A Unified Approach to Beta Moments, Combinatorial Identities, and Random Walks

TL;DR

This paper introduces a unified probabilistic framework linking return probabilities of random walks with moment representations, providing proofs of combinatorial identities and deriving new identities across dimensions.

Contribution

It develops a novel unified approach connecting random walk return probabilities with moments, enabling new combinatorial identities and proofs in multiple dimensions.

Findings

Probabilistic proofs of combinatorial identities involving beta and gamma functions.

New combinatorial identities derived for arbitrary dimensions.

Framework applicable to diverse disciplines studying random walks.

Abstract

The study of random walks has increasingly been popular across diverse disciplines such as statistics, mathematics, quantum physics, where they are used to model paths consisting of successive random steps in a mathematical space. A fundamental quantity of interest is the probability that a simple symmetric random walk returns to the origin after 2n steps. In this paper, we develop a unified probabilistic approach that connects the return probabilities in arbitrary dimensions with moment representations. Using this framework, we provide probabilistic proofs of several combinatorial identities involving beta and gamma functions, and derive new combinatorial identities in general dimensions.