Abstract
We present several inequalities for the Ramanujan generalized modular equation function \(\mu _{a}(r)=\pi F(a,1-a;1;1-r^2)/\) \([2\sin (\pi a)F(a,1-a;1;r^2)]\) with \(a\in (0,1/2]\) and \(r\in (0,1)\), and provide an infinite product formula for \(\mu _{1/4}(r)\), where \(F(a,b;c;x)={}_{2}F_{1}(a,b;c;x)\) is the Gaussian hypergeometric function.
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The research was supported by the Natural Science Foundation of China under Grants 61673169, 11371125, 11601485, and 11401191, the Tianyuan Special Founds of the Natural Science Foundation of China under Grant 11626101, and the Natural Science Foundation of the Department of Education of Zhejiang Province under Grant Y201635325.
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Wang, MK., Li, YM. & Chu, YM. Inequalities and infinite product formula for Ramanujan generalized modular equation function. Ramanujan J 46, 189–200 (2018). https://doi.org/10.1007/s11139-017-9888-3
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DOI: https://doi.org/10.1007/s11139-017-9888-3
Keywords
- Gaussian hypergeometric function
- Ramanujan generalized modular equation
- Quadratic transformation
- Infinite product