New Bessel Identities from Laguerre Polynomials

TL;DR

This paper demonstrates how asymptotic approximations of Laguerre polynomials by Bessel functions can generate new Bessel identities, including a novel generalization of Sonine's identity.

Contribution

It introduces a method to derive Bessel identities from Laguerre polynomial identities using asymptotic approximations, including a new generalization of Sonine's identity.

Findings

Derived new Bessel identities from Laguerre polynomial identities.

Generalized Sonine's identity using asymptotic analysis.

Illustrated the method with multiple examples.

Abstract

For large order, Laguerre polynomials can be approximated by Bessel functions near the origin. This can be used to turn many Laguerre identities into corresponding identities for Bessel functions. We will illustrate this idea with a number of examples. In particular, we will derive a generalization of a identity due to Sonine, which appears to be new.