  The Stirling numbers of type $B$ of the second kind count signed set
partitions. In this paper we provide new combinatorial and analytical
identities regarding these numbers as well as Broder's $r$-version of these
numbers. Among these identities one can find recursions, explicit formulas
based on the inclusion-exclusion principle, and also exponential generating
functions.
  These Stirling numbers can be considered as members of a wider family of
triangles of numbers that are characterized using results of Comtet and
Lancaster.
  We generalize these theorems, which present equivalent conditions for a
triangle of numbers to be a triangle of generalized Stirling numbers, to the
case of the $q,r$-poly Stirling numbers, which are $q$-analogues of the
restricted Stirling numbers defined by Broder and having a polynomial value
appearing in their defining recursion. There are two ways to do this and these
ways are related by a nice identity.
