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 the corresponding sequence of orthogonal polynomials. The paper expresses the Christoffel-Darboux kernel as a sum of products of Hankel integral operators. For ψ ∈ L ∞ (iR), 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 φ1 Γ φ2 ) where Γ φ1 and Γ φ2 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 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.