A Family of Continued Fractions for Ratios of Zeta Values

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≥ 2 and a suitably chosen polynomial R, we obtain a continued fraction with partial numerators bₙ=-n²ᵐ and partial denominators aₙ given by a simple rule, and we prove that its value is \[ ( 1{R(0)}+∑ₖ₌₁^∞ 1{(k+1)^m R(k)R(k-1)})⁻¹. \] By selecting R appropriately, this family produces closed forms such as \[ -1/ζ(4)+4ζ(2)-8, 2/2ζ(5)+6ζ(3)-9, 2/2ζ(5)-2ζ(3)+1. \]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 $c=1$.