The Fibonacci--Redheffer matrix and its properties
Aristides V. Doumas, Panayiotis J. Psarrakos
TL;DR
This work defines the Fibonacci–Redheffer matrix $F_R(n)$ as a Fibonacci analogue of Redheffer’s matrix and analyzes its determinant and spectrum. The determinant is explicitly given by $\det(F_R(n)) = n!_{F}\sum_{k=1}^{n}\frac{\mu(k)}{F_k}$, connecting Fibonacci factorials with Möbius sums, and the partial sums $\sum_{k=1}^{n}\frac{\mu(k)}{F_k}$ converge to a negative constant $C$, yielding a precise large-$n$ asymptotic $\det(F_R(n))\sim C_0\,\phi^{n(n+1)/2}5^{-n/2}$. Spectral analysis shows all eigenvalues are real and simple, none equals any $F_k$, and the eigenvalues satisfy sharp bounds relative to Fibonacci numbers; the largest eigenvalue grows with rate $\rho(F_R(n))\sim \frac{\phi^{n}}{\sqrt{5}}$, with trace asymptotics $\operatorname{tr}(F_R(n))\sim \frac{\phi^{n+2}}{\sqrt{5}}$. The Generalizations section extends the framework to arbitrary Redheffer-type matrices $A_R(n)$ built from a positive sequence $\{a_j\}$, deriving a universal determinant formula and several asymptotic regimes, including concrete results for $a_j=j^p$ and $a_j=j/\ln^2 j$, and a perturbation variant of the $(1,1)$-entry that ties singularity to the negative constant $C$. The paper also connects Hessenberg determinant recurrences to a polynomial $\chi_{F_R(n)}(z)$, and discusses implications related to the Riemann hypothesis via analogous structures for the classical Redheffer matrix.
Abstract
A Redheffer--type matrix with Fibonacci entries is defined, and the determinant and spectral properties of this matrix are studied. Also, more general Redheffer--type matrices are considered and intriguing number-theoretic examples are illustrated. Furthermore, several asymptotic results are discussed and a new expression related to the Riemann hypothesis is presented.