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
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.
The notation jn(τ) is derived from Zagier[10] and Tn refers to the relevant McKay–Thompson series.
Note that, as first noticed by J. McKay, the coefficient of the linear term of j(τ) almost equals 196883, which is the degree of the smallest nontrivial irreducible representation of the monster group, a relationship called monstrous moonshine.
Define Then the two modular functions and sequences are related by if the series converges and the sign chosen appropriately, though squaring both sides easily removes the ambiguity.
The first belongs to a family of formulas which were rigorously proven by the Chudnovsky brothers in 1989[11] and later used to calculate 10 trillion digits of π in 2011.
[3] Using Zagier's notation[10] for the modular function of level 2, Note that the coefficient of the linear term of j2A(τ) is one more than 4371 which is the smallest degree greater than 1 of the irreducible representations of the Baby Monster group.
Define, Then, if the series converges and the sign chosen appropriately.
Examples: The first formula, found by Ramanujan and mentioned at the start of the article, belongs to a family proven by D. Bailey and the Borwein brothers in a 1989 paper.
[13] Define, where 782 is the smallest degree greater than 1 of the irreducible representations of the Fischer group Fi23 and, Examples: Define, where the first is the 24th power of the Weber modular function
And, Examples: Define, and, where the first is the product of the central binomial coefficients and the Apéry numbers (OEIS: A005258)[9] Examples: In 2002, Takeshi Sato[7] established the first results for levels above 4.
It involved Apéry numbers which were first used to establish the irrationality of
Another similarity between levels 6 and 10 is J. Conway and S. Norton showed there are linear relations between the McKay–Thompson series Tn,[14] one of which was, or using the above eta quotients jn, A similar relation exists for level 10.
Let, The three sequences involve the product of the central binomial coefficients
Note that the second sequence, α2(k) is also the number of 2n-step polygons on a cubic lattice.
then, as well as, though the formulas using the complements apparently do not yet have a rigorous proof.
For the other modular functions, Define and, Example: No pi formula has yet been found using j7B.
The modular functions can be related as, if the series converges and signs chosen appropriately.
Let, where the first is the product of the central binomial coefficients and OEIS: A006077 (though with different signs).
Furthermore, there are also linear relations between these, or using the above eta quotients jn, Let, their complements, and, though closed forms are not yet known for the last three sequences.
The modular functions can be related as,[15] if the series converges.
In fact, it can also be observed that, Since the exponent has a fractional part, the sign of the square root must be chosen appropriately though it is less an issue when jn is positive.
Define the McKay–Thompson series of class 11A, or sequence (OEIS: A128525) and where, and, No closed form in terms of binomial coefficients is yet known for the sequence but it obeys the recurrence relation, with initial conditions s(0) = 1, s(1) = 4.
Example:[16] As pointed out by Cooper,[16] there are analogous sequences for certain higher levels.
R. Steiner found examples using Catalan numbers
, and for this a modular form with a second periodic for k exists: Other similar series are with the last (comments in OEIS: A013709) found by using a linear combination of higher parts of Wallis-Lambert series for
Using the definition of Catalan numbers with the gamma function the first and last for example give the identities ...
The last is also equivalent to, and is related to the fact that, which is a consequence of Stirling's approximation.