On sums of coefficients of Borwein type polynomials over arithmetic progressions

摘要

We obtain asymptotic formulas for sums over arithmetic progressions of coefficients of polynomials of the form Pi(n)(j =1) Pi(p-1)(k=1) (1- q(pj-k))(s), where p is an odd prime and n, s are positive integers. Precisely, let ai denote the coefficient of q(i) in the above polynomial and suppose that b is an integer. We prove that |Sigma(i=b) (mod 2pn) a(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).

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