Some identities involving Prouhet-Thue-Morse sequence and its relatives

Let $s_{k}(n)$ denote the sum of digits of an 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 \begin{equation*} f_{m,n}^{\mathbf u}(x)=\sum_{i=0}^{k^{n}-1}ζ_{k}^{s_{k}(i)}(x+{\mathbf u}(i))^{m}. \end{equation*} defined for sequences ${\bf u}(i)$ satisfying a certain recurrence relation. We prove that computing these polynomials is essentially equivalent with computing their constant term and we find an explicit formula for this number. This allows us to prove several interesting identities involving sums of binary digits. We also prove some related results which are of independent interests and can be seen as further generalizations of certain sums involving Prouhet-Thue-Morse sequence.

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