Heine, Hilbert, Pade, Riemann, and Stieltjes: a John Nuttall's work 25 years later

TL;DR

This paper revisits Nuttall's 1986 work on generalized Jacobi polynomials, analyzing their asymptotic behavior using modern techniques involving Riemann-Hilbert problems, orthogonality, and complex analysis.

Contribution

It provides a modern reinterpretation of Nuttall's results by applying Riemann-Hilbert analysis and complex orthogonality to generalized Jacobi polynomials.

Findings

Asymptotic behavior of generalized Jacobi polynomials analyzed

Connection established between orthogonality and Riemann-Hilbert methods

Insights into algebraic functions with branch points

Abstract

In 1986 J. Nuttall published in Constructive Approximation the paper "Asymptotics of generalized Jacobi polynomials", where with his usual insight he studied the behavior of the denominators ("generalized Jacobi polynomials") and the remainders of the Pade approximants to a special class of algebraic functions with 3 branch points. 25 years later we try to look at this problem from a modern perspective. On one hand, the generalized Jacobi polynomials constitute an instance of the so-called Heine-Stieltjes polynomials, i.e. they are solutions of linear ODE with polynomial coefficients. On the other, they satisfy complex orthogonality relations, and thus are suitable for the Riemann-Hilbert asymptotic analysis. Along with the names mentioned in the title, this paper features also a special appearance by Riemann surfaces, quadratic differentials, compact sets of minimal capacity, special functions and other characters.