Unimodality polynomials and generalized Pascal triangles

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.