Communications in Mathematics and Applications Vol. 13, No. 3, pp. 1237–1247, 2022 ISSN 0975-8607 (online); 0976-5905 (print) Published by RGN Publications http://www.rgnpublications.com DOI: 10.26713/cma.v13i3.1619 Research Article The Combination of Bernstein Polynomials with Positive Functions Based on a Positive Parameter s Ali J. Mohammad and Rafah F. Katham* Department of Mathematics, University of Basrah, Basrah, Iraq *Corresponding author: rafah.fouad@gmail.com Received: July 2, 2021 Accepted: August 9, 2022 Abstract. This paper deals with a sequence of the combination of Bernstein polynomials with a positive function τ and based on a parameter s >− 1 2 . These polynomials have preserved the functions 1 and τ. First, the convergence theorem for this sequence is studied for a function f ∈ C[0, 1]. Next, the rate of convergence theorem for these polynomials is descript by using the first, second modulus of continuous and Ditzian-Totik modulus of smoothness. Also, the Quantitative Voronovskaja and Grüss-Voronovskaja are obtained. Finally, two numerical examples are given for these polynomials by chosen a test function f ∈ C[0, 1] and two functions for τ to show that the effect of the different values of s and the different chosen functions τ. Keywords. Bernstein polynomials, Modulus of smoothness, Quantitative Voronovskaja, Grüss- Voronovskaja Mathematics Subject Classification (2020). 41A10, 41A25, 41A36 Copyright © 2022 Ali J. Mohammad and Rafah F. Katham. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction For a function f ∈ C[0, 1], the well-known Bernstein polynomials [1] are defined as: B n ( f ; x) = n X k=0 b n,k ( x) f  k n ¶ (1.1) where b n ( x) = ( n k ) x k (1 − x) n−k and x ∈ [0, 1].