The q-Stirling numbers (polynomials) of the second kind have been investigated and applied in a variety of research subjects including, even, the q-analogue of Bernstein polynomials. The $(p,q)$-Stirling numbers (polynomials) of the second kind have been studied, particularly, in relation to combinatorics. In this paper, we aim to introduce new $(p,q)$-Stirling polynomials of the second kind which are shown to be fit for the $(p,q)$-analogue of Bernstein polynomials. We also present some interesting identities involving the $(p,q)$-binomial coefficients. We further discuss certain vanishing identities associated with the q-and $(p,q)$-Stirling polynomials of the second kind.

Fractional calculus is allowing integrals and derivatives of any positive order (the term `fractional' kept only for historical reasons), which can be considered a branch of mathematical physics which mainly deals with integro-differential equations, where integrals are of convolution form with weakly singular kernels of power-law type. In recent decades fractional calculus has won more and more interest in applications in several fields of applied sciences. In this line, our main object to investigate image formulas of generalized fractional hypergeometric operators involving the product of Mathieu-type series and generalized Mittag-Leffler function. We also consider some interesting special cases of derived results by specializing suitable value of the parameters.

In this manuscript, we construct a complex (p, q) Lorentz-Schurer operator for q > p > 1 and discuss the approximation properties on a compact disk. Based on the Voronovskaja’s type theorem and exact orders, we also obtain quantitative estimate by the (p, q) Lorentz-Schurer operator attached to analytic functions in the compact disk.