Ramanujan–Sato Series
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Ramanujan–Sato series generalizes Ramanujan's pi formulas such as, :\frac = \frac \sum_^\infty \frac \frac to the form :\frac = \sum_^\infty s(k) \frac by using other well-defined
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
s of
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s s(k) obeying a certain
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
, sequences which may be expressed in terms of
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s \tbinom, and A,B,C employing
modular form In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...
s 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 In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example is the subgroup of invertible integer matrices of determinant 1 in which the off-diag ...
\Gamma_0(n), while G. Almkvist has experimentally found numerous other examples also with a general method using
differential operators In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
. Levels ''1–4A'' were given by Ramanujan (1914), level ''5'' by H. H. Chan and S. Cooper (2012), ''6A'' by Chan, Tanigawa, Yang, and Zudilin, ''6B'' by Sato (2002), ''6C'' by H. Chan, S. Chan, and Z. Liu (2004), ''6D'' by H. Chan and H. Verrill (2009), level ''7'' by S. Cooper (2012), part of level ''8'' by Almkvist and Guillera (2012), part of level ''10'' by Y. Yang, and the rest by H. H. Chan and S. Cooper. The notation ''j''''n''(''τ'') is derived from Zagier and ''T''''n'' refers to the relevan
McKay–Thompson series


Level 1

Examples for levels 1–4 were given by Ramanujan in his 1917 paper. Given q=e^ as in the rest of this article. Let, :\begin j(\tau) &= \left(\frac\right)^3 = \frac + 744 + 196884q + 21493760q^2 +\cdots\\ j^*(\tau) &= 432\,\frac = \frac - 120 + 10260q - 901120q^2 + \cdots \end with the j-function ''j''(''τ''),
Eisenstein series Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generalize ...
''E''4, and
Dedekind eta function In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string ...
''η''(''τ''). The first expansion is the McKay–Thompson series of class 1A () with a(0) = 744. 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 In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
of the
monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group; it has order :    : = 2463205976112133171923293 ...
, a relationship called
monstrous moonshine In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular the ''j'' function. The initial numerical observation was made by John McKay in 1978, ...
. Similar phenomena will be observed in the other levels. Define :s_(k)=\binom\binom\binom=1, 120, 83160, 81681600,\ldots () :s_(k)=\sum_^k\binom\binom\binom\binom(-432)^ =1, -312, 114264, -44196288,\ldots Then the two modular functions and sequences are related by :\sum_^\infty s_(k)\,\frac= \pm \sum_^\infty s_(k)\,\frac if the series converges and the sign chosen appropriately, though squaring both sides easily removes the ambiguity. Analogous relationships exist for the higher levels. Examples: :\frac = 12\,\boldsymbol\,\sum_^\infty s_(k)\,\frac,\quad j\left(\frac\right)=-640320^3=-262537412640768000 :\frac = 24\,\boldsymbol\,\sum_^\infty s_(k)\,\frac,\quad j^*\left(\frac\right) = -432\,U_^=-432\left(\frac\right)^ where 645=43\times15, and U_n is a fundamental unit. The first belongs to a family of formulas which were rigorously proven by the Chudnovsky brothers in 1989 and later used to calculate 10 trillion digits of π in 2011. The second formula, and the ones for higher levels, was established by H.H. Chan and S. Cooper in 2012.


Level 2

Using Zagier's notation for the modular function of level 2, :\begin j_(\tau) &=\left(\left(\frac\right)^+2^6 \left(\frac\right)^\right)^2 = \frac + 104 + 4372q + 96256q^2 + 1240002q^3+\cdots \\ j_(\tau) &= \left(\frac\right)^ = \frac - 24 + 276q - 2048q^2 + 11202q^3 - \cdots \end Note that the coefficient of the linear term of ''j''2A(''τ'') is one more than 4371 which is the smallest degree greater than 1 of the irreducible representations of the Baby Monster group. Define, :s_(k)=\binom\binom\binom=1, 24, 2520, 369600, 63063000,\ldots () :s_(k)=\sum_^k\binom\binom\binom\binom(-64)^=1, -40, 2008, -109120, 6173656,\ldots Then, :\sum_^\infty s_(k)\,\frac= \pm \sum_^\infty s_(k)\,\frac if the series converges and the sign chosen appropriately. Examples: :\frac = 32\sqrt\,\sum_^\infty s_(k)\,\frac,\quad j_\left(\frac\sqrt\right)=396^4=24591257856 :\frac = 16\sqrt\,\sum_^\infty s_(k)\,\frac,\quad j_\left(\frac\sqrt\right)=64\left(\frac\right)^=64\,U_^ 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.


