Combinatorial sums and binomial identities associated with the Beta-type polynomials

Abstract

In this paper, we first provide some functional equations of the generating functions for beta-type polynomials. Using these equations, we derive various identities of the beta-type polynomials and the Bernstein basis functions. We then obtain some novel combinatorial identities involving binomial coefficients and combinatorial sums. We also derive some generalizations of the combinatorics identities which are related to the Gould's identities and sum of binomial coefficients. Next, we present some remarks, comments, and formulas including the combinatorial identities, the Catalan numbers, and the harmonic numbers. Moreover, by applying the classical Young inequality, we derive a combinatorial inequality related to beta polynomials and combinatorial sums. We also give another inequality for the Catalan numbers.

Generating functions for series involving higher powers of inverse binomial coefficients and their applications

“…[6, p. 273]). Here we also note that there are many other different proof of Equation (), See, for instance, earlier works [3, 18, 22, 34, 40] and the references cited in each of these earlier works. With the aid of (), we also have solution of Exercise 30* in Charalambides [6, p. 273].…”

“…See, for instance, related references (cf. [3,18,40]) and the references cited in each of these earlier works.…”

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

“…Assuming that 𝑦 is a random variable of this distribution with 0 ≤ 𝑦 ≤ 1. Thus, the probability of getting completely 𝑏 successes in 𝑣 independent the Bernoulli trials, given by the equation (2) (Chattamvelli & Shanmugam, 2020;Lorentz, 1986;Simsek, 2019;2020;Simsek, 2014;Yalcin & Simsek, 2022).…”

“…If we integrate both sides of the equation ( 2) from 𝑦 = 0 to 𝑦 = 1, the following well-known integral formulas for the Bernstein polynomials can be obtained: & Araci, 2010;Lorentz, 1986;Simsek, 2019;2020;Simsek, 2014;, Yalcin & Simsek, 2022.…”

“…Using the above integral formulas for the Bernstein polynomials, we also gave many formulas and finite sums (Simsek, 2019;2020;Simsek, 2014;Yalcin & Simsek, 2022).…”