 Ramanujan–Sato series

In mathematics, a Ramanujan–Sato series generalizes Ramanujan’s pi formulas such as,

to the form

by using other well-defined sequences of integers formula_3 obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients formula_4, and formula_5 employing modular forms of higher levels.Ramanujan made the enigmatic remark that there were "corresponding theories", but it was only recently that H. H. Chan and S. Cooper found a general approach that used the underlying modular congruence subgroup formula_6, while G. Almkvist has experimentally found numerous other examples also with a general method using differential operators.