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A Fredholm determinant formula for Toeplitz determinants

Alexei Borodin·Andrei Okounkov
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Abstract

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 math.RT/9907127, 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.

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Alexei Borodin, Andrei Okounkov. 1999-07-25. A Fredholm determinant formula for Toeplitz determinants. https://arxiv.org/abs/math/9907165

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