 
Summary: Eventually rational points and eventually mpoints of
linear ordinary di#erential operators
Sergei A. Abramov #
Computer Center of the Russian Academy of Science
Vavilova 40, Moscow 117967, Russia
abramov@ccas.ru,
Abstract
Let L(y) = 0 be a linear homogeneous ordinary di#erential equation with polyno
mial coe#cients. 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 c n (x  a) n
which satisfy L(y) = 0 and whose coe#cient c n  considered as a function of n  has
some ``nice'' properties: for example, c n has an explicit representation in terms of n,
or the sequence (c 0 , c 1 , . . .) 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. (c 0 , c 1 , . . .) is an eventually rational sequence, i.e., c n = R(n) for all large enough
n, where R(n) is a rational function of n;
2. (c 0 , c 1 , . . .) is an eventually msparse sequence, where m # 2, i.e., there exists an
integer N such that
