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 $ψ\in L^\infty (i{\mathbb R})$, let $W(ψ)$ be the Wiener-Hopf operator with symbol $ψ$. The paper gives sufficient conditions on $ψ$ such that $1/\det W(ψ)W(ψ^{-1})=\det (I-Γ_{φ_1}Γ_{φ_2})$ where $Γ_{φ_1}$ and $Γ_{φ_2}$ are Hankel operators that are Hilbert--Schmidt. For certain $ψ$, 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.

Theorems & Definitions (40)

• Definition 1.1
• Example 1.2
• Lemma 2.1
• proof
• Proposition 2.2
• Lemma 2.3
• proof
• Definition 3.1
• Theorem 3.2
• proof