The rational Heun operator and Wilson biorthogonal functions

Abstract

We consider the rational Heun operator defined as the most general second-order q-difference operator which sends any rational function of type \([(n-1)/n]\) to a rational function of type \([n/(n+1)]\). We shall take the poles to be located on the Askey–Wilson grid. It is shown that this operator is related to the one-dimensional degeneration of the Ruijsenaars–van Diejen Hamiltonians. The Wilson biorthogonal functions of type \({_{10}}\Phi _9\) are found to be solutions of a generalized eigenvalue problem involving rational Heun operators of the special “classical” kind.