Generalization and Probabilistic Proofs of Some Combinatorial Identities

TL;DR

This paper uses probabilistic methods to derive and generalize combinatorial identities involving gamma and beta functions, connecting moments of random variables to combinatorial formulas.

Contribution

It introduces a systematic probabilistic approach to derive new combinatorial identities from moments of gamma and beta distributions.

Findings

Derived new identities involving gamma and beta functions

Generalized known combinatorial identities with probabilistic methods

Provided a framework linking moments of random variables to combinatorial formulas

Abstract

Using a probabilistic approach, we derive some interesting combinatorial identities involving gamma and beta functions. These results generalize certain well-known combinatorial identities involving binomial coefficients and special functions. In particular, by studying moments of the difference of two gamma and beta random variables, both in the dependent and independent cases, we obtain new combinatorial identities. This approach provides a systematic method to derive further combinatorial identities from probabilistic transformations.