Asymptotics of type I Hermite-Pad\'e polynomials for semiclassical functions

TL;DR

This paper investigates the asymptotic behavior and zero distribution of type I Hermite-Padé polynomials for semiclassical functions, providing new insights into their differential equations and ratios, especially when functions are powers of a single function.

Contribution

It introduces an approach to analyze the asymptotics of Hermite-Padé polynomials for semiclassical functions, including differential equations and ratio asymptotics, with detailed case studies.

Findings

Asymptotic zero distribution described for semiclassical functions

Differential equations satisfied by polynomials and functions

Ratio asymptotics analyzed for powers of a single function

Abstract

Type I Hermite--Pad\'e polynomials for a set of functions $f_0, f_1, ..., f_s$ at infinity, $Q_{n,0}$, $Q_{n,1}$, ..., $Q_{n,s}$, is defined by the asymptotic condition $$ R_n(z):=\bigl(Q_{n,0}f_0+Q_{n,1}f_1+Q_{n,2}f_2+...+Q_{n,s}f_s\bigr)(z) =\mathcal O (\frac1{z^{s n+s}}), \quad z\to\infty, $$ with the degree of all $Q_{n,k}\leq n$. We describe an approach for finding the asymptotic zero distribution of these polynomials as $n\to \infty$ under the assumption that all $f_j$'s are semiclassical, i.e. their logarithmic derivatives are rational functions. In this situation $R_n$ and $Q_{n,k}f_k$ satisfy the same differential equation with polynomials coefficients. We discuss in more detail the case when $f_k$'s are powers of the same function $f$ ($f_k=f^k$); for illustration, the simplest non trivial situation of $s=2$ and $f$ having two branch points is analyzed in depth. Under these conditions, the ratio or comparative asymptotics of these polynomials is also discussed. From methodological considerations and in order to make the situation clearer, we start our exposition with the better known case of Pad\'e approximants (when $s=1$).