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

For a purely $N$-periodic continued fraction $ξ=[\overline{a_0,a_1,\dots,a_{N-1}}]=[a_0,a_1,\cdots]$, 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 $f_ξ(s)=\sum a_{n+1}\lvert h_n-ξk_n\rvert^s$ 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 $(h_k-ξk_k)$ 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 $π$ and $\ln 2$.