  The first aim of this paper is to construct new generating functions for the
generalized {\lambda}-Stirling type numbers of the second kind, generalized
array type polynomials and generalized Eulerian type polynomials and numbers,
attached to Dirichlet character. We derive various functional equations and
differential equations using these generating functions. The second aim is
provide a novel approach to deriving identities including multiplication
formulas and recurrence relations for these numbers and polynomials using these
functional equations and differential equations. Furthermore, by applying
p-adic Volkenborn integral and Laplace transform, we derive some new identities
for the generalized {\lambda}-Stirling type numbers of the second kind, the
generalized array type polynomials and the generalized Eulerian type
polynomials. We also give many applications related to the class of these
polynomials and numbers.
