DIAGONALIZATION OF CERTAIN INTEGRAL-OPERATORS

We study integral representations of formal inverses of the differenti al operator and the Askey-Wilson divided difference operators in diffe rent polynomial bases. The eigenvalues are related to zeros of Bessel functions, q-Bessel functions, and confluent hypergeometric functions. The eigenfunctions are exponential functions and, in the case of the Askey-Wilson operator, are new types of q-exponential functions. We al so found new q-analogs of the sine and cosine functions. (C) 1994 Acad emic Press, Inc.