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Mehmet Acikgoz

Publications and source records attributed to Mehmet Acikgoz.

At least 19 recordsLinked to original sources
arXiv2025

A new family of q-Bernstein polynomials: Probabilistic viewpoint

In this paper, we introduce a new class of polynomials, called probabilistic q-Bernstein polynomials, alongside their generating function. Assuming Y is a random variable satisfying moment conditions, we use the generating function of these polynomials to establish new relations. These include connections to probabilistic Stirling numbers of the second kind and higher-order probabilistic Bernoulli polynomials associated with Y. Additionally, we derive recurrence and differentiation properties for probabilistic q-Bernstein polynomials. Utilizing Leibniz's formula, we give an identity for the generating function of these polynomials. In the latter part of the paper, we explore applications by choosing appropriate random variables such as Poisson, Bernoulli, Binomial, Geometric, Negative Binomial, and Uniform distributions. This allows us to derive relationships among probabilistic q-Bernstein polynomials, Bell polynomials, Stirling numbers of the second kind, higher-order Frobenius-Euler numbers, and higher-order Bernoulli polynomials. We also present p-adic q-integral and fermionic p-adic q-integral representations for probabilistic q-Bernstein polynomials.

math.CA↗
arXiv2014

On The Properties Of $q$-Bernstein-Type Polynomials

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.

math.NT↗
arXiv2013

Extended fermionic $p$-adic integrals on $\mathbb{Z}_p$

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.

math.NT↗
arXiv2013

Identities involving the $\left(h,q\right)$-Genocchi polynomials and $\left(h,q\right)$-Zeta-type function

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.

math.NT↗
arXiv2013

On the modified q-Genocchi numbers and polynomials and their applications

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.

math.NT↗
arXiv2013

Asymptotic expressions of eigenvalues and fundamental solutions of a discontinuous fourth-order boundary value problem

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.

math.CA↗
arXiv2013

Existence and uniqueness of positive solutions of boundary-value problems for fractional differential equations with p-Laplacian operator

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.

math.CA↗
arXiv2013

Extended Fermionic p-Adic q-Integrals On Zp In Connection With Applications Of Umbral Calculus

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.

math.NT↗
arXiv2013

A note on the modified q-Dedekind sums

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.

math.GM↗
arXiv2013

New Generalization of Eulerian polynomials and their applications

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.

math.NT↗