On indefinite sums weighted by periodic sequences

Abstract

For any integer $q\ge 2$ we provide a formula to express indefinite sums of a sequence $(f(n){)}_{n\ge 0}$ weighted by $q$-periodic sequences in terms of indefinite sums of sequences $(f(qn+p){)}_{n\ge 0}$, where $p\in \{0,\dots ,q-1\}$. When explicit expressions for the latter sums are available, this formula immediately provides explicit expressions for the former sums. We also illustrate this formula through some examples.