Determinant evaluation of banded Toeplitz matrices via bivariate polynomial families

We define three kinds banded Toeplitz matrices via with the upper and lower bandwidths ??x? and ??y?. The determinant evaluation is explicitly given for three kinds banded Toeplitz matrices via bivariate Tribonacci and Delannoy polynomials by using generating function approach and recurrence relations. Moreover perturbed versions of each kinds of the banded Toeplitz matrices by a 2 ? 2 general square matrix at the upper right corner will be explicitly computed.