Abstract
Abstract Let w be a semiclassical weight that is generic in Magnus’s sense, and ( p n ) n = 0 ∞ ({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 ψ ∈ L∞ (iℝ), let W(ψ) be the Wiener-Hopf operator with symbol ψ. We give sufficient conditions on ψ such that 1/ det W(ψ) W(ψ−1) = det(I − Γϕ 1 Γϕ 2) where Γϕ 1 and Γϕ 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 2m F 2m-1. These results extend those of Basor and Chen [2], who obtained 4 F 3 likewise. We include examples where the Wiener–Hopf factors are found explicitly.
Highlights
Definition 1.1. (i) Let φ ∈ L2(0, ∞)
The Hankel matrix corresponding to ν is [ν(j + k)]∞ j,k=0, which gives a densely defined operator in 2(N ∪ {0}). (The connection between Hankel matrices and integral operators is discussed in [29, p. 53].)
We show that trace class Hankel operators on Hardy space H2(C+) have a matrix representation with respect to reproducing kernels on the state space
Summary
Definition 1.1. (i) Let φ ∈ L2(0, ∞). the Hankel operator with scattering function φ is the integral operator. Given a trace class Hankel operator Γ, the spectrum consists of 0 and a sequence of eigenvalues λj, listed according to algebraic multiplicity, such that Fredholm determinants such as det(I + Γφ), using operator theory and tools from linear systems. We introduce an algebra C2 of complex functions on a strip containing iR such that each invertible ψ ∈ C has a Wiener –Hopf factorization ψ(iξ) = ψ−(iξ)ψ+(iξ), and we consider the Wiener–Hopf operator W (ψ) of L2(0, ∞) with symbol ψ. When interpreted with suitable linear systems, these formulas give expansions of det(I − Γφ1 Γφ2 ) in terms of the generalised hypergeometric function 2mF2m−1. These results extend those of Basor and Chen [2], who obtained 4F3 likewise.
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