Some Formulas for Generalized Stirling Numbers

Abstract

In this paper, we look at generalizations of Stirling numbers which arise for arbitrary integer sequences and their $k$-th powers. This can be seen as a complementary strategy to the unified approach suggested in [9]. The investigations of [3] and [14] present a more algebraically oriented approach to generalized Stirling numbers.

In the first and second sections of the paper, we give the corresponding formulas for the generalized Stirling numbers of the second and first kind, respectively. In the third section, we briefly discuss some examples and special cases, and in the last section, we apply the square case to facilitate a counting approach for set partitions of even size.