Level 3

Define, :\begin j_(\tau) &=\left(\left(\frac\right)^+3^3 \left(\frac\right)^\right)^2 = \frac + 42 + 783q + 8672q^2 +65367q^3+\cdots\\ j_(\tau) &= \left(\frac\right)^ = \frac - 12 + 54q - 76q^2 - 243q^3 + 1188q^4 + \cdots\\ \end where 782 is the smallest degree greater than 1 of the irreducible representations of the
Fischer group In the area of modern algebra known as group theory, the Fischer groups are the three sporadic simple groups Fi22, Fi23 and Fi24 introduced by . 3-transposition groups The Fischer groups are named after Bernd Fischer who discovered them ...
''Fi''23 and, :s_(k)=\binom\binom\binom=1, 12, 540, 33600, 2425500,\ldots () :s_(k)=\sum_^k\binom\binom\binom\binom(-27)^=1, -15, 297, -6495, 149481,\ldots Examples: :\frac = 2\,\boldsymbol\,\sum_^\infty s_(k)\,\frac,\quad j_\left(\frac\right) = -300^3 = -27000000 :\frac = \boldsymbol\,\sum_^\infty s_(k)\,\frac,\quad j_\left(\frac\right)=-27\,\left(500+53\sqrt\right)^2=-27\,U_^


Level 4

Define, :\begin j_(\tau)&=\left(\left(\frac\right)^+4^2 \left(\frac\right)^\right)^2 = \left(\frac \right)^ =-\left(\frac \right)^ = \frac + 24+ 276q + 2048q^2 +11202q^3+\cdots\\ j_(\tau) &= \left(\frac\right)^ = \frac -8 + 20q - 62q^3 + 216q^5 - 641q^7 + \ldots\\ \end where the first is the 24th power of the Weber modular function \mathfrak(2\tau). And, :s_(k)=\binom^3=1, 8, 216, 8000, 343000,\ldots () :s_(k)=\sum_^k\binom^3\binom(-16)^= (-1)^k \sum_^k\binom^2\binom^2 =1, -8, 88, -1088, 14296,\ldots () Examples: :\frac = 8\,\boldsymbol\,\sum_^\infty s_(k)\,\frac,\quad j_\left(\frac\right)=-2^9=-512 :\frac = 16\,\boldsymbol\,\sum_^\infty s_(k)\,\frac,\quad j_\left(\frac\right) = -16\,\left(1+\sqrt\right)^4=-16\,U_^


Level 5

Define, :\begin j_(\tau)&=\left(\frac\right)^+5^3 \left(\frac\right)^+22 =\frac + 16 + 134q + 760q^2 +3345q^3+\cdots\\ j_(\tau)&=\left(\frac\right)^= \frac- 6 + 9q + 10q^2 - 30q^3 + 6q^4 + \cdots \end and, :s_(k)=\binom\sum_^k \binom^2\binom =1, 6, 114, 2940, 87570,\ldots :s_(k)=\sum_^k(-1)^\binom^3\binom=1, -5, 35, -275, 2275, -19255,\ldots () where the first is the product of the
central binomial coefficient In mathematics the ''n''th central binomial coefficient is the particular binomial coefficient : = \frac \textn \geq 0. They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle. The first ...
s and the Apéry numbers () Examples: :\frac = \frac\,\boldsymbol\,\sum_^\infty s_(k)\,\frac,\quad j_\left(\frac\right)=-15228=-(18\sqrt)^2 :\frac = \frac\,\boldsymbol\,\sum_^\infty s_(k)\,\frac,\quad j_\left(\frac\right)=-5\sqrt\,\left(\frac\right)^=-5\sqrt\,U_^


