THE RAMANUJAN JOURNAL, Vol. 3, No. 1 (1999), 5-35.

Separation of variables and integral relations for special functions

V. K. KUZNETSOV
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, U.K.
vadim@amsta.leeds.ac.uk

E. K. SKLYANIN
Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191011, Russia
sklyanin@pdmi.ras.ru

Received November 5, 1996; Accepted June 9, 1998

Abstract. 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 $q$-ultaspherical 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 $A_1$ Macdonald polynomials into the eigenfunctions of the separating operator $M$ and vice versa.

Keywords: product formulas, method of separation of variables, orthogonal polynomials, factorization of polynomials

1991 Mathematics Classification: Primary 33D05, 58F07; Secondary 44A20, 47B38, 47G10