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

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 $f_{\xi }(s)=\sum a_{n+1}|h_n-\xi k_n{|}^s$ 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 $k\equiv k_0(\mathrm{mod}N)$ 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 $\ln 2$.