Entry 40

Let \(\color{red}{q = e^{2\pi i\tau}}\). Given \(\tau = \sqrt{-n}\) or \(\tau = \frac{1+\sqrt{-n}}{2}\), then the solutions \(\alpha, \beta,\gamma\) to the following equations,
$$\frac{_2F_1\big(\tfrac12,\tfrac12;1;\,1-\alpha\big)}{_2F_1\big(\tfrac12,\tfrac12;1;\,\alpha\big)}\,i=\color{red}{\sqrt{4}\,\tau}$$$$\frac{_2F_1\big(\tfrac13,\tfrac23;1;\,1-\beta\big)}{_2F_1\big(\tfrac13,\tfrac23;1;\,\beta\big)}\,i=\color{red}{\sqrt{3}\,\tau}$$$$\frac{_2F_1\big(\tfrac14,\tfrac34;1;\,1-\gamma\big)}{_2F_1\big(\tfrac14,\tfrac34;1;\,\gamma\big)}\,i=\color{red}{\sqrt{2}\,\tau}$$ are given by, $$\;\alpha =\frac{16}{\left(\frac{\eta(\tau)}{\eta(4\tau)}\right)^8+16} = \left(\frac{\sqrt2\,\eta(\tau)\,\eta^2(4\tau)}{\eta^3(2\tau)}\right)^8  = \left(\frac{\vartheta_2(q)}{\vartheta_3(q)}\right)^4$$ $$\beta = \frac{27}{\left(\frac{\eta(\tau)}{\eta(3\tau)}\right)^{12}+27} = \left(\frac{3}{\left(\frac{\eta(\tau/3)}{\eta(3\tau)}\right)^3+3}\right)^3 = \left(\frac{c(q)}{a(q)}\right)^3$$ $$\gamma = \frac{64}{\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24}+64} = \left(\frac{8}{\left(\frac{\eta(\tau/2)}{\eta(2\tau)}\right)^8+8}\right)^2 = \left(\frac{f(q)}{d(q)}\right)^2$$ where the functions of \(\color{red}{q = e^{2\pi i\tau}}\) are the Jacobi and Borwein theta functions discussed in Entry 38. Also, \(\alpha= \lambda(2\tau)\) and \(\lambda(\tau)\) is the modular lambda function.