In
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,
:
to the form
:
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
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
, and
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 ...
,
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
as in the rest of this article. Let,
:
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
:
()
:
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. Analogous relationships exist for the higher levels.
Examples:
:
:
where
and
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,
:
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,
:
()
:
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.
Level 3
Define,
:
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,
:
()
:
Examples:
:
:
Level 4
Define,
:
where the first is the 24th power of the
Weber modular function . And,
:
()
:
()
Examples:
:
:
Level 5
Define,
:
and,
:
:
()
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:
:
:
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
. First, define,
:
:
:
:
:
The phenomenon of
being squares or a near-square of the other functions will also be manifested by
. 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,
:
or using the above eta quotients ''j''
''n'',
:
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,
:
(, labeled as ''s''
6 in Cooper's paper)
:
()
:
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
with: first, th
Franel numbers; second, , and third,
. Note that the second sequence, ''α''
2(''k'') is also the number of 2''n''-step polygons on a
cubic lattice. Their complements,
:
:
There are also associated sequences, namely the Apéry numbers,
:
()
the Domb numbers (unsigned) or the number of 2''n''-step polygons on a
diamond lattice,
:
()
and the Almkvist-Zudilin numbers,
:
()
where
:
Identities
The modular functions can be related as,
:
:
if the series converges and the sign chosen appropriately. It can also be observed that,
:
which implies,
:
and similarly using α
3 and α'
3.
Examples
One can use a value for ''j''
6A in three ways. For example, starting with,
:
and noting that
then,
:
as well as,
:
though the formulas using the complements apparently do not yet have a rigorous proof. For the other modular functions,
:
:
:
Level 7
Define
:
()
and,
:
Example:
:
No pi formula has yet been found using ''j''
7B.
Level 8
Modular functions
Levels
are related since they are just powers of the same prime. Define,
:
Just like for level 6, five of these functions have a linear relationship,
:
But this is not one of the nine Conway-Norton-Atkin linear dependencies since
is ''not'' a moonshine function. However, it is related to one as,
:
Sequences
:
:
:
()
:
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,
:
if the series converges and signs chosen appropriately. Note also the different exponent of
from the others.
Examples
Recall that
while
. Hence,
:
:
:
For another level 8 example,
:
Level 9
Define,
:
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,
:
:
where the first is the product of the central binomial coefficients and (though with different signs).
Examples:
:
:
Level 10
Modular functions
Define,
:
:
:
:
:
Just like
, the function
is a square or a near-square of the others. Furthermore, there are also linear relations between these,
:
or using the above eta quotients ''j''
''n'',
:
β sequences
Let,
:
(, labeled as ''s''
10 in Cooper's paper)
:
:
their complements,
:
:
and,
:
:
:
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)]
:
:
if the series converges. In fact, it can also be observed that,
:
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,
:
and noting that
then,
:
as well as,
:
though the ones using the complements do not yet have a rigorous proof. A conjectured formula using one of the last three sequences is,
:
which implies there might be examples for all sequences of level 10.
Level 11
Define the McKay–Thompson series of class 11A,
:
or sequence () and where,
:
and,
:
()
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 ...
,
:
with initial conditions ''s''(0) = 1, ''s''(1) = 4.
Example:
:
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 ...
,
:
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:
:
Other similar series are
:
:
:
:
:
:
:
:
:
:
with the last (comments in ) found by using a linear combination of higher parts of
Wallis-Lambert series for
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
:
...
:
.
The last is also equivalent to,
:
and is related to the fact that,
:
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