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

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 $\arccos(pz)$.