Diagonalization of Certain Integral Operators

Abstract We study integral representations of formal inverses of the differential operator and the Askey-Wilson divided difference operators in different 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 also found new q -analogs of the sine and cosine functions.