Ramanujan–Sato series

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 recently 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.

Level 1Edit

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 E4, and Dedekind eta function η(τ). The first expansion is the McKay–Thompson series of class 1A (OEISA007240) with a(0) = 744. Note that, as first noticed by J. McKay, the coefficient of the linear term of j(τ) almost equals  , which is the degree of the smallest nontrivial irreducible representation of the Monster group. Similar phenomena will be observed in the other levels. Define

  (OEISA001421)
 

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:

 
 

and   is a fundamental unit. 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.[12] The second formula, and the ones for higher levels, was established by H.H. Chan and S. Cooper in 2012.[3]

Level 2Edit

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   which is the smallest degree > 1 of the irreducible representations of the Baby Monster group. Define,

  (OEISA008977)
 

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]

Level 3Edit

Define,

 

where   is the smallest degree > 1 of the irreducible representations of the Fischer group Fi23 and,

  (OEISA184423)
 

Examples:

 
 

Level 4Edit

Define,

 

where the first is the 24th power of the Weber modular function  . And,

  (OEISA002897)
  (OEISA036917)

Examples:

 
 

Level 5Edit

Define,

 

and,

 
  (OEISA229111)

where the first is the product of the central binomial coefficients and the Apéry numbers (OEISA005258)[9]

Examples:

 
 

Level 6Edit

Modular functionsEdit

In 2002, Sato[7] established the first results for level > 4. It involved Apéry numbers which were first used to establish the irrationality of  . First, define,

 
 
 
 
 

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,

 

α SequencesEdit

For the modular function j6A, one can associate it with three different sequences. (A similar situation happens for the level 10 function j10A.) Let,

  (OEISA181418, labeled as s6 in Cooper's paper)
  (OEISA002896)
 

The three sequences involve the product of the central binomial coefficients   with: 1st, the Franel numbers  ; 2nd, OEISA002893, and 3rd, (-1)^k OEISA093388. Note that the second sequence, α2(k) is also the number of 2n-step polygons on a cubic lattice. Their complements,

 
 

There are also associated sequences, namely the Apéry numbers,

  (OEISA005259)

the Domb numbers (unsigned) or the number of 2n-step polygons on a diamond lattice,

  (OEISA002895)

and the Almkvist-Zudilin numbers,

  (OEISA125143)

where  .

IdentitiesEdit

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 α3 and α'3.

ExamplesEdit

One can use a value for j6A 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 7Edit

Define

  (OEISA183204)

and,

 

Example:

 

No pi formula has yet been found using j7B.

Level 8Edit

Define,

 

The expansion of the first is the McKay–Thompson series of class 4B (and is the square root of another function). The fourth is also the square root of another function. Let,

 
 
 

where the first is the product[2] of the central binomial coefficient and a sequence related to an arithmetic-geometric mean (OEISA081085),

Examples:

 
 
 
 

though no pi formula is yet known using j8A(τ).

Level 9Edit

Define,

 

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,

 
 

where the first is the product of the central binomial coefficients and OEISA006077 (though with different signs).

Examples:

 
 

Level 10Edit

Modular functionsEdit

Define,

 
 
 
 
 

Just like the level 6, there are also linear relations between these,

 

or using the above eta quotients jn,

 

β SequencesEdit

Let,

  (OEISA005260, labeled as s10 in Cooper's paper)
 
 

their complements,

 
 

and,

 
 
 

though closed-forms are not yet known for the last three sequences.

IdentitiesEdit

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.

ExamplesEdit

Just like level 6, the level 10 function j10A 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 11Edit

Define the McKay–Thompson series of class 11A,

 

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]

 

Higher levelsEdit

As pointed out by Cooper,[16] there are analogous sequences for certain higher levels.

Similar seriesEdit

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 OEISA013709) found by using a linear combination of higher parts of Wallis-Lambert series for 4/Pi 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

 

...

 .