Level 6


Modular functions

In 2002, Takeshi Sato established the first results for levels above 4. It involved Apéry numbers which were first used to establish the irrationality of \zeta(3). First, define, :\beginj_(\tau) &=\left(\sqrt - \frac\right)^2 = \left(\sqrt + \frac\right)^2 = \left(\sqrt + \frac\right)^2-4 =\frac + 10 + 79q + 352q^2 +\cdots \end :\beginj_(\tau) &= \left(\frac\right)^=\frac + 12 + 78q + 364q^2 + 1365q^3+\cdots \end :\beginj_(\tau) &= \left(\frac\right)^=\frac -6 + 15q -32q^2 + 87q^3-192q^4+\cdots \end :\beginj_(\tau) &= \left(\frac\right)^=\frac -4 - 2q + 28q^2 - 27q^3 - 52q^4+\cdots\end :\beginj_(\tau) &= \left(\frac\right)^=\frac +3 + 6q + 4q^2 - 3q^3 - 12q^4 +\cdots\end The phenomenon of j_ being squares or a near-square of the other functions will also be manifested by j_. Another similarity between levels 6 and 10 is J. Conway and S. Norton showed there are linear relations between the McKay–Thompson series ''T''''n'', one of which was, :T_-T_-T_-T_+2T_ = 0 or using the above eta quotients ''j''''n'', :j_-j_-j_-j_+2j_ = 22 A similar relation exists for level 10.


α Sequences

For the modular function ''j''6A, one can associate it with ''three'' different sequences. (A similar situation happens for the level 10 function ''j''10A.) Let, :\alpha_1(k)=\binom\sum_^k \binom^3 =1, 4, 60, 1120, 24220,\ldots (, labeled as ''s''6 in Cooper's paper) :\alpha_2(k)=\binom\sum_^k \binom\sum_^j\binom^3=\binom\sum_^k \binom^2\binom =1, 6, 90, 1860, 44730,\ldots () :\alpha_3(k)=\binom\sum_^k \binom(-8)^\sum_^j\binom^3 =1, -12, 252, -6240, 167580, -4726512,\ldots The three sequences involve the product of the
central binomial coefficient In mathematics the ''n''th central binomial coefficient is the particular binomial coefficient : = \frac \textn \geq 0. They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle. The first ...
s c(k)=\tbinom with: first, th
Franel numbers
\textstyle\sum_^k \tbinom^3; second, , and third, (-1)^k . Note that the second sequence, ''α''2(''k'') is also the number of 2''n''-step polygons on a cubic lattice. Their complements, :\alpha'_2(k)=\binom\sum_^k \binom(-1)^\sum_^j\binom^3 =1, 2, 42, 620, 12250,\ldots :\alpha'_3(k)=\binom\sum_^k \binom(8)^\sum_^j\binom^3 =1, 20, 636, 23840, 991900,\ldots There are also associated sequences, namely the Apéry numbers, :s_(k)=\sum_^k \binom^2\binom^2 =1, 5, 73, 1445, 33001,\ldots () the Domb numbers (unsigned) or the number of 2''n''-step polygons on a diamond lattice, :s_(k)=(-1)^k \sum_^k \binom^2 \binom \binom =1, -4, 28, -256, 2716,\ldots () and the Almkvist-Zudilin numbers, :s_(k)=\sum_^k (-1)^\,3^\,\frac \binom \binom =1, -3, 9, -3, -279, 2997,\ldots () where :\frac=\binom\binom


Identities

The modular functions can be related as, : P = \sum_^\infty \alpha_1(k)\,\frac = \sum_^\infty \alpha_2(k)\,\frac = \sum_^\infty \alpha_3(k)\,\frac : Q = \sum_^\infty s_(k)\,\frac= \sum_^\infty s_(k)\,\frac= \sum_^\infty s_(k)\,\frac if the series converges and the sign chosen appropriately. It can also be observed that, :P = Q = \sum_^\infty \alpha'_2(k)\,\frac = \sum_^\infty \alpha'_3(k)\,\frac which implies, :\sum_^\infty \alpha_2(k)\,\frac = \sum_^\infty \alpha'_2(k)\,\frac and similarly using α3 and α'3.


Examples

One can use a value for ''j''6A in three ways. For example, starting with, :\Delta=j_\left(\sqrt\right)=198^2-4=\left(140\sqrt\right)^2=39200 and noting that 3\cdot17=51 then, :\begin \frac &= \frac\,\sum_^\infty \alpha_1(k)\,\frac\\ \frac &= \frac\,\sum_^\infty \alpha_2(k)\,\frac\\ \frac &= \frac\,\sum_^\infty \alpha_3(k)\,\frac\\ \end as well as, :\begin \frac &= \frac\,\sum_^\infty \alpha'_2(k)\,\frac\\ \frac &= \frac\,\sum_^\infty \alpha'_3(k)\,\frac\\ \end though the formulas using the complements apparently do not yet have a rigorous proof. For the other modular functions, :\frac = 8\sqrt\,\sum_^\infty s_(k)\,\left(\frac12-\frac+k\right)\left(\frac\right)^, \quad j_\left(\sqrt\right)=\left(\frac\right)^=\phi^ :\frac = \frac12\,\sum_^\infty s_(k)\,\frac, \quad j_\left(\sqrt\right)=32 :\frac = 2\sqrt\,\sum_^\infty s_(k)\,\frac, \quad j_\left(\sqrt\right)=81


