  In this paper we investigate special generalized Bernoulli polynomials with
a,b,c parameters that generalize classical Bernoulli numbers and polynomials.
The present paper deals with some recurrence formulae for the generalization of
poly-Bernoulli numbers and polynomials with a,b,c parameters. Poly-Bernoulli
numbers satisfy certain recurrence relationships which are used in many
computations involving poly-Bernoulli numbers. Obtaining a closed formula for
generalization of poly-Bernoulli numbers with a,b,c paramerers therefore seems
to be a natural and important problem. By using the generalization of
poly-Bernoulli polynomials with a,b,c parameters of negative index we define
symmetrized generalization of poly-Bernoulli polynomials with a,b parameters of
two variables and we prove duality property for them. Also by stirling numbers
of the second kind we will find a closed formula for them. Furthermore we
generalize the Arakawa-Kaneko Zeta functions and by using the Laplace-Mellin
integral, we define generalization of Arakawa-Kaneko Zeta functions with a,b
parameters and we obtain an interpolation formula for the generalization of
poly-Bernoulli numbers and polynomials with a,b parameters. Furthermore we
present a link between this type of Zeta functions and Dirichlet series. By our
interpolation formula, we will interpolate the generalization of Arakawa-Kaneko
Zeta functions with a,b parameters.

Number Theory;Combinatorics