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, $$ Δ_n=\det(w_{j-k})_{j,k=0}^{n-1}:= \det(\int_{|z|=1}\frac{w(z)}{z^{j-k+1}}\frac{dz}{2πi})_{j,k=0}^{n-1}, $$ with the Fisher-Hartwig symbol, $$ w(z)=C(1-z)^{α+iβ}(1-1/z)^{α-iβ},\quad |z|=1, α>-1/2, β\in{\mathbb R} . $$ Here $C$ is the normalisation constant chosen so that $w_0=\frac{1}{2π}.$ 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 $$w(z) = {\rm e}^{tz}\prod_{α=1}^{M}(\frac{z-a_α}{z})^{g_α}.$$