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

S. A. Abramov, Mohammed A. Barkatou

• By definition, the coefficient sequence of a d�Alembertian series � Taylor�s or Laurent�s � satisfies a linear recurrence equation with coefficients in and the corresponding recurrence operator can be factored into first-order factors over (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 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