  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}\zeta_{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.
