Abstract
We study a one-parameter family of polynomial $ J $-fractions whose coefficients depend polynomially on the index and on an integer parameter $ u\ge0 $. After a factorial normalization of the denominator sequence, the difference of two consecutive convergents factors into the fixed central-binomial Apéry-type term $ 1/(n^2\binom{2n}{n}) $ and a rational factor determined by a normalized polynomial family $ P_u $. We construct $ P_u $ by an explicit parameter-raising operator. We then prove a parameter-shift telescoping identity, which allows induction on $ u $ and gives$ X(u) = \binom{2u}{u}^3\frac{\pi^2}{18}+\rho_u, \qquad \rho_u\in \mathbb{Q} . $Thus, the operator identity and the telescoping identity provide the algebraic mechanism behind the evaluation of the whole family.