On the theorem of Skolem-Mahler-Lech (Q803183)

The theorem of Skolem-Mahler-Lech roughly says that zero Taylor coefficients of an exponential polynomial (i.e. a finite sum of products of a polynomial by an exponential function \(e^{\beta x})\) lie in arithmetic progressions. Here the same result is obtained for a larger class of functions, namely the ``exponential'' function becoming power series \(\sum x^n/(n!)^m\) and ``polynomials'' becoming power series of \(\bar{\mathbb Q}[[x]]\) with a greater than or equal to \(1\) \(p\)-adic radius of convergence for an infinite set of primes \(p\) and all embeddings of \(\bar{\mathbb Q}\) in \(\mathbb C_p\). Moreover, in the case where only one product is involved the set of zero coefficients is proved to be finite. Proofs are based on a characterization of \(p\)-adic locally analytic functions by means of their interpolation coefficients. Some gaps in a previous paper of the second author [Expo. Math. 7, 137--187 (1989; Zbl 0695.10006)] on the same subject are filled.