On indefinite sums weighted by periodic sequences

Abstract: For any integer $q\geq 2$ we provide a formula to express indefinite sums of a sequence $(f(n)){n\geq 0}$ weighted by $q$-periodic sequences in terms of indefinite sums of sequences $(f(qn+p)){n\geq 0}$, where $p\in{0,\ldots,q-1}$. 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.