Toeplitz Determinants From Compatibility Conditions

Abstract

In this paper we show, how a straightforward and natural application of a pair of fundamental identities valid for polynomials orthogonal over the unit circle, can be used to calculate the determinant of the finite Toeplitz matrix, $${\Delta }_n=\det (w_{j-k}{)}_{j,k=0}^{n-1}:=\det ({\int }_{|z|=1}\frac{w(z)}{z^{j-k+1}}\frac{dz}{2\pi i}{)}_{j,k=0}^{n-1},$$ with the Fisher-Hartwig symbol, $$w(z)=C(1-z{)}^{\alpha +i\beta }(1-1/z{)}^{\alpha -i\beta },|z|=1,\alpha >-1/2,\beta \in \mathbb{R}.$$ Here $C$ is the normalisation constant chosen so that $w_0=\frac{1}{2\pi }.$ We use the same approach to compute a difference equation for expressions related to the determinants of the symbol $$w(z)={\mathrm{e}}^{t(z+1/z)},$$ a symbol important in the study of random permutations. Finally, we study the analogous equations for the symbol $$w(z)={\mathrm{e}}^{tz}\prod _{\alpha =1}^M(\frac{z-a_{\alpha }}{z}{)}^{g_{\alpha }}.$$

16 pages