zbMATH — the first resource for mathematics

Une généralisation du théorème de Skolem-Mahler-Lech. (A generalization of the Skolem-Mahler-Lech theorem). (French) Zbl 0678.10040
Let K be a commutative field with characteristic zero. If d and r are two natural integers and a(n) a sequence with values in K, define \(A(d,r)=\{n;\quad a(nd+r)=0\}.\) In the paper under review the author proves the following theorem:
Let \(f(z)=\sum a(n)z^ n\) be a formal power series with coefficients in K. Suppose that f satisfies a differential equation, which is linear, homogeneous, with polynomial coefficients and such that 0 and the point at infinity are not singular-irregular. Then there exists a natural integer \(d\neq 0\), such that for every r in [0,d-1]: (a) either A(d,r) contains all large enough integers, (b) or A(d,r) has density zero.
This theorem generalizes the Skolem-Mahler-Lech theorem (for a rational power series the same as above holds where condition (b) is replaced by \((b')\) A(d,r) is finite).
The author notes that a formal power series algebraic on K[z] satisfies the hypothesis of his theorem. He also asks whether condition (b) can be replaced as in the Skolem-Mahler-Lech theorem by \((b')\) A(d,r) is finite.
Reviewer: J.-P.Allouche

11B25 Arithmetic progressions
12H99 Differential and difference algebra
Full Text: DOI