On a Relationship between Pascal Matrix and Vandermonde Matrix

Abstract:

      EI-Mikkawy M obtained that the symmetric Pascal matrix $Q_n$ and the Vandermonde matrix $V_n$ are connected by the equation $Q_n=T_nV_n $, where $T_n$ is a stochastic matrix in [1]. In this paper, a decomposition of the matrix $T_n$ is given via the Stirling matrix of the first kind, and a recurrence relation of the elements of the matrix $ T_n $ is obtained, so an open problem proposed by EI-Mikkawy$^{[2]}$ is solved. Some combinatorial identities are also given.