Recurrence Relations for the Maclaurin Coefficients of Products of Elementary Functions and the Bessel Functions

TL;DR

This paper derives recurrence relations for Maclaurin coefficients of products involving elementary functions and Bessel functions, covering various specific functions in the complex plane.

Contribution

It introduces new recurrence relations for Maclaurin coefficients of products of elementary functions and Bessel functions, expanding analytical tools for these special functions.

Findings

Recurrence relations for products with exponential functions

Relations for products with trigonometric and hyperbolic functions

Extensions to complex plane for various elementary functions

Abstract

In this paper, we investigate recurrence relations for the Maclaurin coefficients of the products of a elementary function and the Bessel function of the first kind $\mathcal{J}(z) = h(z) J_\nu(z)$ and the modified Bessel function of the first kind $\mathcal{I}(z) = h(z) I_\nu(z)$ in the complex plane corresponding to several specific choices of $h(z)$. In particular, we specialize $h(z)$ as $e^{pz}$, $(1-\theta z)^p$, $e^{-p \arctan z}$, $\sin(pz)$, $\cos(pz)$, $\sinh(pz)$, $\cosh(pz)$, $\arcsin(pz)$ and $\arccos(pz)$.