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Let n,k be the positive integers, and let S_{k}(n) be the sums of the k-th power of positive integers up to n. By means of that we consider the evaluation of the sum of more general series by Bernstein polynomials. Additionally we show the reality of our idea with some examples., Comment: 6 pages, submitted

In the present paper, we deal mainly with arithmetic properties of Legendre polynomials by using their orthogonality property. We show that Legendre polynomials are proportional with Bernoulli, Euler, Hermite and Bernstein polynomials.

In the paper, using the extended fermionic $p$-adic integral on $\mathbb{Z}_p$, the authors find some applications of the umbral calculus. From these applications, the authors derive some identities on the weighted Euler numbers and polynomials. In other words, the authors investigate systematically the class of Sheffer sequences in connection with the generating function of the weighted Euler polynomials., Comment: 10 pages

The fundamental objective of this paper is to obtain some interesting properties for $\left(h,q\right)$-Genocchi numbers and polynomials by using the fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ and mentioned in the paper $q$-Bernstein polynomials. By considering the $q$-Euler zeta function defined by T. Kim, which can also be obtained by applying the Mellin transformation to the generating function of $\left(h,q\right)$-Genocchi polynomials, we study $\left(h,q\right)$-Zeta-type function. We derive symmetric properties of $\left(h,q\right)$-Zeta function and from these properties we give symmetric property of $\left(h,q\right)$-Genocchi polynomials., Comment: 14 pages; submitted

The main objective of this paper is to introduce the modified q-Genocchi polynomials and to define their generating function. In the paper, we show new relations, which are explicit formula, derivative formula, multiplication formula, and some others, for mentioned q-Genocchi polynomials. By applying Mellin transformation to the generating function of the modified q-Genocchi polynomials, we define q-Genocchi zeta-type functions which are interpolated by the modified q-Genocchi polynomials at negative integers., Comment: 10 pages; submitted

In the present paper, we obtain new interesting relations and identities of the Apostol-Bernoulli polynomials of higher order, which are derived using a Bernoulli polynomial basis. Finally, by utilizing our method, we also derive formulas for the convolutions of Bernoulli and Euler polynomials, expressed via Apostol-Bernoulli polynomials of higher order., Comment: 8 pages, submitted

In this work, we consider not only a discontinuous boundary-value problem with retarded argument and four supplementary transmission conditions at the two points of discontinuities but also, eigenparameter-dependent boundary conditions and obtain asymptotic formulas for the eigenvalues and corresponding eigenfunctions.

In the present paper, we investigate some interesting properties including several special polynomials arising from Caputo-fractional derivative. From our investigation, we derive a lot of interesting identities of several special polynomials., Comment: 10 pages

In this article, we consider the boundary-value problem of nonlinear fractional differential equation with p-Laplacian operator. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of two parameters are obtained. The uniqueness of positive solution on these parameters is also studied., Comment: 16 pages

In the present paper, we deal with multiple generalized Genocchi numbers and polynomials. Also, we introduce analytic interpolating function for the multiple generalized Genocchi numbers attached to \c{hi} at negative integers in complex plane and we define the multiple Genocchi p-adic L-function. Finally, we derive the value of the partial derivative of our multiple p-adic l-function at s=0., Comment: 10 pages

In the present paper, the fundamental aim is to consider a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order modified Dedekind-type sums related to q-Genocchi polynomials with weight alpha by using fermionic p-adic invariant q-integral on Zp., Comment: 6 pages

The purpose of this paper is to derive some applications of umbral calculus by using extended fermionic p-adic q-integral on Zp. From those applications, we derive some new interesting properties on the new family of Euler numbers and polynomials. That is, a systemic study of the class of Sheffer sequences in connection with generating function of the weighted Euler polynomials are given in the present paper., Comment: 13 pages

In this work, we derive numerous identities for multivariate q-Euler polynomials by using umbral calculus., Comment: 8 pages

In the present paper, our objective is to treat a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order Dedekind-type sums with weight in connection with modified q-Genocchi polynomials with weight alpha by using p-adic invariant q-integral on Zp., Comment: 6 pages, submitted

In the present paper, our goal is to introduce a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order dedekind-type sums with weight alpha related to Extended q-Euler polynomials by using p-adic q-integral in the p-adic integer ring., Comment: 5 pages, submitted

