On sums of coefficients of Borwein type polynomials over arithmetic progressions

We obtain asymptotic formulas for sums over arithmetic progressions of coefficients polynomials 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$$ denote coefficient $$q^i$$ in above polynomial suppose that b integer. prove \Big |\sum _{i\equiv b\ \mathrm{mod}\ 2pn}a_i-\frac{v(b)p^{sn}}{2pn}\Big |\le p^{sn/2},$$ $$v(b)=p-1$$ if divisible by $$v(b)=-1$$ otherwise. This improves a recent result Goswami Pantangi (Ramanujan J, 2021. https://doi.org/10.1007/s11139-020-00352-0 ).