Improvements in calculating the recursion satisfied by a family of determinants

Abstract: A variety of problems can be elegantly solved by identifying the recursion satisfied by the determinants of a family of matrices. In 2016, Jia, Yang, and Li provided a general 6-th order recursion for the family of arbitrary pentdiagonal Toeplitz matrices by using Laplace expansions. Recently, Evans and Hendel showed that this method is potentially generalizable and applied it to prove an outstanding conjecture on resistance distance in linear 3-trees. However, Evans and Hendel left as an open problem the convergence of their procedure in the general case. Hendel has recently proven convergence for such a Laplace-expansion approach for an arbitrary family of square, banded, Toeplitz matrices with $k$ super and sub diagonals for any positive integer $k.$ Hendel also eliminated the computational matrix methods of Evans and Hendel replacing them with a simpler algebraic manipulative system. This note supplements this procedure by showing an improved method to solve the resulting system of several simultaneous equations in families of determinants. This improved procedure, applied to explore an outstanding conjecture of Bareett, Evans, and Francis on the general $k$-linear tree, uncovers several interesting patterns which are presented.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)