On determinant expansions for Hankel operators

Blower, Gordon and Chen, Yang (2020) On determinant expansions for Hankel operators. Concrete Operators, 7 (1). pp. 13-44. ISSN 2299-3282

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Abstract

Let $w$ be a semiclassical weight that is generic in Magnus's sense, and $(p_n)_{n=0}^\infty$ the corresponding sequence of orthogonal polynomials. We express 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 $\psi$. We give 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 ${}_{2m}F_{2m-1}$. These results extend those of Basor and Chen \cite{BasorChen2003}, who obtained ${}_4F_3$ likewise. We include examples where the Wiener--Hopf factors are found explicitly. \par \vskip.1in