Level 7

Define :s_(k)=\sum_^k \binom^2\binom\binom =1, 4, 48, 760, 13840,\ldots () and, :\begin j_(\tau) &=\left(\left(\frac\right)^+7 \left(\frac\right)^\right)^2=\frac +10 + 51q + 204q^2 +681q^3+\cdots\\ j_(\tau)&=\left(\frac\right)^= \frac- 4 + 2q + 8q^2 - 5q^3 - 4q^4 - 10q^5 + \cdots \end Example: :\frac = \frac\,\sum_^\infty s_(k)\, \frac, \quad j_\left(\frac\right) = -22^3+1 = -\left(39\sqrt\right)^2=-10647 No pi formula has yet been found using ''j''7B.


Level 8


Modular functions

Levels 2, 4, 8 are related since they are just powers of the same prime. Define, :\begin j_(\tau) &=\sqrt = \left( \sqrt + \frac \right)^2 -16 = \left( \sqrt - \frac \right)^2 = \left( \sqrt + \frac \right)^2\\ &=\left(\frac\right)^+2^6 \left(\frac\right)^ = \frac + 52q + 834q^3 + 4760q^5 + 24703q^7+\cdots\\ j_(\tau)&= \left(\frac\right)^ =\frac - 12q + 66q^3 - 232q^5 + 639q^7 - 1596q^9 + \cdots\\ j_(\tau)&=\left(\frac\right)^=\frac + 8 + 36q + 128q^2 + 386q^3 +1024q^4+\cdots\\ j_(\tau)&=\left(\frac\right)^=\frac - 8 + 36q - 128q^2 + 386q^3 -1024q^4+\cdots\\ j_(\tau)&=\left(\frac\right)^=\sqrt=\frac + 12q + 66q^3 + 232q^5 + 639q^7+\cdots\\ j_(\tau)&=\left(\frac\right)^ =\frac + 4q + 2q^3 - 8q^5 - q^7 + 20q^9 - 2q^ - 40q^ +\cdots \end Just like for level 6, five of these functions have a linear relationship, :j_-j_-j_-j_+2j_ = 0 But this is not one of the nine Conway-Norton-Atkin linear dependencies since j_ is ''not'' a moonshine function. However, it is related to one as, :j_(\tau) = -j_\Big(\tau+\tfrac12\Big)


Sequences

:s_(k)=\binom\sum_^k 4^\binom\binom^2 =\binom\sum_^k \binom\binom\binom=1, 8, 120, 2240, 47320,\ldots :s_(k)=\binom^3=1, 8, 216, 8000, 343000,\ldots :s_(k)=\sum_^k \binom^2\binom^2 =1, 4, 40, 544, 8536,\ldots () :s_(k)=\sum_^k \binom^3\binom =1, 2, 14, 36, 334,\ldots where the first is the product of the central binomial coefficient and a sequence related to an arithmetic-geometric mean ().


Identities

The modular functions can be related as, : \pm\sum_^\infty s_(k)\,\frac = \sum_^\infty s_(k)\,\frac = \sum_^\infty s_(k)\,\frac = \sum_^\infty (-1)^k s_(k)\,\frac if the series converges and signs chosen appropriately. Note also the different exponent of \left(j_(\tau)\right)^ from the others.


Examples

Recall that j_\left(\tfrac\sqrt\right)=396^4, while j_\left(\tfrac\sqrt\right)=396^2. Hence, :\frac = \frac\,\sum_^\infty s_(k)\,\frac,\qquad j_\left(\frac\sqrt\right)=396^2 :\frac = 2\sqrt\,\sum_^\infty s_(k)\,\frac,\qquad j_\left(\frac\sqrt\right)=4\left(99+13\sqrt\right)^=4U_^2 :\frac = 2\,\sum_^\infty (-1)^k s_(k)\,\frac,\qquad j_\left(\frac\sqrt\right)=4\left(1+\sqrt\right)^=4U_^, For another level 8 example, :\frac = \frac1\sqrt\,\sum_^\infty s_(k)\,\frac,\qquad j_\left(\frac\sqrt\right)=2^6=64


