Tridiagonal real symmetric matrices with a connection to Pascal's triangle and the Fibonacci sequence

Abstract

We explore a certain family $\{A_n{\}}_{n=1}^{\infty }$ of $n\times n$ tridiagonal real symmetric matrices. After deriving a three-term recurrence relation for the characteristic polynomials of this family, we find a closed form solution. The coefficients of these characteristic polynomials turn out to involve the diagonal entries of Pascal's triangle in a tantalizingly predictive manner. Lastly, we explore a relation between the eigenvalues of various members of the family. More specifically, we give a sufficient condition on the values $m,n\in \mathbb{N}$ for when $\text{spec}(A_m)$ is contained in $\text{spec}(A_n)$. We end the paper with a number of open questions, one of which intertwines our characteristic polynomials with the Fibonacci sequence in an intriguing manner involving ellipses.

arXiv comment23 pages, 8 figures. Version 2 was not the submitted version. Version 3 is the submitted version. We added a new subsection on a connection to Chebyshev polynomials of the second kind, and we improved the exposition for journal submission