dorsal/arxiv
View SchemaWeighted error-sum identities for periodic continued fractions and their generalizations
| Authors | Kevin Calderon, Nikita Kalinin |
|---|---|
| Categories | |
| ArXiv ID | 2601.07862vv1 |
| URL | https://arxiv.org/abs/2601.07862 |
| License | http://creativecommons.org/licenses/by/4.0/ |
Abstract
For a purely $N$-periodic continued fraction $\xi=[\overline{a_0,a_1,\dots,a_{N-1}}]=[a_0,a_1,\cdots]$, with $a_k=a_{k+N}$ for all $k\ge 0$, and convergents $h_n/k_n=[a_0,a_1,\dots,a_n]$, we obtain explicit expressions for the weighted error sums $f_\xi(s)=\sum a_{n+1}\lvert h_n-\xi k_n\rvert^s$ for $s>1$. A key observation is that, for each residue class $k_0\in{0,1,\dots,N-1}$, the subsequence of approximation errors $(h_k-\xi k_k)$ with $k\equiv k_0 \pmod N$ forms a geometric progression. In addition, we extend our methods to generalized continued fractions with numerators $(b_n)$, obtaining Euler-type identities and weighted error-sum formulae for $\pi$ and $\ln 2$.
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"abstract": "For a purely $N$-periodic continued fraction $\\xi=[\\overline{a_0,a_1,\\dots,a_{N-1}}]=[a_0,a_1,\\cdots]$, with $a_k=a_{k+N}$ for all $k\\ge 0$, and convergents $h_n/k_n=[a_0,a_1,\\dots,a_n]$, we obtain explicit expressions for the weighted error sums $f_\\xi(s)=\\sum a_{n+1}\\lvert h_n-\\xi k_n\\rvert^s$ for $s\u003e1$. A key observation is that, for each residue class $k_0\\in{0,1,\\dots,N-1}$, the subsequence of approximation errors $(h_k-\\xi k_k)$ with $k\\equiv k_0 \\pmod N$ forms a geometric progression. In addition, we extend our methods to generalized continued fractions with numerators $(b_n)$, obtaining Euler-type identities and weighted error-sum formulae for $\\pi$ and $\\ln 2$.",
"arxiv_id": "2601.07862",
"authors": [
"Kevin Calderon",
"Nikita Kalinin"
],
"categories": [
"math.NT"
],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "Weighted error-sum identities for periodic continued fractions and their generalizations",
"url": "https://arxiv.org/abs/2601.07862",
"version": "v1"
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