A spectral product formula for repunits via a tridiagonal Toeplitz similarity
Johann Verwee
TL;DR
The paper links base-$b$ repunits $R_{n+1}(b)$ to determinants of the tridiagonal matrix $V_n(b)$ and shows that a diagonal similarity reduces $V_n(b)$ to a symmetric Toeplitz form, yielding an explicit spectrum. It derives the eigenvalues $\lambda_k(b)=b+1+2\sqrt{b}\cos(\frac{k\pi}{n+1})$ and a closed spectral product $R_{n+1}(b)=\prod_{k=1}^n\lambda_k(b)$, with a hyperbolic cosine form when $b=e^{2x}$. The authors also provide sharp eigenvector expressions and weighted orthogonality, along with a complete inverse formula for $V_n(b)$ in terms of repunits, via Chebyshev polynomials. Together, these results produce concrete finite identities for repunits, explicit matrix inverses, and connections to Chebyshev theory with potential implications for repunit-type sequences.
Abstract
For $b>0$ and $n\geqslant 1$, we consider the $n\times n$ tridiagonal matrix $V_n(b)$ with diagonal entries $b+1$, superdiagonal entries $1$, and subdiagonal entries $b$. A diagonal similarity reduces $V_n(b)$ to a symmetric tridiagonal Toeplitz matrix and hence makes its spectrum explicit. Since $\det\left(V_n(b)\right)$ equals the geometric sum $1+b+\cdots+b^{n}$, taking determinants yields a finite cosine product evaluation for this quantity. As further consequences, we derive sharp bounds from the extremal eigenvalues, write down explicit eigenvectors with respect to a natural weighted inner product, and obtain a closed formula for $V_n(b)^{-1}$.