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Bayram Çekim

Publications and source records attributed to Bayram Çekim.

9 recordsLinked to original sources
arXiv2025

Finite bivariate biorthogonal I -- Konhauser polynomials

In this paper, a finite set of biorthogonal polynomials in two variables is produced using Konhauser polynomials. Some properties containing operational and integral representation, Laplace transform, fractional calculus operators of this family are studied. Also, computing Fourier transform for the new set, a new family of biorthogonal functions are derived via Parseval's identity. On the other hand, this finite set is modified by adding two new parameters in order to have semigroup property and construct fractional calculus operators. Further, integral equation and integral operator are also derived for the modified version.

math.CA↗
arXiv2025

Finite bivariate biorthogonal $N$-Konhauser polynomials

A new set of finite 2D biorthogonal polynomials is defined using the finite orthogonal polynomials $N_{n}^{(p)}(w)$ and the Konhauser polynomials. We present a connection between this finite 2D biorthogonal set and the generalized Laguerre-Konhauser polynomials. Also, we obtain several applications of finite bivariate biorthogonal $N$-Konhauser polynomials.

math.CA↗
arXiv2024

Finite Bivariate Biorthogonal M-Konhauser Polynomials

In this paper, we construct the pair of finite bivariate biorthogonal M-Konhauser polynomials, reduced to the finite orthogonal polynomials $M_{n}^{(p,q)}(t)$, by choosing appropriate parameters in order to obtain a relation between the Jacobi Konhauser polynomials and this new finite bivariate biorthogonal polynomials $_{K}M_{n;\upsilon}^{(p,q)}(z,t)$ similar to the relation between the classical Jacobi polynomials $P_{n}^{(p,q)}(t)$ and the finite orthogonal polynomials $M_{n}^{(p,q)}(t)$. Several properties like generating function, operational/integral representation are derived and some applications like fractional calculus, Fourier transform and Laplace transform are studied thanks to that new transition relation and the definition of finite bivariate M-Konhauser polynomials.

math.CA↗
arXiv2023

Parametric kinds of generalized Apostol-Bernoulli polynomials and their properties

The purpose of this paper is to define generalized Apostol--Bernoulli polynomials with including a new cosine and sine parametric type of generating function using the quasi-monomiality properties and trigonometric functions. In this study, the Apostol-Bernoulli polynomials with three variable are defined with two new generating functions cosine and sine parameters. Then, we investigate multiplicative and derivative operators, diffrential equations, some summation formulas and partial differential equations for these polynomials. Moreover, we introduce Gould--Hopper--Apostol--Bernoulli type polynomials, Hermite--Appell--Apostol--Bernoulli type polynomials and truncated exponential Apostol--Bernoulli type polynomials. Finally, the special cases of these new polynomials are investigated, and the corresponding results are expressed.

math.CA↗
arXiv2020

A Second Type Of Higher Order Generalised Geometric Polynomials and Higher Order Generalised Euler Polynomials

In this study we introduce a second type of higher order generalised geometric polynomials. This we achieve by examining the generalised stirling numbers $S(n; k;α;β;γ)$ [Hsu & Shiue,1998] for some negative arguments. We study their number theoretic properties, asymptotic properties, and study their combinatorial properties using the notion of barred preferential arrangements. We also proposed a generalisation of the classical Euler polynomials and show how these Euler polynomials are related to the second type of higher order generalised geometric polynomials.

math.CO↗
arXiv2017

Dunkl generalization of Szasz Beta type operators

The goal in the paper is to advertise Dunkl extension of Szasz beta type operators. We initiate approximation features via acknowledged Korovkin and weighted Korovkin theorem and obtain the convergence rate from the point of modulus of continuity, second order modulus of continuity, the Lipschitz class functions, Peetre's K-functional and modulus of weighted continuity by Dunkl generalization of Szasz beta type operators.

math.CA↗
arXiv2017

A Dunkl Analogue of Operators Including Two-variable Hermite polynomials

The aim of this paper is to introduce a Dunkl generalization of the operators including two variable Hermite polynomials which are defined by Krech [14](Krech, G. A note on some positive linear operators associated with the Hermite polynomials, Carpathian J. Math., 32 (1) (2016), 71--77) and to investigate approximating properties for these operators by means of the classical modulus of continuity, second modulus of continuity and Peetre's K-functional.

math.CA↗
arXiv2016

Higher order generalized geometric polynomials

According to generalized Mellin derivative (Kargin), we introduce a new family of polynomials called higher order generalized geometric polynomials. We obtain some properties of them.We discuss their connections to degenerate Bernoulli and Euler polynomials. Furthermore, we find new formulas for the Carlitz's (Carlitz) and Howard's (Howard2) finite sums. Finally, we evaluate several series in closed forms, one of which has the coefficients include values of the Riemann zeta function. Moreover, we calculate some integrals in terms of generalized geometric polynomials.

math.CA↗