On the moment problem and the linear difference equations with periodic coefficients

Abstract

The aim of this paper is to study the problem of moments linked to real sequences that are solutions of a linear difference equation of order r, whose coefficients are variable and periodic of period $p\ge 2$. We generate subsequences of such a sequence which satisfy a new linear recurrence equation with constant coefficients. Therefore, through the distributional formulation of these subsequences, new results related to the corresponding moment problem are established. Furthermore, some open questions on the connection between the moment problem for the original recursive sequence with periodic coefficients, and the moment problem for its related p recursive subsequences with constant coefficients, have been raised. To offer an initial perspective on our problem, the special cases $r=1$, $p=2$ and $r=2$, $p=2$ are exposed, and illustrative numerical examples are provided.