A Combinatorial Approach for q-Analogue of r-Stirling Numbers

We define a q-analogue of r-Stirling numbers of the second kind using their combinatorial interpretation in terms of set partition. Some properties are obtained including recurrence relation, explicit formula and certain symmetric formula. Moreover, a q-analogue of r-Stirling numbers of the first kind is introduced to obtain a q-analogue of the orthogonality and inverse relations of the two kinds of r-Stirling numbers.