Separation of Variables and Integral Relations for Special Functions

We show that the method of separation of variables gives a natural generalization of integral relations for classical special functions of one variable. The approach is illustrated by giving a new proof of the quadratic integral relations for the continuous q-ultraspherical polynomials. The separating integral operator M expressed in terms of the Askey-Wilson operator is studied in detail: apart from writing down the characteristic (separation) equations it satisfies, we find its spectrum, eigenfunctions, inversion, invariants (invariant q-difference operators), and give its interpretation as a fractional q-integration operator. We also give expansions of the A₁ Macdonald polynomials into the eigenfunctions of the separating operator M and vice versa.