Level 9

Define, :\begin j_(\tau) &= \left(j(3\tau)\right)^\frac13 =-6+\left(\frac\right)^6 -27 \left(\frac\right)^6=\frac + 248q^2 + 4124q^5 +34752q^8+\cdots\\ j_(\tau) &= \left(\frac\right)^6 = \frac + 6 + 27q + 86q^2 + 243q^3 + 594q^4+\cdots\\ \end The expansion of the first is the McKay–Thompson series of class 3C (and related to the
cube root In mathematics, a cube root of a number is a number that has the given number as its third power; that is y^3=x. The number of cube roots of a number depends on the number system that is considered. Every real number has exactly one real cub ...
of the j-function), while the second is that of class 9A. Let, :s_(k)=\binom\sum_^k (-3)^\binom\binom\binom =\binom\sum_^k(-3)^\binom\binom\binom = 1, -6, 54, -420, 630,\ldots :s_(k)=\sum_^k\binom^2\sum_^j\binom\binom\binom =1, 3, 27, 309, 4059,\ldots where the first is the product of the central binomial coefficients and (though with different signs). Examples: :\frac = \frac\sum_^\infty s_(k)\,\frac,\quad j_\left(\frac\right)=-960 :\frac = 6\,\boldsymbol\,\sum_^\infty s_(k)\,\frac,\quad j_\left(\frac\right)=-3\sqrt\left(53\sqrt+14\sqrt\right) = -3\sqrt


Level 10


Modular functions

Define, :\beginj_(\tau) &=\left(\sqrt - \frac\right)^2 = \left(\sqrt + \frac\right)^2 = \left(\sqrt + \frac\right)^2-4 =\frac + 4 + 22q + 56q^2 +\cdots \end :\beginj_(\tau) &= \left(\frac\right)^=\frac - 4 + 6q - 8q^2 + 17q^3 - 32q^4 +\cdots \end :\beginj_(\tau) &= \left(\frac\right)^=\frac - 2 - 3q + 6q^2 + 2q^3 + 2q^4+\cdots\end :\beginj_(\tau) &= \left(\frac\right)^=\frac + 6 + 21q + 62q^2 + 162q^3 +\cdots \end :\beginj_(\tau) &= \left(\frac\right)=\frac + 1 + q + 2q^2 + 2q^3 - 2q^4 +\cdots\end Just like j_, the function j_ is a square or a near-square of the others. Furthermore, there are also linear relations between these, :T_-T_-T_-T_+2T_ = 0 or using the above eta quotients ''j''''n'', :j_-j_-j_-j_+2j_ = 6


β sequences

Let, :\beta_(k)=\sum_^k \binom^4 =1, 2, 18, 164, 1810,\ldots (, labeled as ''s''10 in Cooper's paper) :\beta_(k)=\binom\sum_^k \binom^\binom\sum_^j \binom^4 =1, 4, 36, 424, 5716,\ldots :\beta_(k)=\binom\sum_^k \binom^\binom (-4)^\sum_^j \binom^4 =1, -6, 66, -876, 12786,\ldots their complements, :\beta_'(k)=\binom\sum_^k \binom^\binom (-1)^\sum_^j \binom^4 =1, 0, 12, 24, 564, 2784,\ldots :\beta_'(k)=\binom\sum_^k \binom^\binom (4)^\sum_^j \binom^4 =1, 10, 162, 3124, 66994,\ldots and, :s_(k)=1, -2, 10, -68, 514, -4100, 33940,\ldots :s_(k)=1, -1, 1, -1, 1, 23, -263, 1343, -2303,\ldots :s_(k)=1, 3, 25, 267, 3249, 42795, 594145,\ldots though closed forms are not yet known for the last three sequences.


Identities

The modular functions can be related as,S. Cooper, "Level 10 analogues of Ramanujan’s series for 1/", Theorem 4.3, p.85, J. Ramanujan Math. Soc. 27, No.1 (2012) :U = \sum_^\infty \beta_1(k)\,\frac = \sum_^\infty \beta_2(k)\,\frac = \sum_^\infty \beta_3(k)\,\frac :V = \sum_^\infty s_(k)\,\frac = \sum_^\infty s_(k)\,\frac = \sum_^\infty s_(k)\,\frac if the series converges. In fact, it can also be observed that, :U = V =\sum_^\infty \beta_2'(k)\,\frac = \sum_^\infty \beta_3'(k)\,\frac Since the exponent has a
fractional part The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. The latter is defined as the largest integer not greater than , called ''floor'' of or \lfloor x\rfloor. Then, the fractional ...
, the sign of the
square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
must be chosen appropriately though it is less an issue when ''j''''n'' is positive.


