Solving Kepler's equation via nonlinear sequence transformations

TL;DR

This paper explores a novel approach using nonlinear sequence transformations to analyze and accelerate the convergence of the classical Kapteyn series solution of Kepler's equation, providing numerical evidence for its potential Stieltjes series nature.

Contribution

It introduces the use of Levin-type sequence transformations to analyze Kepler's equation, supporting the conjecture that its solution series is a Stieltjes series and developing an algorithm for Debye's polynomials.

Findings

Levin $d$- and Weniger $ ext{delta}$-transformations exhibit exponential convergence.

Numerical evidence supports the Stieltjes series conjecture for Kepler's equation solution.

An effective recursive algorithm for Debye's polynomials is developed.

Abstract

Since more than three centuries Kepler's equation continues to represents an important benchmark for testing new computational techniques. In the present paper, the classical Kapteyn series solution of Kepler's equation originally conceived by Lagrange and Bessel will be revisited from a different perspective, offered by the relatively new and still largely unexplored framework of the so-called nonlinear sequence transformations. The main scope of the paper is to provide numerical evidences supporting the fact that the Kapteyn series solution of Kepler's equation could be a Stieltjes series. To support such a conjecture, two types of Levin-type sequence transformations, namely Levin $d$- and Weniger $\delta$-transformations, will be employed to sum up several wildly divergent series derived by the Debye representation of Bessel functions. As an interesting byproduct of this analysis, an effective recursive algorithm to generate arbitrarily higher-order Debye's polynomials will be developed. Such a "Stieltjeness" conjecture will also be numerically validated by directly employing the Levin-type transformations to accelerate the complex Kapteyn series solution of the Kepler equation. Both $d$- and $\delta$- transformations display exponential convergence, whose rate will be numerically estimated. A few conclusive words, together with some hints for future extensions in the direction of more general class of Kapteyn series, eventually close the paper.