A fredholm determinant formula for Toeplitz determinants

We prove a formula expressing a general n by n Toeplitz determinant as a Fredholm determinant of an operator 1-K acting on l(2)(n, n + 1, ...), where the kernel K admits an integral representation in terms of the symbol of the original Toeplitz matrix. The proof is based on the results of one of the authors, see [14], and a formula due to Gessel which expands any Toeplitz determinant into a series of Schur functions. We also consider 3 examples where the kernel involves the Gauss hypergeometric function and its degenerations.