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 $\mathcal{J}(z)=h(z)J_{\nu }(z)$ 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 $\arccos (pz)$.