Some identities involving the Prouhet–Thue–Morse sequence and its relatives

Abstract

Let s _k (n) be the sum of digits of the expansion of the integer n in base k. Motivated by certain identities of Nieto, and Bateman and Bradley involving sums of the form ${\sum_{i=0}^{2^{n}-1}{(-1)}^{s_{2}(i)}{(x+i)}^{m}}$ for m = n and m = n + 1, we consider the sequence of polynomials

$$f_{m,n}^{\mathbf u}(x)=\sum_{i=0}^{k^{n}-1} \zeta_{k}^{s_{k}(i)} {(x+{\mathbf u}(i))}^{m}.$$

Here, the sequence ${{\mathbf{u}}}$ satisfies the recurrence relation

$$\mathbf{u}(ki+j)=P (\mathbf{u}(i)) +jq\quad\mbox{for} \quad t=0,1,\ldots,k-1,$$

and ${q \in V}$, where V is a finitely dimensional vector space over the field K and ${P\: V \to V}$ is a linear endomorphism. Moreover, ${\zeta_{k} \neq 1}$ is a k-th root of unity. We prove that computing the polynomials ${f_{m,n}^{\mathbf{u}}}$ is essentially equivalent with computing its constant term and we find an explicit formula for this number. This allows us to prove several interesting identities involving the sum of (binary) digits function. We also prove some related results which are of independent interests and can be seen as a further generalization of certain sums involving the Prouhet–Thue–Morse sequence.