On a connection between the Pascal, Stirling and Vandermonde matrices

In this paper, we are going to study some additional relations between the Stirling matrix SnSn and the Pascal matrix PnPn. Also the representation for the matrix TnTn and Tn-1 in terms of snsn and SnSn will be considered. Consequently, this will give an answer to an open problem proposed by EI-Mikkawy [On a connection between the Pascal, Vandermonde and Stirling matrices—II, Appl. Math. Comput. 146 (2003) 759–769].