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 in 2012 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.