Unification of Bessel functions of different orders

TL;DR

This paper introduces a new unifying formula that generates Bessel functions of real orders from integer orders, potentially impacting applied mathematics by simplifying the understanding of Bessel functions across different orders.

Contribution

It proposes and proves a novel unifying formula connecting Bessel functions of integer and real orders, expanding the theoretical framework of Bessel functions.

Findings

New unifying formula for Bessel functions of real and integer orders

Proof of the formula's validity

Applications demonstrating the formula's usefulness

Abstract

We investigate the internal space of Bessel functions which is associated to the group Z of positive and negative integers defining their orders. As a result we propose and prove a new unifying formula (to be added to the huge literature on Bessel functions) generating Bessel functions of real orders out of integer order one's. The unifying formula is expected to be of great use in applied mathematics. Some applications of the formula are given for illustration.