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

TL;DR

This work addresses the challenge of obtaining explicit Maclaurin coefficients for products of elementary functions with Bessel and modified Bessel functions, $\mathcal{J}(z)=h(z)J_\nu(z)$ and $\mathcal{I}(z)=h(z)I_\nu(z)$, in the complex plane. By selecting a broad family of $h(z)$—including $e^{pz}$, $(1-\theta z)^p$, $e^{-p \arctan z}$, trigonometric and hyperbolic functions, and inverse trig forms—the authors derive explicit recurrence relations that generate the Maclaurin coefficients, with clear initial conditions and, in many cases, parity patterns. The main contributions are the complete set of recurrences for both $J_\nu$ and $I_\nu$ cases across all listed $h(z)$, featuring three-term recurrences in simpler cases and higher-order multi-term recurrences with detailed coefficient formulas in the more complex ones. These results provide a versatile analytical toolkit for symbolic and numerical manipulation of composite Bessel expressions, with potential applications in physics and engineering where such products arise.

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 $\mathcal{J}(z) = h(z) J_ν(z)$ and the modified Bessel function of the first kind $\mathcal{I}(z) = h(z) I_ν(z)$ in the complex plane corresponding to several specific choices of $h(z)$. In particular, we specialize $h(z)$ as $e^{pz}$, $(1-θz)^p$, $e^{-p \arctan z}$, $\sin(pz)$, $\cos(pz)$, $\sinh(pz)$, $\cosh(pz)$, $\arcsin(pz)$ and $\arccos(pz)$.

Theorems & Definitions (26)

• Theorem 2.1
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• Theorem 2.10