Stirling numbers and the Pascal matrix function

A Pascal matrix function is introduced by Call and Velleman(1993). A decomposition of Stirling matrices of the first kind and the second kind by using the Pascal matrix function is given by Cheon and Kim(2001). In this paper, we introduce shift Stirling matrices and their decompositions by using the Pascal matrix.Finally, we extend our discussion to generalized Stirling numbers and their matrices, which were studied by He(2013), and Hsu and Shiue(1998). Matrix equations presented in this article reflect the relationships between the first kind, the second kind, and generalized Stirling numbers and the binomial numbers. The expressions are succinct and helpful to find more properties of Stirling numbers.