On a Toeplitz determinant identity of Borodin and Okounkov

Abstract

The authors of the title proved an elegant identity expressing a Toeplitz determinant in terms of the Fredholm determinant of an infinite matrix which (although not described as such) is the product of two Hankel matrices. The proof used combinatorial theory, in particular a theorem of Gessel expressing a Toeplitz determinant as a sum over partitions of products of Schur functions. The purpose of this note is to give two other proofs of the identity. The first uses an identity of the second author for the quotient of Toeplitz determinants in which the same product of Hankel matrices appears and the second, which is more direct and extends the identity to the case of block Toeplitz determinants, consists of carrying the first author's collaborative proof of the strong Szegö limit theorem one step further.