Proof of Convergence of a Laplace Expansion Algorithm For Calculating Recursions Satisfied by a Family of Determinants

In Evans and Hendel's recent proof of an outstanding conjecture on the resistance distances of a family of linear 3-trees, a key technique in the proof was calculating the recursion satisfied by a family of determinants. The underlying algorithm employed to prove the conjecture converged (i.e., terminated) in the particular case studied, and the paper presented an open question on when such a procedure converges in general. This paper proves the convergence of a Laplace expansion procedure for an arbitrary family of determinants of banded, square, Toeplitz matrices. A comparison of the procedure presented in this paper, the paper by Evans and Hendel, and a paper by Jia, Yang, and Li is presented.