Ramanujan's cubic transformation and generalized modular equation

Ramanujan's cubic transformation and generalized modular equation

We study the quotient of hypergeometric functions \begin{equation*} \mu_{a}^*(r)=\frac{\pi}{2\sin{(\pi a)}}\frac{F(a,1-a;1;1-r^3)}{F(a,1-a;1;r^3)} \quad (r\in(0,1)) \end{equation*} in the theory of Ramanujan's generalized modular equation for \(a\in(0,1/2]\), find an infinite product formula fo...

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Veröffentlicht in:arXiv.org 2013-05
Hauptverfasser: Wang, Miaokun, Chu, Yuming, Jiang, Yueping
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Formulas (mathematics)
Hypergeometric functions
Transformations (mathematics)
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description We study the quotient of hypergeometric functions \begin{equation*} \mu_{a}^*(r)=\frac{\pi}{2\sin{(\pi a)}}\frac{F(a,1-a;1;1-r^3)}{F(a,1-a;1;r^3)} \quad (r\in(0,1)) \end{equation*} in the theory of Ramanujan's generalized modular equation for \(a\in(0,1/2]\), find an infinite product formula for \(\mu_{1/3}^*(r)\) by use of the properties of \(\mu_{a}^*(r)\) and Ramanujan's cubic transformation. Besides, a new cubic transformation formula of hypergeometric function is given, which complements the Ramanujan's cubic transformation.
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Transformations (mathematics)
title Ramanujan's cubic transformation and generalized modular equation
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