Asymptotic formulas for determinants of a special class of Toeplitz + Hankel matrices

TL;DR

This paper derives asymptotic formulas for determinants of a special class of Toeplitz plus Hankel matrices with Fisher-Hartwig type symbols, extending previous results to non-even symbols.

Contribution

It generalizes existing asymptotic formulas for Toeplitz + Hankel determinants to include certain non-even symbols, broadening the scope of prior work.

Findings

Derived asymptotic formulas for determinants as matrix size grows

Extended previous results to non-even Fisher-Hartwig symbols

Provided explicit conditions on symbols for asymptotic analysis

Abstract

We compute the asymptotics of the determinants of certain $n\times n$ Toeplitz + Hankel matrices $T_n(a)+H_n(b)$ as $n\to\infty$ with symbols of Fisher-Hartwig type. More specifically we consider the case where $a$ has zeros and poles and where $b$ is related to $a$ in specific ways. Previous results of Deift, Its and Krasovsky dealt with the case where $a$ is even. We are generalizing this in a mild way to certain non-even symbols.