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<p>The level 3 case for Ramanujan-type series has been considered as the most mysterious and the most challenging, out of all possible levels for Ramanujan-type series. This motivates the development of new techniques for constructing Ramanujan-type series of level 3. Chan and Liaw introduced an alternating analogue of the Borwein brothers’ identity for Ramanujan-type series of level 3; subsequently, Chan, Liaw, and Tian formulated another proof of the Chan–Liaw identity, via the use of Ramanujan’s class invariant. Using the elliptic lambda function and the elliptic alpha function, we prove, via a limiting case of the Kummer–Goursat transformation, a new identity for evaluating the summands for alternating Ramanujan-type series of level 3, and we apply this new identity to prove three conjectured formulas for quadratic-irrational, Ramanujan-type series that had been discovered via numerical experiments with Maple in 2012 by Aldawoud. We also apply our identity to prove a new Ramanujan-type series of level 3 with a quartic convergence rate and quartic coefficients.</p>

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Proofs of conjectures on Ramanujan-type series of level 3

  • 期刊论文

  • 发表日期：

    2025年4月8日

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  • John M. Campbell

摘要

The level 3 case for Ramanujan-type series has been considered as the most mysterious and the most challenging, out of all possible levels for Ramanujan-type series. This motivates the development of new techniques for constructing Ramanujan-type series of level 3. Chan and Liaw introduced an alternating analogue of the Borwein brothers’ identity for Ramanujan-type series of level 3; subsequently, Chan, Liaw, and Tian formulated another proof of the Chan–Liaw identity, via the use of Ramanujan’s class invariant. Using the elliptic lambda function and the elliptic alpha function, we prove, via a limiting case of the Kummer–Goursat transformation, a new identity for evaluating the summands for alternating Ramanujan-type series of level 3, and we apply this new identity to prove three conjectured formulas for quadratic-irrational, Ramanujan-type series that had been discovered via numerical experiments with Maple in 2012 by Aldawoud. We also apply our identity to prove a new Ramanujan-type series of level 3 with a quartic convergence rate and quartic coefficients.