M. Ali Özarslan

Szász–Mirakjan type operators providing a better error estimation

Abstract In this work, giving a modification of the well-known Szasz–Mirakjan operators, we prove that the error estimation of our operators is better than that of the classical Szasz–Mirakjan operators. Furthermore, we obtain a Voronovskaya type theorem for these modified operators.

Some generating relations for extended hypergeometric functions via generalized fractional derivative operator

Recently an extension of beta function containing an extra parameter, which proved to be useful earlier, was used to extend the hypergeometric functions (Aslam Chaudhry, 2004 [6]). In this paper, linear and bilinear generating relations for extended hypergeometric functions are obtained by defining the extension of the fractional derivative operator. Some properties of the extended fractional derivative operator are also presented.

Unified Apostol-Bernoulli, Euler and Genocchi polynomials

In this paper, we present a unified family of polynomials including not only the Apostol-Bernoulli, Euler and Genocchi polynomials, but also a general family of polynomials suggested by Ozden et al. [H. Ozden, Y. Simsek, H.M. Srivastava, A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl. 60 (10) (2010) 2779-2787]. We obtain the explicit representation of this unified family, in terms of a Gaussian hypergeometric function. Some symmetry identities and multiplication formula are also given.

Statistical approximation results for Kantorovich-type operators involving some special polynomials

In this paper we introduce a general sequence of Kantorovich-type operators associated with some special polynomials and study their approximation properties. Mainly, we compute the rates of A-statistical convergence of these operators, which are observed to be stronger than that of the operators used commonly in Approximation Theory. Furthermore, we present an rth order generalization of our sequence of Kantorovich-type operators.

Rates of convergence of certain King-type operators for functions with derivative of bounded variation

In this paper, we estimate the rates of pointwise approximation of certain King-type positive linear operators for functions with derivative of bounded variation. We also extend our results to the statistical approximation process via the concept of statistical convergence.

Some families of generating functions for the extended Srivastava polynomials

Abstract Almost four decades ago, H.M. Srivastava considered a general family of univariate polynomials, the Srivastava polynomials, and initiated a systematic investigation for this family [10] . In 2001, B. Gonzalez, J. Matera and H.M. Srivastava extended the Srivastava polynomials by inserting one more parameter [4] . In this study we obtain a family of linear generating functions for these extended polynomials. Some illustrative results including Jacobi, Laguerre and Bessel polynomials are also presented. Furthermore, mixed multilateral and multilinear generating functions are derived for these polynomials.