On the Relation Between Global Properties of Linear Difference and Differential Equations with Polynomial Coefficients, 1

Abstract

This paper is concerned with applications of the Mellin transformation in the study of homogeneous linear differential and difference equations with polynomial coefficients. We begin by considering a differential equation (D) with regular singularities at O and ∞ and arbitrary singularities in the rest of the complex plane, and the difference equation (Δ′) obtained from (D) by a variant of the formal Mellin transformation. We define fundamental systems of solutions of (Δ′), analytic in either a right or a left half plane. by the use of Mellin transforms of microsolutions of (D). The relations between these fundamental systems are expressed in terms of central connection matrices of (D). Second, we study the differential equation (D1) obtained from (D) by means of a formal Laplace transformation and the difference equation (Δ1) obtained from (D1) by a formal Mellin transformation. We use Mellin transforms of "ordinary" solutions of (D1) with moderate growth at ∞ to construct fundamental systems of solutions of (Δ1). The relation between these fundamental systems involves certain Stokes multipliers and a formal monodromy matrix of (D1).