A method for proving Ramanujan's series for \(1/\pi\) (Q2181590)

Ramanujan obtained a list of 17 extraordinary summations for the number $1/\pi$, such as \[ \sum_{n\geq0}\frac{(\frac12)_n^3}{(n!)^3}\,\frac{42n+15}{64^n}=\frac{16}{\pi}. \] This paper explains a general method of establishing these formulas. The basic idea, following a suggestion of J.Wan, is based on the hypergeometric function $F_s(\alpha)={}_2F_1(s^{-1},1-s^{-1}; 1; \alpha)$ and its derivative $\alpha \frac{d}{d\alpha} (F_s(\alpha))$, combined with Legendre's relation and Clausen's identity expressing $(F_s(\alpha))^2$ as a sum. The technique is described for series of the above kind consisting of positive terms and also alternating series. The method relies on a modular transformation of the form \[ F_s(\alpha)=m_s(\alpha,\beta) F_s(\beta), \] where the multiplier $m_s(\alpha, \beta)$ is given explicitly when $s=2, 3, 4, 6$.