On sums of coefficients of Borwein type polynomials over arithmetic progressions

We obtain asymptotic formulas for sums over arithmetic progressions of coefficients of polynomials of the form ∏ _j=1^n∏ _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 |∑ _i≡ b mod 2pna_i-v(b)p^sn/2pn |≤ 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 ).