Quadratic irrational analogues of Ramanujan's series for $1/\ddot{I}€$

John M. Campbell, Shaun Cooper, Dongxi Ye

About 40 years ago Jonathan and Peter Borwein discovered the series identity $$\sum _{n=0}^{\infty }\frac{(-1{)}^n(6n)!}{(3n)!(n!{)}^3}\frac{(A+nB)}{C^{n+1/2}}=\frac{1}{12\ddot{I}€}$$ where…