The last is also equivalent to,

 

and is related to the fact that,

 

which is a consequence of Stirling's approximation.

See alsoEdit

ReferencesEdit

  1. ^ a b Chan, Heng Huat; Chan, Song Heng; Liu, Zhiguo (2004). "Domb's numbers and Ramanujan–Sato type series for 1/π". Advances in Mathematics. 186 (2): 396–410. doi:10.1016/j.aim.2003.07.012.
  2. ^ a b c Almkvist, Gert; Guillera, Jesus (2013). "Ramanujan–Sato-Like Series". In Borwein, J.; Shparlinski, I.; Zudilin, W. (eds.). Number Theory and Related Fields. Springer Proceedings in Mathematics & Statistics. vol 43. New York: Springer. pp. 55–74. doi:10.1007/978-1-4614-6642-0_2. ISBN 978-1-4614-6641-3.
  3. ^ a b c Chan, H. H.; Cooper, S. (2012). "Rational analogues of Ramanujan's series for 1/π" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 153 (2): 361–383. doi:10.1017/S0305004112000254.
  4. ^ Almkvist, G. (2012). "Some conjectured formulas for 1/π coming from polytopes, K3-surfaces and Moonshine". arXiv:1211.6563. Cite journal requires |journal= (help)
  5. ^ Ramanujan, S. (1914). "Modular equations and approximations to π". Quart. J. Math. Oxford. 45.
  6. ^ Chan; Tanigawa; Yang; Zudilin (2011). "New analogues of Clausen's identities arising from the theory of modular forms". Advances in Mathematics. 228 (2): 1294–1314. doi:10.1016/j.aim.2011.06.011.
  7. ^ a b Sato, T. (2002). "Apéry numbers and Ramanujan's series for 1/π". Abstract of a talk presented at the Annual meeting of the Mathematical Society of Japan.
  8. ^ Chan, H.; Verrill, H. (2009). "The Apéry numbers, the Almkvist–Zudilin Numbers, and new series for 1/π". Mathematical Research Letters. 16 (3): 405–420. doi:10.4310/MRL.2009.v16.n3.a3.
  9. ^ a b Cooper, S. (2012). "Sporadic sequences, modular forms and new series for 1/π". Ramanujan Journal. 29 (1–3): 163–183. doi:10.1007/s11139-011-9357-3.
  10. ^ a b Zagier, D. (2000). "Traces of Singular Moduli" (PDF): 15–16. Cite journal requires |journal= (help)
  11. ^ Chudnovsky, David V.; Chudnovsky, Gregory V. (1989), "The Computation of Classical Constants", Proceedings of the National Academy of Sciences of the United States of America, 86 (21): 8178–8182, doi:10.1073/pnas.86.21.8178, ISSN 0027-8424, JSTOR 34831, PMC 298242, PMID 16594075.
  12. ^ Yee, Alexander; Kondo, Shigeru (2011), 10 Trillion Digits of Pi: A Case Study of summing Hypergeometric Series to high precision on Multicore Systems, Technical Report, Computer Science Department, University of Illinois.
  13. ^ Borwein, J. M.; Borwein, P. B.; Bailey, D. H. (1989). "Ramanujan, modular equations, and approximations to pi; Or how to compute one billion digits of pi". Amer. Math. Monthly. 96 (3): 201–219. doi:10.1080/00029890.1989.11972169.
  14. ^ Conway, J.; Norton, S. (1979). "Monstrous Moonshine". Bulletin of the London Mathematical Society. 11 (3): 308–339 [p. 319]. doi:10.1112/blms/11.3.308.
  15. ^ S. Cooper, "Level 10 analogues of Ramanujan’s series for 1/π", Theorem 4.3, p.85, J. Ramanujan Math. Soc. 27, No.1 (2012)
  16. ^ a b Cooper, S. (December 2013). "Ramanujan's theories of elliptic functions to alternative bases, and beyond" (PDF). Askey 80 Conference.

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