On sums of coefficients of Borwein type polynomials over arithmetic progressions

TL;DR

This paper derives asymptotic formulas for sums of coefficients of Borwein-type polynomials over arithmetic progressions, providing bounds that improve previous results and revealing detailed distribution properties of these coefficients.

Contribution

The authors establish sharper asymptotic bounds for sums of polynomial coefficients over arithmetic progressions, advancing understanding of Borwein-type polynomial coefficient distributions.

Findings

Derived asymptotic formulas for coefficient sums over arithmetic progressions.

Proved bounds that improve previous results by Goswami and Pantangi.

Revealed detailed distribution patterns of polynomial coefficients.

Abstract

We obtain asymptotic formulas for sums over arithmetic progressions of coefficients of polynomials of the form $$\prod_{j=1}^n\prod_{k=1}^{p-1}(1-q^{pj-k})^s,$$ where $p$ is an odd prime and $n, s$ are positive integers. Let us denote by $a_i$ the coefficient of $q^i$ in the above polynomial and suppose that $b$ is an integer. We prove that $$\Big|\sum_{i\equiv b\ \text{mod}\ 2pn}a_i-\frac{v(b)p^{sn}}{2pn}\Big|\leq p^{sn/2},$$ where $v(b)=p-1$ if $b$ divisible by $p$ and $v(b)=-1$ otherwise. This improves a recent result of Goswami and Pantangi.