Polynomial solutions to Cauchy problems for complex Bessel operators

Abstract

We investigate Cauchy problems for complex differential equations of the form , where is a Bessel differential operator in the “time variable”, and a linear differential operator in the “space variables”, possibly also involving Bessel operators. We establish conditions for existence and uniqueness of polynomial solutions whenever the Cauchy data is polynomial, and we give explicit formulas for these solutions. When the Cauchy data consists of monomials, these polynomial solutions are analogous to the heat polynomials for the heat equation.