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Analytic Regularization of a Ramanujan Machine Conjecture

Chao Wang

Abstract

We provide a formal analytic derivation of a continued fraction identity for $-π/4$ recently conjectured by the Ramanujan Machine~\cite{Raayoni2021}. By utilizing the contiguous relations of the Gauss hypergeometric function ${}_2F_1(a, b; c; z)$, we establish that the conjectured polynomial architecture is a regularized representation of the transcendental ratio $\mathcal{R}(1/2, 0, 1/2; -1)$. Through an explicit equivalence transformation $\mathcal{T}$ defined by a linear scaling sequence, we map the Gaussian unit-denominator expansion to the conjectured form, thereby recovering the quadratic partial numerators $(n-1)^2$ and linear partial denominators $-(2n-1)$. Convergence is rigorously established via the limit-periodicity of the transformed coefficients, which reside on the Worpitzky boundary $L=1/4$.

Analytic Regularization of a Ramanujan Machine Conjecture

Abstract

We provide a formal analytic derivation of a continued fraction identity for recently conjectured by the Ramanujan Machine~\cite{Raayoni2021}. By utilizing the contiguous relations of the Gauss hypergeometric function , we establish that the conjectured polynomial architecture is a regularized representation of the transcendental ratio . Through an explicit equivalence transformation defined by a linear scaling sequence, we map the Gaussian unit-denominator expansion to the conjectured form, thereby recovering the quadratic partial numerators and linear partial denominators . Convergence is rigorously established via the limit-periodicity of the transformed coefficients, which reside on the Worpitzky boundary .
Paper Structure (7 sections, 17 equations, 1 table)

This paper contains 7 sections, 17 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proof of the Main Result
  3. Analytic Correspondence and Hypergeometric Evaluation
  4. Phase 2: Explicit Equivalence Transformation
  5. Phase 3: Analytic Convergence and Asymptotic Stability
  6. Numerical Convergence Analysis
  7. Discussion and Conclusion

Theorems & Definitions (1)

  • Conjecture 1.1: Raayoni2021