In this paper, we focus on the q-Genocchi numbers and polynomials. We shall introduce new identities of the q-Genocchi numbers and polynomials by using the fermionic p-adic integral on Zp which are very important in the study of Frobenius-Genocchi numbers and polynomials. Also, we give Cauchy-integral formula for the q-Genocchi polynomials and moreover by using measure theory on p-adic integral we derive the distribution formula q-Genocchi polynomials. Finally, we present a new definition of q-Zeta-type function by using Mellin transformation which is the interpolation function of the q-Genocchi polynomials at negative integers., Comment: 9 pages

In this paper, we will deal with some new formulae for two product Genocchi polynomials together with both Euler polynomials and Bernoulli polynomials. We get some applications for Genocchi polynomials. Our applications possess a number of interesting properties to study in Theory of Analytic numbers which we express in the present paper., Comment: 10 pages, submitted

In the present paper, we deal with a fourth-order boundary value problem problem with eigenparameter dependent boundary conditions and transmission conditions at a interior point. A self-adjoint linear operator A is defined in a suitable Hilbert space H such that the eigenvalues of such a problem coincide with those of A. Following Mukhtarov and his students methods [2,4,6] we obtain asymptotic formulae for its eigenvalues and fundamental solutions. Our applications possess a number of interesting properties for studying in boundary value problems which we state in this paper., Comment: 11 pages. arXiv admin note: substantial text overlap with arXiv:1208.5395

In the present paper, we introduce Eulerian polynomials with a and b parameters and give the definition of them. By using the definition of generating function for our polynomials, we derive some new identities in Theory of Analytic Numbers. Also, we give relations between Eulerian polynomials with a and b parameters, Bernstein polynomials, Poly-logarithm function, Bernoulli numbers and Euler numbers. Moreover, we see that our polynomials at a =-1 are related to Euler-Zeta function at negative inetegers. Finally, we get Witt's formula for new generalization of Eulerian polynomials which we express in this paper., Comment: 10 pages, submitted

In the present paper, we effect Dirichlet's type of twisted Eulerian polynomials by using p-adic fermionic q-integral on the p-adic integer ring. Also, we introduce some new interesting identities for them. As a result of them, by using contour integral on the generating function of Dirichlet's type of twisted Eulerian polynomials and so we define twisted Eulerian-L-function which interpolates of Dirichlet's type of Eulerian polynomials at negative integers which we state in this paper., Comment: 8 pages, submitted

In the present paper, we introduce Eulerian polynomials attached to by using p-adic q-integral on Zp . Also, we give new interesting identities via the generating functions of Dirichlet's type of Eulerian polynomials. After, by applying Mellin transformation to this generating function of Dirichlet' type of Eulerian polynomials, we derive L-function for Eulerian polynomials which interpolates of Dirichlet's type of Eulerian polynomials at negative integers., Comment: 8 pages, submitted

We give some new identities for (h; q)-Genocchi numbers and polynomials by means of the fermionic p-adic q-integral on Zp and the weighted q-Bernstein polynomials., Comment: 8 pages, submitted

In the present paper, we investigate special generalized q-Euler numbers and polynomials. Some earlier results of T. Kim in terms of q-Euler polynomials with weight alpha can be deduced. For presentation of our formulas we apply the method of generating function and p-adic q-integral representation on Zp. We summarize our results as follows. In section 2, by using combinatorial techniques we present two formulas for q-Euler numbers with weight alpha. In section 3, we derive distribution formula (Multiplication Theorem) for Dirichlet type of q-Euler numbers and polynomials with weight . Moreover we define partial Dirichlet type zeta function and Dirichlet q-L-function, and obtain some interesting combinatorial identities for interpolating our new definitions. In addition, we derive behavior of the Dirichlet type of q-Euler L-function with weight alpha, Lq (s; x j) at s = 0. Furthermore by using second kind stirling numbers, we obtain an explicit formula for Dirichlet type q-Euler numbers with weight alpha. Moreover a novel formula for q-Euler-Zeta function with weight in terms of nested series of E;q (n j) is derived . In section 4, by introducing p-adic Dirichlet type of q-Euler measure with weight, and we obtain some combinatorial relations, which interpolate our previous results. In section 5, which is the main section of our paper. As an application, we introduce a novel concept of dynamics of the zeros of analytically continued q-Euler polynomials with weight alpha., Comment: 15 pages, submitted

In the present paper, we deal with Fourier-transformation of Frobenius-Euler polynomials. We shall give its applications by using infinite series. Our applications possess interesting properties which we state in this paper., Comment: 7 pages, submitted

The fundamental aim of this paper is to define weighted q-Hardy-littlewood-type maximal operator by means of fermionic p-adic q-invariant distribution on Zp . Also, we derive some interesting properties concerning this type maximal operator., Comment: 8 pages, submitted. arXiv admin note: substantial text overlap with arXiv:1202.1969

