Weighted error-sum identities for periodic continued fractions and their generalizations

TL;DR

This paper derives explicit formulas for weighted error sums in periodic continued fractions, revealing geometric progressions in approximation errors and extending results to generalized fractions involving important constants like pi and ln 2.

Contribution

It provides explicit expressions for weighted error sums in periodic continued fractions and extends these methods to generalized fractions involving fundamental constants.

Findings

Error subsequences form geometric progressions

Explicit formulas for weighted error sums are derived

Extensions to generalized continued fractions for pi and ln 2

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 $\ln 2$.