Abstract. 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 parameters 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 polyBernoulli polynomials with a; b; c 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, define generalization of Arakawa-Kaneko Zeta functions with a,b,c parameters and obtain an interpolation formula for the generalization of poly-Bernoulli numbers and polynomials with a,b,c 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,c parameters.Mathematics subject classification (2010): 11B73, 11A07.

In this paper, we establish more properties of generalized poly-Euler polynomials with three parameters and we investigate a kind of symmetrized generalization of poly-Euler polynomials. Moreover, we introduce a more general form of multipoly-Euler polynomials and obtain some identities parallel to those of the generalized poly-Euler polynomials.

Let G = V E be a simple connected graph. The distance between two vertices of G is defined to be the length of the shortest path between the two vertices. There are topological indices assigned to G and based on the distance function which are invariant under the action of the automorphism group of G. Some important indices assigned to G are the Wiener, Szeged, and PI index which we will find them for a certain chemical graph called dendrimer nanostar.

Abstract. In the present paper, we introduce modified Dirichlet's type of twisted q -Euler polynomials with weight α . We apply the method of generating function and p -adic q -integral representation on Z p , which are exploited to derive further classes of q -Euler numbers and polynomials. Our new generating function possess a number of interesting properties which we state in this paper.

In this paper we extend the Dirichlet integral formula of Lobachevsky.