What question did this study set out to answer?

The aim is to present the Judsan Formula, a hypergeometric framework for 1/π series with variable convergence rates.

July 12, 20260 citationsOpen Access

The Judsan Formula: A Unified 3F₂ Framework for 1/π Series with Tunable Convergence

Key Points

The aim is to present the Judsan Formula, a hypergeometric framework for 1/π series with variable convergence rates.
Developed a unified 3F₂ framework containing Ramanujan-type series for 1/π.
Established tunable convergence speed for parameters s > 1 and level ℓ in natural numbers.
Performed numerical verification for various special cases of the formula.
Showed that as s approaches infinity, the convergence speed increases significantly.
Identified special cases such as Chudnovsky's and Ramanujan's formulas that fit within this framework.
Proved that under the Complex Multiplication condition, coefficients are algebraic numbers.

Abstract

I present the Judsan Formula, a unified 3F₂ hypergeometric framework that contains a broad class of Ramanujan-type series for 1/π as special cases. For any real parameter s > 1 and level ℓ ∈ N, I construct a series with tunable convergence speed. The Chudnovsky formula (ℓ=6, s=163/4), Ramanujan's 9801 formula (ℓ=4, s=58/4), and the level 3 Ramanujan formula (ℓ=3, s=9) appear as special cases. When s satisfies the Complex Multiplication condition, the coefficients α and β are exact algebraic numbers in Q̄. As s → ∞, the convergence becomes arbitrarily fast. This is the first known 1/π formula with tunable convergence speed. Numerical verification is provided for all cases.

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Cite This Study

Judsan Niyakaran (Fri,) studied this question.

synapsesocial.com/papers/6a532fb44f7abc118aded1a5 https://doi.org/https://doi.org/10.5281/zenodo.21296438

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