Investigation into the role of the Bessel function order in the Fourier-Bessel series and the Hankel Transform

This work focuses on estimating the number of terms of a Fourier-Bessel series of order $p^{\mathrm{\prime }}$ required to get within a certain error of a Bessel function of a fixed order $p$ where $p\mathrm{\ne }{p}^{\mathrm{\prime }}$. Our approach consists of two steps: one, constructing an invariant over $n$ of the $n^{\text{th}}$ order Hankel transform; and two, observing the effect of expanding a suitably scaled Bessel function of a fixed order $p$ in its Fourier-Bessel series of order $p^{\mathrm{\prime }}$. We demonstrate a new error metric to simplify the error computations. Further, we generate an empirical model using numerical simulations and examine its capabilities in predicting the number of terms required.