We construct the linear positive operators generated by the q-Dunkl generalization of the exponential function. We have approximation properties of the operators via a universal Korovkin-type theorem and a weighted Korovkin-type theorem. The rate of convergence of the operators for functions belonging to the Lipschitz class is presented. We obtain the rate of convergence by means of the classical, second order, and weighted modulus of continuity, respectively, as well.

The purpose of the paper is to introduce Stancu‐type linear positive operators generated by Dunkl generalization of exponential function. We present approximation properties with the help of well‐known Korovkin‐type theorem and weighted Korovkin‐type theorem and also acquire the rate of convergence in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre's K‐functional, and second‐order modulus of continuity by Dunkl analogue of Szász operators. Copyright © 2015 John Wiley & Sons, Ltd.

Approximation and some other basic properties of various linear and nonlinear operators are potentially useful in many different areas of researches in the mathematical, physical, and engineering sciences. Motivated essentially by this aspect of approximation theory, our present study systematically investigates the approximation and other associated properties of a class of the Szász-Mirakjan-type operators, which are introduced here by using an extension of the familiar Beta function. We propose to establish moments of these extended Szász-Mirakjan Beta-type operators and estimate various convergence results with the help of the second modulus of smoothness and the classical modulus of continuity. We also investigate convergence via functions which belong to the Lipschitz class. Finally, we prove a Voronovskaja-type approximation theorem for the extended Szász-Mirakjan Beta-type operators.

In this paper, the matrix extension of the multivariable Humbert polynomials is introduced. Various families of linear, multilinear and multilateral generating matrix functions of these matrix polynomials are presented. Miscellaneous applications are also discussed.

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