Inequalities and infinite product formula for Ramanujan generalized modular equation function

• DOI: 10.1007/s11139-017-9888-3
• Publication Date: 2017-03-20T06:28:05Z
• Authors: Miao-Kun Wang; Yong-Min Li; Yu-Ming Chu
• Keywords: Infinite product; Modular equation; Ramanujan tau function

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.