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

TL;DR

This paper explores identities involving the Prouhet-Thue-Morse sequence and related sums, providing explicit formulas and generalizations for polynomial sums with digit sum weights, extending previous identities and uncovering new mathematical relations.

Contribution

It introduces a new class of polynomials linked to digit sum sequences, proves their properties, and derives explicit formulas and identities involving the Prouhet-Thue-Morse sequence.

Findings

Explicit formula for the constant term of the polynomials.

New identities involving binary digit sums.

Generalizations of sums related to the Prouhet-Thue-Morse sequence.

Abstract

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.