Sur une proprieté des polynômes de Stirling

Abstract

In this article, we give a positive answer to a question posed in 1960 by D.S. Mitrinović and R.S. Mitrinović (see: D.S. Mitrinović et R.S. Mitrinović, Tableaux qui fournissent des polynômes de Stirling, Publications de la Faculté d'Electronique, série: Mathématiques et physique, 34, (1960).1-23.) concerned the Stirling numbers of the first kind $s(n,k).$ We prove that for all $k\ge 2$ there exist an integer $m_k$ and a primitive polynomial $P_k(x)$ in $\mathbb{Z}[x]$ such that for all $n\ge k$, $s(n,n-k)=\frac{1}{m_k}\left(\genfrac{}{}{0ex}{}{n}{k+1}\right){\left(n(n-1)\right)}^{\mathrm{mod}(k,2)}{P}_k(n)$. Moreover for all $k\ge 1$, $P_{2k}(0)=P_{2k+1}(0)$.

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