  The main object of this paper is to investigate a new class of the
generalized Hurwitz type poly-Bernoulli numbers and polynomials from which we
derive some algorithms for evaluating the Hurwitz type poly-Bernoulli numbers
and polynomials. By introducing a new generalization of the Stirling numbers of
the second kind, we succeed to establish some combinatorial formulas for the
generalized Hurwitz type poly-Bernoulli numbers and polynomials with negative
upper indices. Moreover, we give a connection between the generalized Stirling
numbers of the second kind and graph theory.
