Mehmet Ali Özarslan

Extension of gamma, beta and hypergeometric functions

The main object of this paper is to present generalizations of gamma, beta and hypergeometric functions. Some recurrence relations, transformation formulas, operation formulas and integral representations are obtained for these new generalizations.

MKZ Type Operators Providing a Better Estimation on [1/2,1)

In the present paper, we introduce a modification of the Meyer-Konig and Zeller (MKZ) operators which preserve the test functions f0(x) = 1 and f2(x)= x2, and we show that this modification provides a better estimation than the classical MKZ operators on the interval [1/2, 1) with respect to the modulus of continuity and the Lipschitz class functionals. Furthermore, we present the r-th order generalization of our operators and study their approximation properties.

Some families of generating functions for a certain class of three-variable polynomials

The present investigation is a continuation of the works initiated by Srivastava [A contour integral involving Foxs H-function, Indian J. Math. 14 (1972), pp. 1–6] and followed by several papers such as (among others) [A. Altın, E. Erkuş, and M.A. Özarslan, Families of linear generating functions for polynomials in two variables, Integral Transforms Spec. Funct. 17 (2006), pp. 315–320; B. González, J. Matera, and H.M. Srivastava, Some q-generating functions and associated generalized hypergeometric polynomials, Math. Comput. Modelling 34 (2001), pp. 133–175; S.-D. Lin, Y.-S. Chao, and H.M. Srivastava, Some families of hypergeometric polynomials and associated integral representations, J. Math. Anal. Appl. 294 (2004), pp. 399–411; S.-D. Lin, H.M. Srivastava, and P.-Y. Wang, Some families of hypergeometric transformations and generating relations, Math. Comput. Modelling 36 (2002), pp. 445–459; E. Özergin, M.A. Özarslan, and H.M. Srivastava, Some families of generating functions for a class of bivariate polynomials, Math. Comput. Modelling 50 (2009), pp. 1113–1120]. In this study, we introduce a family of three-variable polynomials and derive a number of two-sided linear generating functions between these polynomials and another family of two-variable polynomials. Furthermore, mixed multilateral and multilinear generating functions are derived for these polynomials. Several other results of the above-mentioned types are also considered.

q-Bernstein-Schurer-Kantorovich Operators

In the present paper, we introduce the q-Bernstein-Schurer-Kantorovich operators. We give the Korovkin-type approximation theorem and obtain the rate of convergence of this approximation by means of the first and the second modulus of continuity. Moreover, we compute the order of convergence of the operators in terms of the elements of Lipschitz class functions and the modulus of continuity of the derivative of the function.MSC: 41A10, 41A25, 41A36.

Some generalized Lagrange-based Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials

In this paper, we introduce a general family of Lagrange-based Apostol-type polynomials thereby unifying the Lagrange-based Apostol-Bernoulli and the Lagrange-based Apostol-Genocchi polynomials. We also define Lagrange-based Apostol-Euler polynomials via the generating function. In terms of these generalizations, we find new and useful relations between the unified family and the Apostol-Euler polynomials. We also derive their explicit representations and list some basic properties of each of them. Further relations between the above-mentioned polynomials, including a family of bilinear and bilateral generating functions, are given. Moreover, a generating relation involving the Stirling numbers of the second kind is derived.

The extended Mittag-Leffler function and its properties

In this paper, we present the extended Mittag-Leffler functions by using the extended Beta functions (Chaudhry et al. in Appl. Math. Comput. 159:589-602, 2004) and obtain some integral representations of them. The Mellin transform of these functions is given in terms of generalized Wright hypergeometric functions. Furthermore, we show that the extended fractional derivative (Özarslan and Özergin in Math. Comput. Model. 52:1825-1833, 2010) of the usual Mittag-Leffler function gives the extended Mittag-Leffler function. Finally, we present some relationships between these functions and the Laguerre polynomials and Whittaker functions.

A-statistical approximation of generalized Szász―Mirakjan―Beta operators

Abstract In this paper we prove a Korovkin type approximation theorem and obtain the rate of convergence of the generalized Szasz–Mirakjan–Beta operators by means of modulus of continuity and elements of Lipschitz class. Furthermore we give the A -statistical approximation theorem for these operators and investigate the case which provides the best estimation.

Some families of generating functions for a class of bivariate polynomials

The main purpose of this paper is to present various families of generating functions for a class of polynomials in two variables. Furthermore, several general classes of bilinear, bilateral or mixed multilateral generating functions are obtained for these polynomials.

I-convergence theorems for a class of k-positive linear operators

In this paper, we obtain some approximation theorems for k- positive linear operators defined on the space of analytical functions on the unit disc, via I-convergence. Some concluding remarks which includes A-statistical convergence are also given.

Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials

In this paper, we introduce a unified family of Hermite-based Apostol-Bernoulli, Euler and Genocchi polynomials. We obtain some symmetry identities between these polynomials and the generalized sum of integer powers. We give explicit closed-form formulae for this unified family. Furthermore, we prove a finite series relation between this unification and 3d-Hermite polynomials.MSC:11B68, 33C05.