Entry 89

The modular lambda function \(\lambda(\tau)\)$$\lambda(\tau) = \left(\frac{\sqrt{2}\,\eta(\tfrac{\tau}{2})\,\eta^2(2\tau)}{\eta^3(\tau)}\right)^8$$ given by Mathematica as ModularLambda[tau] can solve, $$\frac{_2F_1\big(\tfrac12,\tfrac12,1,1-\lambda(\tau)\big)}{_2F_1\big(\tfrac12,\tfrac12,1,\lambda(\tau)\big)}  = -\tau\sqrt{-1}$$ For the special case \(\tau=\sqrt{-11}\) $$\frac{_2F_1\big(\tfrac12,\tfrac12,1,1-u\big)}{_2F_1\big(\tfrac12,\tfrac12,1,u\big)}  = \sqrt{11}$$ we can can use the tribonacci constant \(T\), the real root of \(T^3-T^2-T-1=0\). The solution in terms of \(T\) is $$u=\lambda(\tau)=\frac14\left(2-\sqrt{\frac{2T+15}{2T+1}}\right) = \frac14\left(2-\sqrt{\frac{17S+2}{3S+2}}\right)= 0.00047741\dots$$ or in terms of \(S\) as the nice nested cube roots $$S = \frac1{T-1}=\sqrt[3]{\frac12+\sqrt[3]{\frac12+\sqrt[3]{\frac12+\sqrt[3]{\frac12+\dots}}}}$$ the cubic version of the golden ratio's $$\phi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}$$