Analytic Proof of a Quartic Continued Fraction Identity for $8/π^2$ via Operator Decoupling
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866915869503782912 |
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| author | Wang, Chao |
| author_facet | Wang, Chao |
| contents | 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 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 $f_n = A_n - (8/π^2)\,B_n$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03027 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Analytic Proof of a Quartic Continued Fraction Identity for $8/π^2$ via Operator Decoupling Wang, Chao General Mathematics 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 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 $f_n = A_n - (8/π^2)\,B_n$, 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. |
| title | Analytic Proof of a Quartic Continued Fraction Identity for $8/π^2$ via Operator Decoupling |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2602.03027 |