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