Proof of a conjecture about unimodal polynomials
The author investigates the well-known conjecture that the polynomial \(\prod^{r}_{\nu =1}(1-t^{n\nu})/(1-t^{\nu})\) is symmetric unimodal if n is even and \(r\geq 1\), or n is odd and \(r\geq 11\). It is proved that conjecture is true for \(3\leq n\leq 20\) and for \(n=100\) and \(n=101\). The method of proof makes use of a clever induction for ``large'' r, say \(r\geq r_ 0(n)\), where \(r_ 0(n)\approx 10\) for large n. As by-product in the course of proof, two elegant inequalities for Chebyshev polynomials are obtained, namely \(| U_{n-1}(x)| \leq n^{x^ 2}\) and \(| T_ n'(x)| \leq n^{1+x^ 2}(| x| \leq 1).\)
scientific article; zbMATH DE number 4110695
scientific article