In the following, we provide
another proof (Theorem 3.1 below) of recent results of Harris-Sibuya, using some
elementary commutative algebra. Our purpose is to give a uniform treatment for
their results which also permits some generalization. We note that the study of
differential equations under the hypothesis that the solutions satisfy an algebraic
relation is not new. Fano, among others, made a systematic study of this situation in
the last century. Also Lamé equations in which two solutions have a rational
function as their product have proved to be a good source of examples for unusual
arithmetic behavior. But in the case of Harris-Sibuya, as well as the present
paper, the solutions need not be solutions of the same linear equation. In the
treatment below the differential equation only enters in dilineating a type of
recursion.