The present paper deals with Bernstein polynomials and Frobenius-Euler numbers and polynomials. We apply the method of generating function and fermionic p-adic integral representation on Zp, which are exploited to derive further classes of Bernstein polynomials and Frobenius-Euler numbers and polynomials. To be more precise we summarize our results as follows, we obtain some combinatorial relations between Frobenius-Euler numbers and polynomials. Furthermore, we derive an integral representation of Bernstein polynomials of degree n on Zp . Also we deduce a fermionic p-adic integral representation of product Bernstein polynomials of different degrees n1, n2,...on Zp and show that it can be written with Frobenius-Euler numbers which yields a deeper insight into the effectiveness of this type of generalizations. Our applications possess a number of interesting properties which we state in this paper., Comment: 8 pages, submitted

The objective of this paper is to derive some interesting properties of Genocchi, Euler and Bernstein polynomials by means of the orthogonality of Hermite polynomials., Comment: This paper has been withdrawn by the author due to a crucial sign error in equation 1

The aim of this paper is to give a new approach to modified q-Bernstein polynomials for functions of two variables. By using these type polynomials, we derive recurrence formulas and some new interesting identities related to the second kind Stirling numbers and generalized Bernoulli polynomials. Moreover, we give the generating function and interpolation function of these modified q-Bernstein polynomials of two variables and also give the derivatives of these polynomials and their generating function., Comment: 11 pages

In this paper, we shall investigate and analyse a new study on the best simultaneous approximation in the context of linear 2-normed spaces inspired by Elumalai and his coworkers in Elumalai. The basis of this investigation is to extend and refinement the definition of the classical aproximation, best approximation and some related concepts to linear 2-normed spaces., Comment: 8 pages

In this work, we consider the generating function of Kim's q-Euler polynomials and introduce new generalization of q-Genocchi polynomials and numbers of higher order. Also, we give surprising identities for studying in Analytic Numbers Theory and especially in Mathematical Physics. Moreover, by applying q-Mellin transformation to generating function of q-Genocchi polynomials of higher order and so we define q-Hurwitz-Zeta type function which interpolates of this polynomials at negative integers., Comment: 10 pages

In the present paper, we analyse analytic continuation of weighted q-Genocchi numbers and polynomials. A novel formula for weighted q-Genocchi- Zeta function {\zeta}G,q (s | {\alpha}) in terms of nested series of {\zeta}G,q (n | {\alpha}) is derived. Moreover, we introduce a novel concept of dynamics of the zeros of analytically continued weighted q-Genocchi polynomials., Comment: 5 pages, submitted

In this paper, we deal with q-Euler numbers and q-Bernoulli numbers. We derive some interesting relations for q-Euler numbers and polynomials by using their generating function and derivative operator. Also, we show between the q-Euler numbers and q-Bernoulli numbers via the p-adic q-integral in the p-adic integer ring., Comment: 9 pages, submitted

In this paper, the authors deal with the $q$-Genocchi numbers and polynomials with weight zero. They discover some interesting relations via the $p$-adic $q$-integral on $\mathbb{Z}_{p}$ and familiar basis Bernstein polynomials. Finally, the authors show that the $p$-adic $\log$ gamma functions are associated with the $q$-Genocchi numbers and polynomials with weight zero., Comment: 8 pages

The essential aim of this paper is to define weighted q-Hardy-littlewood-type maximal operator by means of p-adic q-invariant distribution on Zp. Moreover, we give some interesting properties concerning this type maximal operator., Comment: 7 pages

The q-calculus theory is a novel theory that is based on finite difference re-scaling. The rapid development of q-calculus has led to the discovery of new generalizations of q-Euler polynomials involving q-integers. The present paper deals with the modified Dirichlet's type of twisted q-Euler polynomials with weight alpha. We apply the method of generating function and p-adic q-integral representation on Zp, which are exploited to derive further classes of q-Euler numbers and polynomials. To be more precise we summarize our results as follows, we obtain some combinatorial relations between modified Dirichlet's type of twisted q-Euler numbers and polynomials with weight alpha. Furthermore we derive witt's type formula and Distribution formula (Multiplication theorem) for modified Dirichlet's type of twisted q-Euler numbers and polynomials with weight alpha. In section three, by applying Mellin transformation we define q-analogue of modified twisted q-l-functions of Dirichlet's type and also we deduce that it can be written as modified Dirichlet's type of twisted q-Euler polynomials with weight alpha. Moreover we will find a link between modified twisted Hurwitz-zeta function and q-analogue of modified twisted q-l-functions of Dirichlet's type which yields a deeper insight into the effectiveness of this type of generalizations. In addition we consider q-analogue of partial zeta function and we derive behavior of the modified q-Euler L-function at s = 0. In final section, we construct p-adic twisted Euler q-L function with weight alpha and interpolate Dirichlet's type of twisted q-Euler polynomials with weight alpha at negative integers. Our new generating function possess a number of interesting properties which we state in this paper., Comment: 14 pages

