Some Darling-Siegert relationships connected with random flights

TL;DR

This paper derives key integral identities involving Bessel functions, interprets them probabilistically through random walks, and extends previous work on Darling-Siegert relationships in the context of random flights.

Contribution

It introduces new integral identities with probabilistic interpretations related to random walks and extends prior results on Darling-Siegert relationships in random flight analysis.

Findings

Derived four integral identities involving Bessel functions.

Connected identities to combinatorial formulas and random walk models.

Extended previous work on Darling-Siegert relationships in random flights.

Abstract

We derive in detail four important results on integrals of Bessel functions from which three combinatorial identities are extracted. We present the probabilistic interpretation of these identities in terms of different types of random walks, including asymmetric ones. This work extends the results of a previous paper concerning the Darling-Siegert interpretation of similar formulas emerging in the analysis of random flights.