Some combinatorial applications of the q-Riordan matrix

다국어 초록 (Multilingual Abstract)

Recently, the authors developed a q-analogue for Riordan matrices by means of Eulerian generating functions of the form g(z)=@?<SUB>n≥0</SUB>g<SUB>n</SUB>z<SUP>n</SUP>/n!<SUB>q</SUB> where n!<SUB>q</SUB> 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)-Bessel numbers of both types. By setting r=0 and letting 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.