Some Finite Summation Identities Comprising Binomial Coefficients for Integrals of the Bernstein Polynomials and Their Applications

Abstract

Certain finite sums, including the Catalan numbers, factorial functions and binomial coefficients, and their computational formulas are of indispensable importance both in probability and statistics applications and in other branches of science. The primary aim of this article is to give the integral representation of the finite sum containing the products of the Bernstein polynomials, given in Theorem 2.1 in our article Simsek and Yardimci (2016), by applying the Beta function and the Euler gamma functions, also known as the factorial function. Other aims of this article are to reveal new finite sum formulas containing binomial coefficients by analyzing and unifying this integral representation. Finally, some relations among these sums, binomial coefficients, and the Catalan numbers are given.