Gamma function, Beta function and combinatorial identities

Recently, lot of developments and extension of gamma and beta function have been studied by many researchers due to their nice properties and variety of applications in different field of science. The aim of this paper is to investigate some generalized inequalities associated with extended beta and gamma functions.

How Enumerative Combinatorics met Special Functions, thanks to Joe Gillis

The aim of this paper is to obtain a new range of modified gamma and beta $k$ function. The author present a brief survey of a new range of modified gamma and beta $k$-function, first and second summation relation, various functional, symmetric, Mellian transform, and integral representation are obtained. Furthermore, mean, variance and the moment generating function for the beta Distribution is studied.

In this paper we establish plenty of number theoretic and combinatoric identities involving generalized Bernoulli and Stirling numbers of both kinds. These formulas are deduced from Pascal type matrix representations of Bernoulli and Stirling numbers. For this we define and factorize a modified Pascal matrix corresponding to Bernoulli and Stirling cases.

We introduce new generalizations of the Gamma and the Beta functions. Their properties are investigated and known results are obtained as particular cases.

In this article we obtain a general polynomial identity in $k$ variables, where $k\geq 2$ is an arbitrary positive integer. We use this identity to give a closed-form expression for the entries of the powers of a $k \times k$ matrix. Finally, we use these results to derive various combinatorial identities.

This note is dedicated to Professor Gould. The aim is to show how the identities in his book "Combinatorial Identities" can be used to obtain identities for Fibonacci and Lucas polynomials. In turn these identities allow to derive a wealth of numerical identities for Fibonacci and Lucas numbers.

We survey combinatorial interpretations of some dozen identities for the double factorial such as, for instance, (2n-2)!! + Sum_{k=2}^{n} (2n-1)!!(2k-4)!!/(2k-1)!! = (2n-1)!!. Our methods are mostly bijective.

We give a combinatorial interpretation for the hypergeometric functions associated with tuples of rational numbers.

In this paper, we first quickly review the basics of an algebro-geometric method of Karaji's L-summing technique in today's modern language of algebra. Then, we also review the theory of Gosper's algorithm as a decision procedure for obtaining the indefinite sums involving hypergeometric terms. Then, we show that how one can use Gosper's algorithm equipped with the L-summing method to obtain a class of combinatorial identities associated with a given algebraic identity.