Euler-Cauchy Type Differential Difference Equations

Abstract. Since Dickman (1930), it has been known that differential difference equations arise in the study of certain problems in analytic number theory. Here, we examine a class of advanced argument differential difference equations analogous to Euler-Cauchy ordinary differential equations. Solutions of certain equations of this type have arisen as adjoint functions in sieve theory, and in the problem of finding the asymptotic average length of the longest cycle of a random permutation. Subject to mild assumptions, each of our equations is shown to have a unique solution which is analytic in the right half-plane. The solution has a compact representation as an exponential of a Hellinger type integro-differential operator acting on a monomial.