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

Abstract

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.

arXiv comment12 pages. Submitted to special issue of the Fibonacci Quarterly honoring Curtis Cooper and Arthur Benjamin. Revision based on remarks of an anonymous referee. Flow of text greatly improved by breaking up lemmas, cross-referencing definitions and notation conventions, and pairing lemma and example statements