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

AbstractLet L(y)=0 be a linear homogeneous ordinary differential equation with polynomial coefficients. One of the general problems connected with such an equation is to find all points a (ordinary or singular) and all formal power series ∑n=0∞cn(x−a)n which satisfy L(y)=0 and whose coefficient 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≠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 coefficients have been solved completely