On the Determinants of the Square-Type Stirling Matrix and Bell Matrix

Abstract

We study determinants of the square-type Stirling matrix $S^{\ast }$ and the square-type Bell matrix $B^{\ast }$ . For this purpose, we prove that $S^{\ast }$ and $B^{\ast }$ have LU factorizations $S^{\ast }=L_S{U}_S$ and $B^{\ast }=L_B{U}_B$ where the diagonal entries of $U_S$ are $k^{k-1}$ , while those of $U_B$ are $k!$ ( $k\ge 1$ ).