  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 Szeg\"o limit theorem one step further.
