$\texttt{ChisholmD.wl}$- Automated rational approximant for bi-variate series

TL;DR

This paper introduces an automated Mathematica package for evaluating diagonal Chisholm rational approximants of two-variable series, enhancing their applicability for analytic continuation and convergence acceleration in physics-related functions.

Contribution

The paper presents the first automated tool for diagonal Chisholm approximants, including modifications for evaluation around arbitrary points, improving their utility in complex analysis.

Findings

Effective for analytic continuation of two-variable functions

Improves convergence acceleration in series evaluation

Applicable to hypergeometric series in physics contexts

Abstract

The Chisholm rational approximant is a natural generalization to two variables of the well-known single variable Pad\'e approximant, and has the advantage of reducing to the latter when one of the variables is set equals to 0. We present, to our knowledge, the first automated Mathematica package to evaluate diagonal Chisholm approximants of two variable series. For the moment, the package can only be used to evaluate diagonal approximants i.e. the maximum powers of both the variables, in both the numerator and the denominator, is equal to some integer $M$. We further modify the original method so as to allow us to evaluate the approximants around some general point $(x,y)$ not necessarily $(0,0)$. Using the approximants around general point $(x,y)$, allows us to get a better estimate of the result when the point of evaluation is far from $(0,0)$. Several examples of the elementary functions have been studied which shows that the approximants can be useful for analytic continuation and convergence acceleration purposes. We continue our study using various examples of two variable hypergeometric series, $\mathrm{Li}_{2,2}(x,y)$ etc that arise in particle physics and in the study of critical phenomena in condensed matter physics. The demonstration of the package is discussed in detail and the Mathematica package is provided as an ancillary file.