The three missing terms in Ramanujan’s septic theta function identity-Reference-Cited by-同舟云学术

The three missing terms in Ramanujan’s septic theta function identity

Published:2023-02-23 Issue:4 Volume:60 Page:885-911
ISSN:1382-4090
Container-title:The Ramanujan Journal
language:en
Short-container-title:Ramanujan J

Author:

Rebák Örs^ORCID

Abstract

AbstractOn page 206 in his lost notebook, Ramanujan recorded the following enigmatic identity for his theta function

$$\varphi (q)$$

φ ( q ) :

$$\begin{aligned} \varphi (e^{-7\pi \sqrt{7}}) = 7^{-3/4}\varphi (e^{-\pi \sqrt{7}})\big \{1 + (\quad )^{2/7} + (\quad )^{2/7} + (\quad )^{2/7}\big \}. \end{aligned}$$

φ ( e - 7 π 7 ) = 7 - 3 / 4 φ ( e - π 7 ) { 1 + ( ) 2 / 7 + ( ) 2 / 7 + ( ) 2 / 7 } . We give the three missing terms. In addition, we calculate the class invariant

$$G_{343}$$

G 343 and further special values of

$$\varphi (e^{-n\pi }),$$

φ ( e - n π ) , for

$$n = 7, 21, 35,$$

n = 7 , 21 , 35 , and 49.

Funder

UiT The Arctic University of Norway

Publisher

Springer Science and Business Media LLC

Subject

Algebra and Number Theory

Link

https://link.springer.com/content/pdf/10.1007/s11139-022-00584-2.pdf

Reference38 articles.

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4. Berndt, B.C.: Ramanujan’s Notebooks. Part III. Springer, New York (1991)

5. Berndt, B.C.: Ramanujan’s theory of theta-function. In: Ram Murty, M. (ed.) Theta Functions, From the Classical to the Modern, Centre de Recherches Mathématiques Proceedings and Lecture Notes, vol. 1, pp. 1–63. American Mathematical Society, Providence (1993)

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