Uniform bounds, zero separation and monotonicity for the regular Coulomb wave functions

This paper begins by deriving the uniform bounds for the regular Coulomb wave function $F_{\ell,\eta}$ and its derivative $F_{\ell,\eta}'$. We then examine detailed zero configurations of $F_{\ell,\eta}$ and $F_{\ell+1,\eta}$, extending insights into the earlier work that was restricted to $\ell>-3/2$. Our investigation also includes an analysis of the monotonicity of the zeros of $F_{\ell,\eta}$ with respect to parameters $\ell$ and $\eta$, respectively. Furthermore, we expand our exploration to associated orthogonal polynomials, as well as the functions involving both $F_{\ell,\eta}$ and $F_{\ell,\eta}'$. Finally, we explore the breakdown of the Sturm separation theorem by means of the zeros of associated orthogonal polynomials.