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A MODULAR PROOF OF TWO OF RAMANUJAN’S FORMULAE FOR $1/\unicode[STIX]{x1D70B}$

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 February 2019

YUE ZHAO Open the ORCID record for YUE ZHAO [Opens in a new window]
YUE ZHAO*
Affiliation:
ECSE Department, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180, USA email zhaoy11@rpi.edu
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Abstract

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In this article, we give new proofs of two of Ramanujan’s $1/\unicode[STIX]{x1D70B}$ formulae

$$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70B}}=\frac{2\sqrt{2}}{99^{2}}\mathop{\sum }_{m=0}^{\infty }(26390m+1103)\frac{(4m)!}{396^{4m}(m!)^{4}}\end{eqnarray}$$
and
$$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70B}}=\frac{2}{84^{2}}\mathop{\sum }_{m=0}^{\infty }(21460m+1123)\frac{(-1)^{m}(4m)!}{(84\sqrt{2})^{4m}(m!)^{4}}\end{eqnarray}$$
 using the theory of modular forms. The method can also be used to prove other classical $1/\unicode[STIX]{x1D70B}$ formulae.

Keywords

Ramanujan seriestheta seriesDedekind eta functionsmodular forms

MSC classification

Primary: 11F03: Modular and automorphic functions
Secondary: 11F20: Dedekind eta function, Dedekind sums 11F27: Theta series; Weil representation; theta correspondences

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 109 , Issue 1 , August 2020 , pp. 131 - 144
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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