The square lattice Ising model on the rectangle III: Hankel and Toeplitz determinants

Based on the results obtained in [Hucht, J. Phys. A: Math. Theor. 50, 065201 (2017)], we show that partition function of anisotropic square lattice Ising model $L \times M$ rectangle, with open boundary conditions both directions, is given by determinant a $M/2 M/2$ Hankel matrix, equivalently can be written as Pfaffian skew-symmetric $M Toeplitz matrix. The $M-1$ independent matrix elements matrices are Fourier coefficients certain symbol function, which ratio two characteristic polynomials. These polynomials associated to different directions system, encode respective conditions, and directly related through symmetry considered under exchange directions. generalized other well suited for analysis finite-size scaling functions critical limit using Szeg\H{o}'s theorem.