Toeplitz Determinants From Compatibility Conditions

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, $$w(z)=C(1-z)^{\alpha+i\beta}(1-1/z)^{\alpha-i\beta},\quad |z|=1, \alpha>-1/2, \beta\in{\mathbb R} .$$ 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 $$w(z) = {\rm e}^{t(z+1/z)},$$ a symbol important in the study of random permutations. Finally, we study the analogous equations for the symbol $$w(z) = {\rm e}^{tz}\prod_{\alpha=1}^{M}(\frac{z-a_{\alpha}}{z})^{g_{\alpha}}.$$