Exponential generating functions for the associated Bessel functions

TL;DR

This paper introduces exponential generating functions for associated Bessel functions, exploring their properties, relations, and presenting a Rodrigues formula, thus extending the mathematical understanding of these functions.

Contribution

It provides new exponential generating functions for associated Bessel functions and analyzes their independence and sequence structure, which was not previously established.

Findings

Two new types of exponential generating functions are derived.

Associated Bessel functions with different parameters are shown to be dependent or independent.

A Rodrigues formula for associated Bessel functions is established.

Abstract

Similar to the associated Legendre functions, the differential equation for the associated Bessel functions $B_{l,m}(x)$ is introduced so that its form remains invariant under the transformation $l\rightarrow -l-1$. A Rodrigues formula for the associated Bessel functions as squared integrable solutions in both regions $l<0$ and $l\geq 0$ is presented. The functions with the same $m$ but with different positive and negative values of $l$ are not independent of each other, while the functions with the same $l+m$ ($l-m$) but with different values of $l$ and $m$ are independent of each other. So, all the functions $B_{l,m}(x)$ may be taken into account as the union of the increasing (decreasing) infinite sequences with respect to $l$. It is shown that two new different types of exponential generating functions are attributed to the associated Bessel functions corresponding to these rearranged sequences.