Toeplitz Determinants From Compatibility Conditions

Details Toeplitz Determinants From Compatibility Conditions Toeplitz Determinants From Compatibility Conditions

2004-02-23

arxiv.org/abs/math/0402362v1

Mathematics

Classical Analysis and ODEs

16 pages

Scientific paper

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 $$w(z) = {\rm e}^{tz}\prod_{\alpha=1}^{M}(\frac{z-a_{\alpha}}{z})^{g_{\alpha}}.$$

Affiliated with

Also associated with

No associations

Rating

Toeplitz Determinants From Compatibility Conditions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Toeplitz Determinants From Compatibility Conditions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Toeplitz Determinants From Compatibility Conditions will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-349729