Gamma function, Beta function and combinatorial identities

Using a probabilistic approach, we derive several interesting identities involving beta functions. Our results generalize certain well-known combinatorial identities involving binomial coefficients and gamma functions.

Given two combinatorial identities proved earlier, a new set of variations of these combinatorial identities is listed and proved with the integral representation method. Some identities from literature are shown to be special cases of these new identities.

The aim of this note is to provide a new identity connected with the Gauss hypergeometric function. This is achieved using results of certain combinatorial identities and a hypergeometric function approach.

This paper describes a method to find a connection between combinatorial identities and hypergeometric series with a number of examples. Combinatorial identities can often be written as hypergeometric series with unit argument. In a number of cases these hypergeometric series are balanced and can be reduced to a simpler form. In this paper some combinatorial identities are proved using this method assuming that the results in the tables of Prudnikov et al. [12] are proven wit...

The harmonic numbers and generalized harmonic numbers appear frequently in many diverse areas such as combinatorial problems, many expressions involving special functions in analytic number theory and analysis of algorithms. The aim of this paper is to derive some identities involving generalized harmonic numbers and generalized harmonic functions from the beta functions F(x)= B( x+1, n+1), ( n=0,1,2,..) using elementary methods.

We give combinatorial proofs for some identities involving binomial sums that have no closed form.

In the process of studying a conjecture of Holly M. Green and Martin W. Liebeck, we obtain two interesting identities by elementary methods, one is a combinatorial identity, and the other is a number theoretic identity.

A probability method is provided to prove three classes of combinatorial identities. The method is extremely simple, only one step after the proper probability setup.

We prove astonishing identities generated by compositions of positive integers. In passing, we obtain two new identities for Stirling numbers of the first kind. In the two last sections we clarify an algebraic sense of these identities and obtain several other structural close identities.

This short note deals with some applications of the Beta function