Inequalities and infinite product formula for Ramanujan generalized modular equation function
Let \[ F(a, b ; c ; x)= {}_2F_{1}(a, b ; c ; x)=\sum_{n=0}^{\infty} \frac{(a, n)(b, n)}{(c, n)} \frac{x^{n}}{n !} \quad(|x|<1) \] be the Gaussian hypergeometric function. Let also \(a \in(0,1 / 2], r \in(0,1)\) and \(p>0 .\) Then the Ramanujan generalized modular equation with signature \(1 / a\) and degree \(p\) is given by \[ \frac{F\left(a, 1-a ; 1 ; 1-s^{2}\right)}{F\left(a, 1-a ; 1 ; s^{2}\right)}=p \frac{F\left(a, 1-a ; 1 ; 1-r^{2}\right)}{F\left(a, 1-a ; 1 ; r^{2}\right)} \] Making use of the decreasing homeomorphism \(\mu_{a}:(0,1) \rightarrow(0, \infty)\) defined by \[ \mu_{a}(r)=\frac{\pi}{2 \sin (\pi a)} \frac{F\left(a, 1-a ; 1 ; 1-r^{2}\right)}{F\left(a, 1-a ; 1 ; r^{2}\right)} \] we obtain the Ramanujan generalized modular equation function. The authors introduce several inequalities for \(\mu_{a}(r)\) and provide an infinite product formula for \(\mu_{1/4}(r)\) .
scientific article; zbMATH DE number 6879666
scientific article