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In mathematics, 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 called ...
s of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
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 paramete ...
, sequences which may be expressed in terms of
binomial coefficients 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 ...
\tbinom, and A,B,C employing
modular forms In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of ...
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 \Gamma_0(n), while G. Almkvist has
experimentally An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs when ...
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 Don Bernard Zagier (born 29 June 1951) is an American-German mathematician whose main area of work is number theory. He is currently one of the directors of the Max-Planck-Institut für Mathematik, Max Planck Institute for Mathematics in Bonn, Ger ...
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 general ...
''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 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, having order    2463205976112133171923293141475 ...
. 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 A base unit (also referred to as a fundamental unit) is a unit adopted for measurement of a ''base quantity''. A base quantity is one of a conventionally chosen subset of physical quantities, where no quantity in the subset can be expressed in ter ...
. 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 In the area of modern algebra known as group theory, the baby monster group ''B'' (or, more simply, the baby monster) is a sporadic simple group of order :   241313567211131719233147 : = 4154781481226426191177580544000000 : = 4, ...
. 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 In mathematics, the Weber modular functions are a family of three functions ''f'', ''f''1, and ''f''2,''f'', ''f''1 and ''f''2 are not modular functions (per the Wikipedia definition), but every modular function is a rational function in ''f'', '' ...
\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 = \prod\limits_^\frac \textn \geq 0. They are called central since they show up exactly in the middle of the even-numbered rows in Pascal ...
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, 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) &=j_(\tau)+\frac-2 = j_(\tau)+\frac+16 = j_(\tau)+\frac+14 =\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 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


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 = \prod\limits_^\frac \textn \geq 0. They are called central since they show up exactly in the middle of the even-numbered rows in Pascal ...
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 The diamond cubic crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. While the first known example was diamond, other elements in group 14 also adopt this structure, including α-tin, the sem ...
, :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

Define, :\begin j_(\tau)&=\left(j_(2\tau)\right)^\frac12=\frac + 52q + 834q^3 + 4760q^5 + 24703q^7+\cdots\\ &= \left(\left(\frac\right)^+4 \left(\frac\right)^\right)^2 = \left(\left(\frac\right)^ - 4 \left(\frac\right)^\right)^2\\ 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(j_(2\tau)\right)^\frac12=\left(\frac\right)^=\frac + 12q + 66q^3 + 232q^5 + 639q^7+\cdots \end The expansion of the first is the McKay–Thompson series of class 4B (and is the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
of another function). The fourth is also the square root of another function. Let, :s_(k)=\binom\sum_^k 4^\binom\binom^2 =\binom\sum_^k \binom\binom\binom=1, 8, 120, 2240, 47320,\ldots :s_(k)=(-1)^\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 (), Examples: :\frac = \frac\,\sum_^\infty s_(k)\,\frac,\qquad j_\left(\frac\sqrt\right)=396^2=156816 :\frac = \frac\,\sum_^\infty s_(k)\, \frac :\frac = 2\sqrt\,\sum_^\infty s_(k)\,\frac,\qquad j_\left(\frac\sqrt\right)=4\left(1+\sqrt\right)^,\quad j_\left(\frac\sqrt\right)=4\left(99+13\sqrt\right)^=4U_^2 :\frac = \frac\,\sum_^\infty s_(k)\,\frac,\qquad j_\left(\frac\sqrt\right)=64 though no pi formula is yet known using ''j''8A(''τ'').


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 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) &=j_(\tau)+\frac+8 = j_(\tau)+\frac+6 = j_(\tau)+\frac-2 =\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 the level 6, 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 sign of the square root 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 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 paramete ...
, :(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 In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Cata ...
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 (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory ...
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 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 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 but less p ...
.


See also

*
Chudnovsky algorithm The Chudnovsky algorithm is a fast method for calculating the digits of , based on Ramanujan’s formulae. It was published by the Chudnovsky brothers in 1988. It was used in the world record calculations of 2.7 trillion digits of in December ...
* Borwein's algorithm


References


External links


Franel numbersMcKay–Thompson series
{{DEFAULTSORT:Ramanujan-Sato series Mathematical series Pi algorithms Srinivasa Ramanujan