A Difference Differential Equation of Euler–Cauchy Type

We study a class of advanced argument linear difference differential equations analogous to Euler–Cauchy ordinary differential equations. Solutions of two equations of this type have arisen as adjoint functions in sieve theory, and they are also of use in control theory. Here we study the problem in a general setting. Subject to mild assumptions, each of our equations is shown to have a unique solution which is analytic in the right half-plane. In some cases the solution is a polynomial, and in others it has an asymptotic expansion. Finally, the solution is shown to have a representation as an exponential of a Hellinger type integro-differential operator acting on a monomial.