Guardat en:
| Autor principal: | |
|---|---|
| Format: | Preprint |
| Publicat: |
2026
|
| Matèries: | |
| Accés en línia: | https://arxiv.org/abs/2602.03027 |
| Etiquetes: |
Afegir etiqueta
Sense etiquetes, Sigues el primer a etiquetar aquest registre!
|
Taula de continguts:
- 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.