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A Formal Proof of a Continued Fraction Conjecture for $π$ Originating from the Ramanujan Machine

Chao Wang

TL;DR

The paper provides a rigorous analytic proof of a Ramanujan Machine conjecture that expresses $-\\pi/4$ as a noncanonical continued fraction. It links the identity to the ratio of contiguous Gauss hypergeometric functions via a hypergeometric kernel $\\mathcal{R}(a,b,c;z)$ and uses a three-phase scheme: kernel derivation, an explicit equivalence transformation with a scaling sequence $\\{r_n\\}$ to impose $b_n=-(3n-2)$, and asymptotic regularization that shows the integer numerators $a_n=-(n-1)(2n-5)$ form a symbolically minimal attractor while preserving convergence. The analysis confirms absolute convergence by the Worpitzky criterion with limit $L=-2/9$ and uses the Thiele-Perron theorem to fix the branch corresponding to $-\\pi/4$, with numerical verification supporting the theoretical results. The work bridges heuristic algorithmic induction with classical hypergeometric theory, offering a general framework for validating Experimental Mathematics identities through discrete transformations and stability analysis.

Abstract

We provide a formal analytic proof for a class of non-canonical polynomial continued fractions representing π/4, originally conjectured by the Ramanujan Machine using algorithmic induction [4]. By establishing an explicit correspondence with the ratio of contiguous Gaussian hypergeometric functions 2F1(a, b; c; z), we show that these identities can be derived via a discrete sequence of equivalence transformations. We further prove that the conjectured integer coefficients constitute a symbolically minimal realization of the underlying analytic kernel. Stability analysis confirms that the resulting limit-periodic structures reside strictly within the Worpitzky convergence disk, ensuring absolute convergence. This work demonstrates that such algorithmically discovered identities are not isolated numerical artifacts, but are deeply rooted in the classical theory of hypergeometric transformations.

A Formal Proof of a Continued Fraction Conjecture for $π$ Originating from the Ramanujan Machine

TL;DR

The paper provides a rigorous analytic proof of a Ramanujan Machine conjecture that expresses as a noncanonical continued fraction. It links the identity to the ratio of contiguous Gauss hypergeometric functions via a hypergeometric kernel and uses a three-phase scheme: kernel derivation, an explicit equivalence transformation with a scaling sequence to impose , and asymptotic regularization that shows the integer numerators form a symbolically minimal attractor while preserving convergence. The analysis confirms absolute convergence by the Worpitzky criterion with limit and uses the Thiele-Perron theorem to fix the branch corresponding to , with numerical verification supporting the theoretical results. The work bridges heuristic algorithmic induction with classical hypergeometric theory, offering a general framework for validating Experimental Mathematics identities through discrete transformations and stability analysis.

Abstract

We provide a formal analytic proof for a class of non-canonical polynomial continued fractions representing π/4, originally conjectured by the Ramanujan Machine using algorithmic induction [4]. By establishing an explicit correspondence with the ratio of contiguous Gaussian hypergeometric functions 2F1(a, b; c; z), we show that these identities can be derived via a discrete sequence of equivalence transformations. We further prove that the conjectured integer coefficients constitute a symbolically minimal realization of the underlying analytic kernel. Stability analysis confirms that the resulting limit-periodic structures reside strictly within the Worpitzky convergence disk, ensuring absolute convergence. This work demonstrates that such algorithmically discovered identities are not isolated numerical artifacts, but are deeply rooted in the classical theory of hypergeometric transformations.
Paper Structure (15 sections, 14 equations, 1 table)

This paper contains 15 sections, 14 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proof of the Main Result
  3. Phase 1: Derivation of the Hypergeometric Kernel
  4. Phase 2: Explicit Equivalence Transformation and Scaling Dynamics
  5. Selection of the Scaling Sequence
  6. Derivation of the Exact Analytic Numerators
  7. Phase 3: Asymptotic Regularization and Limit-Periodicity
  8. Asymptotic Comparison
  9. Convergence and the Worpitzky Criterion
  10. Branch Selection via Thiele-Perron Theorem
  11. Detailed Asymptotic and Convergence Analysis
  12. Second-Order Asymptotic Expansion of $\rho_n$
  13. The Convergence Factor
  14. Numerical Verification
  15. Conclusion

Theorems & Definitions (1)

  • Conjecture 1.1