Eventually rational and m-sparse points of linear ordinary differential operators with polynomial coefficients

Let L(y) = 0 be a linear homogeneous ordinary di erential equation with polynomial coefcients. One of the general problems connected with such an equation is to nd all points a (ordinary or singular) and all formal power series ∑∞ n=0 cn(x − a) which satisfy L(y) = 0 and whose coe cient cn — considered as a function of n — has some ‘nice’ properties: for example, cn has an explicit representation in terms of n, or the sequence (c0; c1; : : :) has many zero elements, and so on. It is possible that such properties appear only eventually (i.e., only for large enough n). We consider two particular cases: 1. (c0; c1; : : :) is an eventually rational sequence, i.e., cn = R(n) for all large enough n, where R(n) is a rational function of n; 2. (c0; c1; : : :) is an eventually m-sparse sequence, where m¿2, i.e., there exists an integer N such that (cn 6= 0)⇒ (n ≡ N (modm)) for all large enough n. Note that those two problems were previously solved only ‘for all n’ rather than ‘for n large enough’, although similar problems connected with polynomial and hypergeometric sequences of coe cients have been solved completely. c © 2000 Published by Elsevier Science B.V. All rights reserved.