A q, r-analogue for the Stirling numbers of the second kind of Coxeter groups of type B

A generalization of the Stirling numbers of the second kind of type B is given in two different directions. One generalization is via their q-analogue and the other one uses r distinguished elements. Both directions are explained and proved in a combinatorial way using generalized restricted growth words which we define here for type B. Moreover, we present their ordinary and exponential generating functions, where the exponential generating function is also used to present the r-variant as connection constants between two bases of ℝ[x].