On (p,q)-analogue of divided differences and Bernstein operators

Abstract

Abstract. In this paper, (p, q)-calculus is applied to construct (p, q)-analogue of divided differences. Another equivalent form of (p, q)-Bernstein operators which generalize the Phillips q-Bernstein polynomials are defined in terms of (p, q)-divided differences. It is shown that these operators reproduce constant as well as linear test functions. Further, we show that the difference of two consecutive (p, q)-Bernstein polynomials of a function f can be expressed in terms of second-order divided differences of f .

Certain Identities Associated with (p,q)-Binomial Coefficients and (p,q)-Stirling Polynomials of the Second Kind

“…The proof is omitted. The interested reader may be referred (for example) to ( [16], p. 268) and [73] (see also [74]). .…”

“…Here and throughout, we assume that 0 < q < p ≤ 1. Mursaleen et al [73] defined (p, q)-differences, recursively, as follows: For any function h :…”

“…They [73] established the following relation between the Newton's divided difference and the (p, q)-differences:…”

“…Mursaleen et al [85,86] (see also ([87], p. 2), ( [73], p. 2), [88]) presented the (p, q)-analogue of Bernstein polynomials:…”

Some approximation results for (p,q)-Lupaş-Schurer operators

“…Besides this, we also refer to the reader some recent papers on (p, q)-calculus in approximation theory: e.g. [1], [6], [7], [8], [10], [11], [12], [16], [17] and [18]. Before proceeding further, we recall significant definitions and notations on the concept of (p, q)-calculus.…”