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 <a href="/abs/math.RT/9907127">math.RT/9907127</a>, 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.