Ramanujan–Sato series
Unveiling math links by Ramanujan & Shimura in number theory / From Wikipedia, the free encyclopedia
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In mathematics, a Ramanujan–Sato series[1][2] generalizes Ramanujan’s pi formulas such as,
to the form
by using other well-defined sequences of integers obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients , 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 ,[3] while G. Almkvist has experimentally found numerous other examples also with a general method using differential operators.[4]
Levels 1–4A were given by Ramanujan (1914),[5] level 5 by H. H. Chan and S. Cooper (2012),[3] 6A by Chan, Tanigawa, Yang, and Zudilin,[6] 6B by Sato (2002),[7] 6C by H. Chan, S. Chan, and Z. Liu (2004),[1] 6D by H. Chan and H. Verrill (2009),[8] level 7 by S. Cooper (2012),[9] part of level 8 by Almkvist and Guillera (2012),[2] part of level 10 by Y. Yang, and the rest by H. H. Chan and S. Cooper.
The notation jn(τ) is derived from Zagier[10] and Tn refers to the relevant McKay–Thompson series.