On the sum of the values of a polynomial at natural numbers which form a decreasing arithmetic progression

TL;DR

This paper investigates when sums of polynomial values at decreasing arithmetic sequences are themselves polynomial, characterizes the relevant polynomial spaces, and explores generalizations via formal power series, inspired by a historical formula.

Contribution

It characterizes polynomial spaces where sums over decreasing arithmetic progressions are polynomial, with detailed analysis for the case d=2 and generalizations using formal power series.

Findings

Characterization of polynomial spaces ${\mathscr{E}}_d$ for sums to be polynomial

Detailed study of the case d=2

Generalization of results via formal power series

Abstract

The purpose of this paper consists to study the sums of the type $P(n) + P(n - d) + P(n - 2 d) + \dots$, where $P$ is a real polynomial, $d$ is a positive integer and the sum stops at the value of $P$ at the smallest natural number of the form $(n - k d)$ ($k \in \mathbb{N}$). Precisely, for a given $d$, we characterize the $\mathbb{R}$-vector space ${\mathscr{E}}_d$ constituting of the real polynomials $P$ for which the above sum is polynomial in $n$. The case $d = 2$ is studied in more details. In the last part of the paper, we approach the problem through formal power series; this inspires us to generalize the spaces $\mathscr{E}_d$ and the underlying results. Also, it should be pointed out that the paper is motivated by the curious formula: $n^2 + (n - 2)^2 + (n - 4)^2 + \dots = \frac{n (n + 1) (n + 2)}{6}$, due to Ibn al-Banna al-Marrakushi (around 1290).