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

• URL: http://arxiv.org/abs/2601.08461v1
• Date: Tue, 13 Jan 2026 11:38:44 GMT
• Title: A Formal Proof of a Continued Fraction Conjecture for $π$ Originating from the Ramanujan Machine
• Abstract summary: We provide a formal analytic proof for a class of non-canonical continued fractions representing /4.<n>We show that these identities can be derived via a discrete sequence of equivalence transformations.<n>This work demonstrates that such algorithmically discovered identities are not isolated numerical artifacts, but are deeply rooted in the classical theory of hypergeometric transformations.
• Score: 2.579369112489757
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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