Polynomial Continued Fractions: A Proved Logarithmic Ladder, a 4/π Casoratian Identity, and 482 Irrational Constants
Papanokechi
Abstract
We prove that the polynomial continued fractions PCF(−kn², (k+1)n+k) and PCF(−n(2n−3), 3n+1) converge respectively to 1/ln(k/(k−1)) for every real k > 1 and to 4/π, and from the second family we derive a rapidly convergent identity for π/4. The key tool is a combination of closed-form convergents and a discrete Casoratian telescope for the underlying three-term recurrence. This differs from the classical Gauss and Brouncker formulas in that the evaluation is obtained directly inside the recurrence, with error O(2^{−N}/N^{7/2}) for the resulting series. As a computational application, we certify 482 quadratic generalized continued fractions against a basis of 15 standard constants using Arb ball arithmetic at 10,000-bit precision and depth N=5000. These computations yield 1500+ correct digits and rigorous irrationality statements for 482 previously unclassified constants. We also prove that the 4/π family cannot be expressed as a Gauss ₂F₁ continued fraction, because its term ratio is forced into a ₂F₃ form over ℚ(√5). Code and verification scripts available at https://github.com/papanokechi/pcf-research (with a dedicated companion repository at https://github.com/papanokechi/pcf-casoratian-identities). Comments welcome. The author invites technical assessment from specialists in polynomial-continued-fraction irrationality methods, hypergeometric identities, and PSL2(ℤ) modular obstructions. Feedback of highest value: prior-art pointers (especially earlier closed-form treatments of the PCF(−kn², (k+1)n+k) family or of the 4/π Casoratian telescope), gaps or strengthenings in the logarithmic-ladder convergence argument, and alternative derivations of the 4/π identity that do not route through the ₂F₃ over ℚ(√5) obstruction. Independent reproductions of the Arb-interval-arithmetic certification at depth N=5000 are also of interest. Contact: GitHub Discussions (preferred for technical exchange) or email [email protected].