Analytic Proof of a Quartic Continued Fraction Identity for $8/\pi^2$ via Operator Decoupling

TL;DR

This paper provides a rigorous analytic proof of a complex continued fraction identity for 8/π², using operator decoupling and algebraic decomposition to connect it to binomial series and confirm its convergence.

Contribution

It introduces an operator-theoretic method to prove a generalized continued fraction identity for a transcendental constant, advancing automated conjecture verification.

Findings

Established the continued fraction identity for 8/π².

Connected the continued fraction to a binomial series involving arcsin.

Confirmed absolute convergence using Pincherle's Theorem.

Abstract

We present a rigorous analytic proof of a generalized continued fraction (GCF) identity for the transcendental constant $8/\pi^2$, a result recently conjectured via the algorithmic framework of the Ramanujan Machine. Distinct from canonical GCFs derived from classical hypergeometric series, the identity at hand features a complex polynomial architecture characterized by quartic partial numerators. Our approach utilizes an algebraic decomposition of the second-order shift operator $\mathcal{L} = \mathcal{T}^2 - b_n \mathcal{T} - a_n$ into a coupled first-order system. This decomposition enables an exact mapping of the higher-order recurrence to a cascaded system, from which the continued fraction is identified as the reciprocal of a binomial series for $(\arcsin)^2$ involving central binomial coefficients. The convergence is established through Pincherle's Theorem: the true minimal solution of the associated difference equation is $f_n = A_n - (8/\pi^2)\,B_n$, which satisfies $f_n/B_n \to 0$, confirming absolute convergence of the continued fraction. This work provides a systematic operator-theoretic methodology for verifying automated conjectures of transcendental constants with high-degree polynomial coefficients.