Abstract

The Ramanujan Machine project has catalogued many conjectural continued fraction identities involving zeta constants.
In this paper we present a unified construction that produces a family of continued fractions whose limits are rational combinations of zeta values.
For any integer $m\ge 2$ and a suitably chosen polynomial $R$, we obtain a continued fraction with partial numerators $b_n=-n^{2m}$ and partial denominators $a_n$ given by a simple rule, and we prove that its value is
  \[
  \Bigl( \frac1{R(0)}+\sum_{k=1}^\infty \frac1{(k+1)^m R(k)R(k-1)}\Bigr)^{-1}.
  \]
  By selecting $R$ appropriately, this family produces closed forms such as
  \[
  -\frac{1}{\zeta(4)+4\zeta(2)-8},\qquad
  \frac{2}{2\zeta(5)+6\zeta(3)-9},\qquad
  \frac{2}{2\zeta(5)-2\zeta(3)+1}.
  \]
The proof uses an explicit factorial--polynomial closed form for the numerator sequence, a telescoping difference equation, and elementary partial fraction techniques.
The results confirm several identities proposed by the Ramanujan Machine (after correcting suspected typographical errors in the original conjectures).
We also present a natural one‑parameter extension that yields families depending on an additional scale parameter $c$, which contains the original family as the special case $c=1$.

Notes (English)

Methods (English)

LLM Declaration

The author used LLM in this article. The interaction history is provided in AI_Interaction_Logs.zip.