# On determinant expansions for Hankel operators

## TL;DR

This paper explores determinant expansions related to Hankel operators and orthogonal polynomials, providing new formulas and conditions for Wiener-Hopf operators, with applications to hypergeometric functions and explicit examples.

## Contribution

It introduces new determinant formulas for Hankel operators and Wiener-Hopf operators, extending previous results and providing explicit factorization examples.

## Key findings

- Determinant of Wiener-Hopf operators expressed as Hankel operator products.
- Conditions under which these determinants relate to hypergeometric functions.
- Explicit examples of Wiener-Hopf factorization for specific symbols.

## Abstract

Let $w$ be a semiclassical weight which is generic in Magnus's sense, and $(p_n)_{n=0}^\infty$ the corresponding sequence of orthogonal polynomials. The paper expresses the Christoffel--Darboux kernel as a sum of products of Hankel integral operators. For $\psi\in L^\infty (i{\mathbb R})$, let $W(\psi )$ be the Wiener-Hopf operator with symbol $\psi$. The paper gives sufficient conditions on $\psi$ such that $1/\det W(\psi )W(\psi^{-1})=\det (I-\Gamma_{\phi_1}\Gamma_{\phi_2})$ where $\Gamma_{\phi_1}$ and $\Gamma_{\phi_2}$ are Hankel operators that are Hilbert--Schmidt. For certain $\psi$, Barnes's integral leads to an expansion of this determinant in terms of the generalised hypergeometric ${}_nF_m$. These results extend those of Basor and Chen [2], who obtained ${}_4F_3$ likewise. The paper includes examples where the Wiener--Hopf factors are found explicitly.