Stirling matrix via Pascal matrix Original Research Article

The Pascal-type matrices obtained from the Stirling numbers of the first kind s(n,k) and of the second kind S(n,k) are studied, respectively. It is shown that these matrices can be factorized by the Pascal matrices. Also the LDU-factorization of a Vandermonde matrix of the form Vn(x,x+1,…,x+n−1) for any real number x is obtained. Furthermore, some well-known combinatorial identities are obtained from the matrix representation of the Stirling numbers, and these matrices are generalized in one or two variables.