A Family of Continued Fractions with Closed Forms

In a previous work, the author proved a conjectured continued fraction identity catalogued by the Ramanujan Machine project. The identity involved the constant 1/ (1- 2). In this paper we show that the original continued fraction is a special case of a large family that can be evaluated in closed form. We derive a two-parameter family of continued fractions \ a₀+b₁a₁+b₂{a₂+b₃{a₃+}}, \ with \ aₙ= (a₀2+t) n²+ (3a₀2+t) n+a₀, bₙ=-a₀ t2\, n² (n+1) ², \ and prove that its limit is \ a₀ r²2 (r+ (1-r) (1-r) ), r=2ta₀. \ The proof uses an explicit closed form for the numerator sequence Pₙ, a telescoping difference equation, and elementary integration. We also discuss the analytic continuation of the formula to complex parameters and outline higher-order analogues that lead to combinations of logarithms and rational functions.