The rapid development of q-calculus has led to the discovery of new generalizations of Bernstein polynomials and Genocchi polynomials involving q-integers. The present paper deals with weighted q-Bernstein polynomials and q-Genocchi numbers with weight alpha and beta. We apply the method of generating function and p-adic q-integral representation on Zp, which are exploited to derive further classes of Bernstein polynomials and q-Genocchi numbers and polynomials. To be more precise we summarize our results as follows, we obtain some combinatorial relations between q-Genocchi numbers and polynomials with weight alpha and beta. Furthermore, we derive an integral representation of weighted q-Bernstein polynomials of degree n on Zp. Also we deduce a fermionic p-adic q-integral representation of product weighted q-Bernstein polynomials of different degrees n1,n2,...on Zp and show that it can be written with q-Genocchi numbers with weight alpha and beta which yields a deeper insight into the effectiveness of this type of generalizations. Our new generating function possess a number of interesting properties which we state in this paper., Comment: 10 pages

The fundamental aim of this paper is to describe q-Analogue of p-adic log gamma functions with weight alpha and beta. Moreover, we give relationship between p-adic q-log gamma funtions with weight ({\alpha}, {\beta}) and q-extension of Genocchi numbers with weight alpha and beta and modified q Euler numbers with weight {\alpha}, Comment: 6 pages

The present paper deals with unification of the multiple twisted Euler and Genocchi numbers and polynomials associated with p-adic q-integral on Zp at q = 1. Some earlier results of Ozden's papers in terms of unification of the multiple twisted Euler and Genocchi numbers and polynomials associated with p-adic q-integral on Zp at q = 1 can be deduced. We apply the method of generating function and p-adic q-integral representation on Zp, which are exploited to derive further classes of Euler polynomials and Genocchi polynomials. To be more precise we summarize our results as follows, we obtain some relations between H.Ozden's generating function and fermionic p-adic q-integral on Zp at q = 1. Furthermore we derive Witt's type formula for the unification of twisted Euler and Genocchi polynomials. Also we derive distribution formula (Multiplication Theorem) for multiple twisted Euler and Genocchi numbers and polynomials associated with p-adic q-integral on Zp at q = 1 which yields a deeper insight into the effectiveness of this type of generalizations. Furthermore we define unification of multiple twisted zeta function and we obtain an interpolation formula between unification of multiple twisted zeta function and unification of the multiple twisted Euler and Genocchi numbers at negative integer. Our new generating function possess a number of interesting properties which we state in this paper., Comment: 13 pages

The fundamental aim of this paper is to introduce and investigate a new property of quasi 2-normed space based on a question given by C. Park (2006) [2] for the completion quasi 2-normed space. Finally, we also find an answer for a question Park's., Comment: 7 pages

The aim of this paper is to give a new approach to modified $q$-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling numbers and generalized Bernoulli polynomials are derived. Moreover, the generating function, interpolation function of these polynomials of several variables and also the derivatives of these polynomials and their generating function are given. Finally, we get new interesting identities of modified $q$-Bernoulli numbers and $q$-Euler numbers applying $p$-adic $q$-integral representation on $\mathbb {Z}_p$ and $p$-adic fermionic $q$-invariant integral on $\mathbb {Z}_p$, respectively, to the inverse of $q$-Bernstein polynomials., Comment: 17 pages, some theorems added to new version

The main object of this paper is to construct a new generating function of the (q-) Bernstein type polynomials. We establish elementary properties of this function. By using this generating function, we derive recurrence relation and derivative of the (q-) Bernstein type polynomials. We also give relations between the (q-) Bernstein type polynomials, Hermite polynomials, Bernoulli polynomials of higher-order and the second kind Stirling numbers. By applying Mellin transformation to this generating function, we define interpolation of the (q-) Bernstein type polynomials. Moreover, we give some applications and questions on approximations of (q-) Bernstein type polynomials, moments of some distributions in Statistics.