Some Theorems on the q-Analogue of the Generalized Stirling Numbers

In this paper, we establish more properties for the q-analogue of the unied generalization of Stirling numbers including the vertical and horizontal recurrence relations, and the rational generating function. This generating function plays an important role in deriving one of the explicit formulas in symmetric function form which will be used in giving combinatorial interpretations of the q-analogue in the context of 0-1 tableau. Moreover, using the combinatorics of 0-1 tableaux, we obtain certain generalization of Carlitz identity.