The Akiyama-Tanigawa matrix and related combinatorial identities

In this paper, we define the generalized Akiyama-Tanigawa matrix and investigate its relationship with the r-Stirling numbers of the second kind. Also, we give the generating function for any column or row of the generalized Akiyama-Tanigawa matrix. As a result, we find some new identities about r-Stirling numbers of the second kind. In addition, some applications are given.