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

By definition, the coefficient sequence c = (cn) 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 z0 ∈ C of L is a d’Alembertian series, then the expansion of u(z) is of the same type at any ordinary ∗Supported by RFBR grant 07-01-00482-a.