On sums of coefficients of Borwein type polynomials over arithmetic progressions (Q2168696)

In this paper the authors consider the class of polynomials of the form \(\prod_{j=1}^n\prod_{k=1}^{p-1}(1-q^{pj-k})\), where \(p\) is an odd prime and \(n\) and \(s\) are positive integers. Let \(a_i\) denote the coefficient of \(q^i\) in the above polynomial and let \(S_{d,b}=\sum_{i\equiv b(\text{mod d)}} a_i\), where \(d\) is a positive integer divisible by \(p\) and \(b\) is an integer. Obviously, \(S_{d,b}\) represents a sum of \(a_i\) over an arithmetic progression. The estimation of \(S_{d,b}\) for several values of \(d\) is closely connected with the so-called Borwein conjectures [\textit{G. E. Andrews}, J. Symb. Comput. 20, No. 5--6, 487--501 (1995; Zbl 0849.68062)]. For instance, estimates of \(S_{d,b}\) for \(d=3n\) and \(d=pn\) are obtained recently. The main result of this paper concerns arithmetic progressions with a larger common difference \(d=2pn\). From the authors' abstract: ``We prove that \(|\sum_{i\equiv b(\text{mod 2pn)}} a_i-\frac{v(b)p^{sn}}{2pn}|\le p^{sn/2}\), where \(v(b)=p-1\) if \(b\) divisible by \(p\) and \(v(b)=-1\) otherwise.''