On determinant expansions for Hankel operators

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 $\ddot{I}ˆ\in L^{\infty }(i\mathbb{R})$, let $W(\ddot{I}ˆ)$ be the Wiener-Hopf operator with symbol $\ddot{I}ˆ$. The paper gives sufficient conditions on $\ddot{I}ˆ$ such that $1/\det W(\ddot{I}ˆ)W(\ddot{I}{ˆ}^{-1})=\det (I-\hat{I}{“}_{\ddot{I}{•}_1}\hat{I}{“}_{\ddot{I}{•}_2})$ where $\hat{I}{“}_{\ddot{I}{•}_1}$ and $\hat{I}{“}_{\ddot{I}{•}_2}$ are Hankel operators that are Hilbert--Schmidt. For certain $\ddot{I}ˆ$, Barnes's integral leads to an expansion of this determinant in terms of the generalised hypergeometric ${}_n{F}_m$. These results extend those of Basor and Chen [2], who obtained ${}_4{F}_3$ likewise. The paper includes examples where the Wiener--Hopf factors are found explicitly.

30 pages