A Fredholm determinant formula for Toeplitz determinants

as the Fredholm determinant of an operator 1−K acting on l2({n, n+1, . . . }), where the kernel K = K(φ) admits an integral representation in terms of φ. The answer is affirmative and the construction of the kernel is explained below. We give two versions of the result: an algebraic one, which holds in the suitable algebra of formal power series, and an analytic one. In order to minimize the amount of analysis, we make a rather restrictive analyticity assumption on the function φ. One should be able to relax this assumption, see Remark 3. Our proof is a direct application of two results due to I. Gessel [7] and one of the authors [13], respectively. We also consider 3 examples in which the kernel K can be expressed in classical special functions. First two of them have been worked out in [1, 15, 2, 10].