On a connection between the Pascal, Stirling and Vandermonde matrices (Q2381822)

The authors study some relations between the Stirling matrix \(S_n\) and the Pascal matrix \(P_n\), in addition to the ones that appeared in the work of \textit{M. El-Mikkawy} [Appl. Math. Comp. 145, 23--32 (2003; Zbl 1045.15013)]. They also consider the representation of the stochastic matrix \(T_n\) and \(T_n^{-1}\) in terms of the Stirling number \(s_n\) and \(S_n\). Hence, they give an answer to an open problem proposed by \textit{M. El-Mikkawy} [Appl. Math. Comput. 146, 759--769 (2003; Zbl 1045.15014)].