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. Precisely, let \(a_i\) denote the coefficient of \(q^i\) in the above polynomial and suppose that b is an integer. We prove that \( \Big |\sum _{i\equiv b\ \mathrm{mod}\ 2pn}a_i-\frac{v(b)p^{sn}}{2pn}\Big |\le p^{sn/2},\) where \(v(b)=p-1\) if b divisible by p and \(v(b)=-1\) otherwise. This improves a recent result of Goswami and Pantangi (Ramanujan J, 2021. https://doi.org/10.1007/s11139-020-00352-0).
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This work was supported in part by the National Science Foundation of China (11771280)
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Li, J., Yu, X. On sums of coefficients of Borwein type polynomials over arithmetic progressions. Ramanujan J 59, 143–155 (2022). https://doi.org/10.1007/s11139-021-00512-w
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DOI: https://doi.org/10.1007/s11139-021-00512-w