Fredholm Determinant evaluations of the Ising Model diagonal correlations and their \lambda - generalisation

The diagonal spin-spin correlations of the square lattice Ising model, originally expressed as Toeplitz determinants, are given by two distinct Fredholm determinants - one with an integral operator having an Appell function kernel and another with a summation operator having a Gauss hypergeometric function kernel. Either determinant allows for a Neumann expansion possessing a natural \lambda - parameter generalisation and we prove that both expansions are in fact equal, implying a continuous and a discrete representation of the form factors. Our proof employs an extension of the classic study by Geronimo and Case, applying scattering theory to orthogonal polynomial systems on the unit circle, to the bi-orthogonal situation.