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Analytic Proof of a Quartic Continued Fraction Identity for $8/π^2$ via Operator Factorization

Chao Wang

TL;DR

The paper addresses the problem of proving a RAMANUJAN-Machine conjecture: a quartic generalized continued fraction equaling $8/\\pi^2$. It develops an operator-theoretic approach that factorizes the second-order recurrence into a cascaded first-order system via auxiliary sequences $c_n$ and $d_n$, yielding explicit numerator and denominator trajectories and an Apéry-like summation. The key result is $S=\\sum_{m=1}^{\\infty} \frac{2^{m}}{m^{2} \binom{2m}{m}} = \pi^{2}/8$, which together with Pincherle's criterion gives the continued fraction value $K=8/\\pi^{2}$. This work provides a rigorous, systematic framework for verifying automated conjectures about transcendental constants with high-degree polynomial coefficients, with potential broad applicability to operator-factorization methods in discrete analysis.

Abstract

We present a rigorous analytic proof of a generalized continued fraction (GCF) identity for the transcendental constant $8/π^2$, 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 an Apéry-like summation involving central binomial coefficients. The convergence is established through Pincherle's Theorem, confirming that the numerator sequence constitutes the minimal solution to the associated difference equation. This work provides a systematic operator-theoretic methodology for verifying automated conjectures of transcendental constants with high-degree polynomial coefficients.

Analytic Proof of a Quartic Continued Fraction Identity for $8/π^2$ via Operator Factorization

TL;DR

The paper addresses the problem of proving a RAMANUJAN-Machine conjecture: a quartic generalized continued fraction equaling . It develops an operator-theoretic approach that factorizes the second-order recurrence into a cascaded first-order system via auxiliary sequences and , yielding explicit numerator and denominator trajectories and an Apéry-like summation. The key result is , which together with Pincherle's criterion gives the continued fraction value . This work provides a rigorous, systematic framework for verifying automated conjectures about transcendental constants with high-degree polynomial coefficients, with potential broad applicability to operator-factorization methods in discrete analysis.

Abstract

We present a rigorous analytic proof of a generalized continued fraction (GCF) identity for the transcendental constant , 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 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 an Apéry-like summation involving central binomial coefficients. The convergence is established through Pincherle's Theorem, confirming that the numerator sequence constitutes the minimal solution to the associated difference equation. This work provides a systematic operator-theoretic methodology for verifying automated conjectures of transcendental constants with high-degree polynomial coefficients.
Paper Structure (16 sections, 4 theorems, 35 equations)

This paper contains 16 sections, 4 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the Main Result
  4. Phase 1: Shift Operator Factorization
  5. Phase 2: Analytic Construction of the Solution via Operator Decoupling
  6. Derivation of the Denominator $\{B_n\}$
  7. Derivation of the Numerator $\{A_n\}$
  8. Series Representation of the Convergent Ratio
  9. Phase 3: Asymptotic Evaluation and Transcendental Summation
  10. Detailed Algebraic Reduction of the General Term
  11. Summation and Identification of the Apéry-like Series
  12. Transcendental Evaluation and Pincherle's Criterion
  13. Convergence Analysis and the Theory of Minimal Solutions
  14. Pincherle’s Criterion: The General Framework
  15. Proof of Convergence for the Ramanujan GCF
  16. ...and 1 more sections

Key Result

Theorem 2.1

The numerator sequence $\{A_n\}$ and the denominator sequence $\{B_n\}$ of the $n$-th convergent satisfy the same second-order linear homogeneous difference equation: The sequences are uniquely determined by their respective initial frames:

Theorems & Definitions (8)

  • Conjecture 1
  • Theorem 2.1: Fundamental Recurrence Formulas
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Theorem 4.1: Pincherle's Theorem Gautschi1967
  • Proposition 4.2: Identification of the Solution Classes
  • proof