Approximation properties of ( p, q)-variant of Stancu-Schurer and Kantorovich-Stancu-Schurer operators Abdul Wafi, Nadeem Rao ∗ Department of Mathematics, Jamia Millia Islamia, New Delhi-110 025, India Abstract In this article, we have introduced ( p, q)-variant of Stancu-Schurer and Kantorovich-Stancu-Schurer operators and discussed the rate of convergence, Korovkin type theorem for these operators. Keywords: ( p, q)-integers, ( p, q)-Bernstein operators, ( p, q)-Kantorovich operators, ( p, q)-Stancu-Schurer, Korovvkin. 2010 Mathematics Subject Classification 41A10, 41A25, 41A36, 41A36 1. Introduction In 1885, Weirstrass gave a very famous result known as Weierstrass approximation theorem which plays an important role in the development of approximation theory. It was considered to be typical untill Bernstein supplied a very short and simple proof of it. Berntein[1] considered polynomials defined as B n ( f ; x)= n ∑ k=0 P n,k (x) f  k n  , k = 0, 1, 2, ..., n = 1, 2, 3, ... where, P n,k (x)= ( n k ) x k (1 − x) n−k and x ∈ [0, 1]. It is a powerful tool for numerical analysis, computer added geometric design(CAGD) and solutions of differential equations. But, these polynomials were restricted to con- tinuous functions only. So, in order to approximate integrable functions, Kantorovich[2] generalized Bernstein polynomials as follows K n ( f ; x)=(n + 1) n ∑ k=0 P n,k (x) k+1 n+1  k n+1 f (t )dt k = 0, 1, 2, 3, ..., n = 1, 2, 3, .... For the last two decades, the application of q-calculus emerged as a new area in the field of approximation theory. Motivated by the application of q-calculus, Lupas[8] introduced a sequence of Bernstein polynomials based on q-integer. Another form of q-Bernstein operators was given by Philips[9]. Several researchers in- troduced different type of operators based on q-integers ([3]-[7]). Recently, Mursaleen et al and Acar applied ( p, q)-calculus in approximation theory and introduced ( p, q)-analogue of Bernstein operators[10], Bernstein- Kantorovich operators[14], Bernstein-Stancu operators[11] and Szasz-Mirakjan operators[15] respectively. We recall some basic notions based on ( p, q)-integers[10]. Let 0 < q ≤ p < 1. Then, ( p, q)-integers for non negative integers n, k [k] p,q = p k − q k p − q , ∗ Corresponding author Email addresses: awafi@jmi.ac.in (Abdul Wafi), nadeemrao1990@gmail.com (Nadeem Rao) Preprint submitted to Elsevier February 14, 2022 arXiv:1508.01852v2 [math.NT] 22 Aug 2015