A Note on Wiener-hopf Determinants and the Borodin-okounkov Identity

Recently, a beautiful identity due to Borodin and Okounkov was proved for Toeplitz determinants which shows how one can write a Toeplitz determinant as a Fredholm determinant. In this note we generalize this to the Wiener-Hopf case. The proof in the Wiener-Hopf case follows identically with the second one given in [1]. We include it here for completeness sake and because the nature of the identity is slightly different in the continuous verses discrete convolution setting. In the Wiener-Hopf case we begin with a Fredholm determinant on a finite interval and then show how this can be written as a Fredholm determinant of an operator defined on L of a half-line. The point is the second operator has a very “small” kernel and thus higher order approximations (as a function of the length of the finite interval) can be found. We now state the analogue of the identity and then apply it to a particular case to show how error estimates can be computed. In the future we hope to refine the estimates given here, apply this identity to other important examples, and also extend it to other operators. We consider the Fredholm determinant of the finite Wiener-Hopf operator