A spectral product formula for repunits via a tridiagonal Toeplitz similarity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917186244706304 |
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| author | Verwee, Johann |
| author_facet | Verwee, Johann |
| contents | 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 $V_n(b)^{-1}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_02615 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A spectral product formula for repunits via a tridiagonal Toeplitz similarity Verwee, Johann Spectral Theory Number Theory 11B83 (primary), 15A18, 33C45 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 $V_n(b)^{-1}$. |
| title | A spectral product formula for repunits via a tridiagonal Toeplitz similarity |
| topic | Spectral Theory Number Theory 11B83 (primary), 15A18, 33C45 |
| url | https://arxiv.org/abs/2601.02615 |