On the Relation between Global Properties of Linear Difference and Differential Equations with Polynomial Coefficients, II

Abstract

This paper is the second one in a series of three articles dealing with applications of the Mellin transformation to the theory of linear differential and difference equations with polynomial coefficients. In the previous part, we studied the case of a differential equation having at most regular singularities atOand ∞ and arbitrary singularities in the rest of the complex plane. This second part is concerned with differential equations having a regular singularity at ∞ and an irregular one at the origin of the complex plane. Using particular types of Mellin (or Pincherle) transforms of appropriate solutions of the differential equation, we construct two fundamental systems of solutions of an associated difference equation. Both fundamental systems admit the same asymptotic representation as the variable tends to ∞, one in the right half plane and the other in the left half plane. We discuss the corresponding connection problem and its relation to the Stokes phenomenon of the differential equation.