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 ${\displaystyle s(k)}$ obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients ${\displaystyle {\tbinom {n}{k}}}$, and 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 ${\displaystyle \Gamma _{0}(n)}$, while G. Almkvist has experimentally found numerous other examples also with a general method using differential operators.

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