Some combinatorial applications of the q-Riordan matrix

Recently, the authors developed a q  -analogue for Riordan matrices by means of Eulerian generating functions of the form g(z)=∑n≥0gnzn/n!qg(z)=∑n≥0gnzn/n!q where n!qn!q is the q-factorial. We apply this concept to give q  -analogues of some familiar objects from the set partitions with double restrictions on blocks, namely the (r,s)(r,s)-Bessel numbers of both types. By setting r=0r=0 and letting s→∞s→∞, these numbers may be reduced to the q-Stirling numbers of both kinds. Several algebraic formulas for the q-analogues are also derived using combinatorial methods together with the concept of q-Riordan matrices. In particular, q-analogues of the classical Bessel numbers of both kinds and their combinatorial interpretations are obtained.