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-ln 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 \[ a_0+{b_1}{a_1+{b_2}{a_2+{b_3}{a_3+⋯}}}, \] with \[ a_n=(a_0/2+t)n^2+(3a_0/2+t)n+a_0, b_n=-a_0 t/2\,n^2(n+1)^2, \] and prove that its limit is \[ a_0 r^2/2(r+(1-r)ln(1-r)), r=2t/a_0. \] 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.