Examples

Just like level 6, the level 10 function ''j''10A can be used in three ways. Starting with, :j_\left(\sqrt\right) = 76^2 = 5776 and noting that 5\cdot19=95 then, :\begin \frac &= \frac\,\sum_^\infty \beta_1(k)\,\frac\\ \frac &= \frac\,\sum_^\infty \beta_2(k)\,\frac\\ \frac &= \frac\,\,\sum_^\infty \beta_3(k)\,\,\frac\\ \end as well as, :\begin \frac &= \frac\,\sum_^\infty \beta_2'(k)\,\frac\\ \frac &= \frac\,\sum_^\infty \beta_3'(k)\,\frac \end though the ones using the complements do not yet have a rigorous proof. A conjectured formula using one of the last three sequences is, :\frac = \frac\,\sum_^\infty s_(k)\frac,\quad j_\left(\frac\right) = -5^2 which implies there might be examples for all sequences of level 10.


Level 11

Define the McKay–Thompson series of class 11A, :j_(\tau)= (1+3F)^3+\left(\frac+3\sqrt\right)^2=\frac + 6 + 17q + 46q^2 + 116q^3 +\cdots or sequence () and where, :F = \frac and, :s_(k) = 1, 4, 28, 268, 3004, 36784, 476476,\ldots () No closed form in terms of binomial coefficients is yet known for the sequence but it obeys the
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
, :(k + 1)^3 s_ = 2(2k + 1)\left(5k^2 + 5k + 2\right)s_k - 8k\left(7k^2 + 1\right)s_ + 22k(k - 1)(2k - 1)s_ with initial conditions ''s''(0) = 1, ''s''(1) = 4. Example: :\frac=\frac\sum_^\infty s_(k)\,\frac,\quad j_\left(\frac\right)=-44


Higher levels

As pointed out by Cooper, there are analogous sequences for certain higher levels.


Similar series

R. Steiner found examples using
Catalan numbers The Catalan numbers are a sequence of natural numbers that occur in various Enumeration, counting problems, often involving recursion, recursively defined objects. They are named after Eugène Charles Catalan, Eugène Catalan, though they were p ...
C_k , :\frac = \sum_^\infty \left(2C_\right)^2 \frac\qquad z \in \Z,\quad n\ge2,\quad n \in \N and for this a
modular form In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...
with a second periodic for ''k'' exists: :k=\frac,\qquad k=\frac Other similar series are :\frac = \sum_^\infty \left(2C_\right)^2 \frac :\frac = \sum_^\infty \left(2C_\right)^2 \frac \qquad z \in \Z :\frac = \sum_^\infty \left(2C_\right)^2 \frac :\frac = \sum_^\infty \left(2C_\right)^2 \frac :\frac = \sum_^\infty \left(2C_\right)^2 \frac :\frac = \sum_^\infty \left(2C_\right)^2 \frac :\frac = \sum_^\infty \left(2C_\right)^2 \frac :\frac = \sum_^\infty \left(2C_\right)^2 \frac :\frac = \sum_^\infty \left(2C_\right)^2 \frac :\frac = \sum_^\infty \left(2C_k\right)^2 \frac with the last (comments in ) found by using a linear combination of higher parts of Wallis-Lambert series for \tfrac and Euler series for the circumference of an ellipse. Using the definition of Catalan numbers with the
gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
the first and last for example give the identities :\frac14 = \sum_^\infty ^2 \left(4zk-(4n-3)z+4^\right)\qquad z \in \Z,\quad n\ge2,\quad n \in \N ... :4 = \sum_^\infty ^2 (k+1). The last is also equivalent to, :\frac = \frac14 \sum_^\infty \frac\, \frac and is related to the fact that, : \lim_ \frac = \pi which is a consequence of
Stirling's approximation In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related ...
.


See also

*
Chudnovsky algorithm The Chudnovsky algorithm is a fast method for calculating the digits of , based on Ramanujan's formulae. Published by the Chudnovsky brothers in 1988, it was used to calculate to a billion decimal places. It was used in the world record calcu ...
* Borwein's algorithm


References


External links


Franel numbersMcKay–Thompson series
{{DEFAULTSORT:Ramanujan-Sato series Series (mathematics) Pi algorithms Srinivasa Ramanujan