statement holds with T oda replaced by KdV. In this context, this paper determines the analog s of the Bessel and Airy potentials.