Ramanujan’s Theory of Elliptic Functions to the Cubic Base

Heng Huat Chan; Zhi Guo Liu

doi:10.1007/s00025-025-02399-1

Ramanujan’s Theory of Elliptic Functions to the Cubic Base

Heng Huat Chan^*
, Zhi Guo Liu

^*Corresponding author for this work

School of Mathematical Sciences

Shandong University

Research output: Contribution to journal › Article › peer-review

Abstract

In this article, we develop Ramanujan’s theory of elliptic functions to the cubic base using Jacobi’s theta functions. Our new approach does not involve the theta series discovered by J. M. Borwein and P. B. Borwein, Goursat’s transformation formulas for the hypergeometric series, analogue of Gauss’ AGM and the theory of modular forms.

Original language	English
Article number	81
Journal	Results in Mathematics
Volume	80
Issue number	3
DOIs	https://doi.org/10.1007/s00025-025-02399-1
State	Published - May 2025

Keywords

Theta functions
cubic elliptic functions
hypergeometric functions

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10.1007/s00025-025-02399-1

Cite this

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abstract = "In this article, we develop Ramanujan{\textquoteright}s theory of elliptic functions to the cubic base using Jacobi{\textquoteright}s theta functions. Our new approach does not involve the theta series discovered by J. M. Borwein and P. B. Borwein, Goursat{\textquoteright}s transformation formulas for the hypergeometric series, analogue of Gauss{\textquoteright} AGM and the theory of modular forms.",

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KW - Theta functions

KW - cubic elliptic functions

KW - hypergeometric functions

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Ramanujan’s Theory of Elliptic Functions to the Cubic Base

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Keywords

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