Unimodality of regular partition polynomials

Let n, p and j be integers. Define \(\begin{aligned} R_{n,p,j}(q):=\prod _{k=0}^{n}(1+q^{pk+1})(1+q^{pk+2})\cdots (1+q^{pk+j}). \end{aligned}\) $$\begin{array}{c}\hfill R_{n,p,j}(q):=\prod _{k=0}^n(1+q^{pk+1})(1+q^{pk+2})\cdots (1+q^{pk+j}).\end{array}$$ The coefficients of the polynomial \(R_{n,p,j}(q)\) $R_{n,p,j}(q)$ count certain regular partition. Recently, Dong and Ji studied unimodality of the polynomials \(R_{n,p,p-1}(q)\) $R_{n,p,p-1}(q)$ . As an extension, in this paper, we give a criterion for unimodality of the polynomials \( R_{n,p,j}(q)\) $R_{n,p,j}(q)$ for \(p \ge 6\) $p\ge 6$ and \(\lceil \frac{p+1}{2}\rceil \le j\le p-1.\) $\lceil \frac{p+1}{2}\rceil \le j\le p-1.$ In particular, using our criterion and Mathematica, we obtain that \(R_{n,p,j}(q)\) $R_{n,p,j}(q)$ is unimodal for \(n\ge 3\) $n\ge 3$ if \(6\le p \le 15\) $6\le p\le 15$ and \(\lceil \frac{p+1}{2}\rceil \le j\le p-1.\) $\lceil \frac{p+1}{2}\rceil \le j\le p-1.$