Settling some sum suppositions (Q2416468)

In the paper under review, the authors prove some identities involving the sum of base $b$-digits function $s_b(n)$, where $b>1$ is an integer. The easiest to state is Theorem 5, which asserts that $$ \sum_{n=0}^{b^N-1} \zeta^{s_b(n)}f(s_b(n)+ny)=0$$ holds for all $b$th roots unity $\zeta$, any real number $y$ and any polynomial $f$ of degree $<N$. In particular, the paper settles some conjectures from \textit{J. Byszewski} and \textit{M. Ulas} [Acta Math. Hung. 147, No. 2, 438--456 (2015; Zbl 1374.11018)]. The method reduces to clever manipulations with generating functions.