Revisiting the Coulomb problem: A novel representation of the confluent hypergeometric function as an infinite sum of discrete Bessel functions

TL;DR

This paper introduces a new, numerically stable representation of the confluent hypergeometric function as an infinite sum of Bessel functions, derived via a tridiagonal basis approach to the Coulomb problem.

Contribution

It presents a novel representation of the confluent hypergeometric function using discrete Bessel functions, improving convergence and numerical stability.

Findings

New representation is more rapidly convergent than existing formulas.

The approach effectively solves the Coulomb problem for continuum states.

Provides a stable numerical method for special function evaluation.

Abstract

We use the tridiagonal representation approach to solve the radial Schr\"odinger equation for the continuum scattering states of the Coulomb problem in a complete basis set of discrete Bessel functions. Consequently, we obtain a new representation of the confluent hypergeometric function as an infinite sum of Bessel functions, which is numerically very stable and more rapidly convergent than another well-known formula.