D’Alembertian series solutions at ordinary points of LODE with polynomial coefficients

Abstract

By definition, the coefficient sequence c = ( c n ) of a d’Alembertian series — Taylor’s or Laurent’s — satisfies a linear recurrence equation with coefficients in C ( n ) and the corresponding recurrence operator can be factored into first-order factors over C ( n ) (if this operator is of order 1, then the series is hypergeometric). Let L be a linear differential operator with polynomial coefficients. We prove that if the expansion of an analytic solution u ( z ) of the equation L ( y ) = 0 at an ordinary (i.e., non-singular) point z 0 ∈ C of L is a d’Alembertian series, then the expansion of u ( z ) is of the same type at any ordinary point. All such solutions are of a simple form. However the situation can be different at singular points.