Generalization and Probabilistic Proofs of Some Combinatorial Identities

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.