Unimodality polynomials and generalized Pascal triangles

Abstract

In this paper, we show that if \(P(x)=\sum_{k=0}^{m}a_{k}x^{k}\) is a polynomial with nondecreasing, nonnegative coefficients, then the coefficients sequence of \(P(x^{s}+\cdots +x+1)\) is unimodal for each integer \(s\geq 1\). This paper is an extension of Boros and Moll's result ``A criterion for unimodality'', who proved that the polynomial \(P(x+1)\) is unimodal.