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Serkan Araci

Publications and source records attributed to Serkan Araci.

At least 19 recordsLinked to original sources
arXiv2025

Gauss-Appell polynomials: An umbral calculus approach

This article aims to reinforce the broad applicability of the umbral approach to address complex mathematical challenges and contribute to various scientific and engineering endeavors. The umbral methods are used to reformulate the theoretical framework of special functions and provide powerful techniques for uncovering new extensions and relationships among these functions. This research article introduces an innovative class of special polynomials, specifically the Gauss-Appell polynomials. The fundamental attributes of this versatile family of special polynomials are outlined, including generating relations, explicit representations, and differential recurrence relations. Certain examples of the particular members that belong to the class of Gauss-Appell polynomials are also considered.

math.CA↗
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↗
arXiv2020

Certain results for unified Apostol type-truncated exponential-Gould-Hopper polynomials and their relatives

The present article aims to introduce a unified family of the Apostol type-truncated exponential-Gould-Hopper polynomials and to characterize its properties via generating functions. A unified presentation of the generating function for the Apostol type-truncated exponential-Gould-Hopper polynomials is established and its applications are given. By the use of operational techniques, the quasi-monomial properties for the unified family are proved. Several explicit representations and multiplication formulas related to these polynomials are obtained. Some general symmetric identities involving multiple power sums and Hurwitz-Lerch zeta functions are established by applying different analytical means on generating functions.

math.GM↗
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

Some New Symmetric Identities for the q-Zeta Type Functions

The main object of this paper is to obtain several symmetric properties of the q-Zeta type functions. As applications of these properties, we give some new interesting identities for the modified q-Genocchi polynomials. Finally, our applications are shown to lead to a number of interesting results which we state in the present paper.

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↗