数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (4): 1482-1496.doi: 10.1007/s10473-025-0413-2

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PROOFS OF CONJECTURES ON RAMANUJAN-TYPE SERIES OF LEVEL 3

John M. CAMPBELL   

  1. Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada
  • 收稿日期:2024-04-19 修回日期:2024-08-12 出版日期:2025-10-10 发布日期:2025-10-10

PROOFS OF CONJECTURES ON RAMANUJAN-TYPE SERIES OF LEVEL 3

John M. CAMPBELL   

  1. Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada
  • Received:2024-04-19 Revised:2024-08-12 Online:2025-10-10 Published:2025-10-10
  • About author:John M. CAMPBELL, E-mail: jmaxwellcampbell@gmail.com

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.

关键词: Ramanujan-type series, complete elliptic integral, modular relation

Abstract: 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.

Key words: Ramanujan-type series, complete elliptic integral, modular relation