Une propriété arithmétique de certains opérateurs différentiels. (An arithmetic property of some differential operators) (Q1081639)

Define \(D=d/dx\) and let \(L(D)=\sum^{s}_{k=0}A_ k(x) D^ k\in {\mathbb{Q}}[x] [D]\) be a linear differential operator with polynomial coefficients. G. Polya for \(L(D)=D\) and D. Cantor for \(L(D)=P(xD)\), \(P\in {\mathbb{Q}}[x]\), proved that if \(u\in {\mathbb{Z}}[[ x]]\) is such that L(D)(u) is a rational function, then u is also a rational function. In this paper, we prove that the same property is true for the following class of linear differential operators: \(L(D)=\sum^{s}_{k=0}A_ k(x) D^ k\) with \(A_ 0\) a nonzero constant, \(A_ s(0)\neq 0\), and for \(k\geq 1\), the degree of \(A_ k\) is less than k-1.