Asymptotic formulas for determinants of a sum of finite Toeplitz and   Hankel matrices
  The purpose of this paper is to describe asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices with singular generating functions. The formulas are similar to those of the analogous problem for finite Toeplitz matrices for a certain class of symbols. However, the appearance of the Hankel matrices changes the nature of the asymptotics in some instances depending on the location of the singularities. Several concrete examples are also described in the paper. 
Estelle L. Basor and Torsten Ehrhardt