The main goal of the present paper is to construct some families of the Carlitz's $q$-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz's $q$-Bernoulli polynomials and numbers with weight ($_{p}$. We then define the modified degenerate Carlitz's $q$-Bernoulli polynomials and numbers with weight ($alpha ,beta $) and obtain some recurrence relations and other identities...

In this work we give an alterative proof of one of basic properties of zonal polynomials and generalised it for Jack polynomials

The notion of a descent polynomial, function in enumerative combinatorics that counts permutations with specific properties, enjoys revived recent research interest due to its connection other important notions combinatorics, viz. peak polynomials and symmetric functions. We define the dm(I,n) as generalization polynomial we prove for any positive integer m, this is n sufficiently large (simila...

In this paper we establish several polynomials similar to Bernstein's polynomials and several refinements of  Hermite-Hadamard inequality for convex functions.

let g = (v, e) be a simple graph. hosoya polynomial of g isd(u,v)h(g, x) = {u,v}v(g)x , where, d(u ,v) denotes the distance between vertices uand v. as is the case with other graph polynomials, such as chromatic, independence anddomination polynomial, it is natural to study the roots of hosoya polynomial of a graph. inthis paper we study the roots of hosoya polynomials of some specific graphs.

in this paper, we give some determinantal and permanental representations of generalized lucas polynomials, which are a general form of generalized bivariate lucas p-polynomials, ordinary lucas and perrin sequences etc., by using various hessenberg matrices. in addition, we show that determinant and permanent of these hessenberg matrices can be obtained by using combinations. then we show, the ...

The edge detour index polynomials were recently introduced for computing the edge detour indices. In this paper we find relations among edge detour polynomials for the 2-dimensional graph of $TUC_4C_8(S)$ in a Euclidean plane and $TUC4C8(S)$ nanotorus.

In this work we study numbers and polynomials generated by two type of composition of generating functions and get their explicit formulae. Furthermore we state an improvementof the composita formulae's given in [6] and [3], using the new composita formula's we construct a variety of combinatorics identities. This study go alone to dene new family of generalized Bernoulli polynomials which incl...