Sur une propriet\'e des polyn\^omes de Stirling

Abstract

In this article, we give a positive answer to a question posed in 1960 by D.S. Mitrinovi\'{c} and R.S. Mitrinovi\'{c} (see: D.S. Mitrinovi\'{c} et R.S. Mitrinovi\'{c}, Tableaux qui fournissent des polyn\^{o}mes de Stirling, Publications de la Facult\'{e} d'Electronique, s\'{e}rie: Math\'{e}matiques et physique, 34, (1960).1-23.) concerned the Stirling numbers of the first kind $s(n,k).$ We prove that for all $k\geq 2$ there exist an integer $m_{k}$ and a primitive polynomial $P_{k}(x)$ in $\mathbb{Z}[x]$ such that for all $n\geq k$, $s(n,n-k)=\frac{1}{m_{k}}\binom{n}{k+1}\left(n(n-1)\right) ^{\mathop{\rm mod}\nolimits (k,2)}P_{k}(n)$. Moreover for all $k\geq1$, $P_{2k}(0)=P_{2k+1}(0)$.