A q‐Erkuş–Srivastava polynomials operator

Abstract

Chan et al. (Integral Transforms Spec. Funct. 12 (2), 139–148, (2001)) constructed a multivariable extension of the Lagrange polynomials, popularly known as the Chan–Chyan–Srivastava polynomials. Altin and Erkuş (Integral Transforms Spec. Funct. 17, 239–244, (2006)) proposed Lagrange–Hermite polynomials in several variables. Erkuş and Srivastava (Integral Transforms Spec. Funct., 17, 267–273, (2006)) presented an unification (and generalization) of the Chan–Chyan–Srivastava polynomials and the multivariable Lagrange–Hermite polynomials, called as Erkuş–Srivastava polynomials. Duman (Taiwanese J. Math. 12 (2), 539‐543, (2008)) defined a ‐analogue of these generalized multivariable polynomials. Inspired by these studies, we construct a linear positive operator by means of the ‐Erkuş–Srivastava multivariable polynomials and study the Korovkin‐type theorems and the rate of convergence of these operators by using summability techniques of weighted ‐statistical convergence and the power series method. We also define a th‐order Taylor generalization of the multivariable polynomials operator and investigate the approximation of th‐order continuously differentiable Lipschitz class elements. Finally, we define the bivariate case of ‐Erkuş–Srivastava multivariable polynomials operator and study its ‐statistical convergence by using four‐dimensional matrix transformation.