0029: Part 8, Series Reversions

Part 8:  Series Reversion for the Moonshine Functions

By Tito Piezas III

I. Introduction

II. Series Reversion for other Moonshine Functions

III. Cullen’s and Gosper’s Approximations

I.  Introduction

It is well-known that the j-function, j(τ), at τ = (1+√-163)/2, assumes the nice integer value, and its reciprocal, Notice that the coefficients of  and  are related as,

7442 – 196884 = 356652

7443 – 2∙744∙196884 + 21493760 = 140361152

and so on.  But not so well-known are the infinite series, Similarly, between  and ,

7442 + 196884 = 750420

7443 + 2∙744∙196884 + 167975456 = 872769632

(In Modular Matrix Models (p.20) by Y.He and V. Jejjala, they explain different ones between  and  as due to Voiculescu polynomials.  For example, 7443 + 3∙744∙196884 + 21493760 = 872769632, etc.)  The third and fourth series are just series reversions.  Simply put, if x is a series in y, then a reversion is to express y as a series in x.  (Or the reciprocal of x or y, as the case may be.)  The examples above are just particular cases of the general forms:

    j = j(τ) = 1/q + 744 + 196884q + 21493760q2 + ….               A000521

  1/j = q - 744q2 + 356652q3 - 140361152q4 + ….                      A066395

  1/q = j - 744 -196884(1/j) - 167975456(1/j)2 - …                     A178451

    q = (1/j) + 744(1/j)2 + 750420(1/j)3 + 872769632(1/j)4 + ….    A091406

where, as in the rest of the article, q = exp(2πiτ).  While j(τ) is a function of a complex variable, for τ = (1+√-d)/2 with d > 0, then q is the real number q = -exp(-π√d).  Thus, for the familiar,

eπ√163 ≈ 6403203 + 744

it is equally valid to say it is the j-function truncated at two terms, or its series reversion for 1/q also truncated at two terms.  But the latter interpretation allows us to generalize the close approximation for π using the natural logarithm of rationals, indefinitely since the series for 1/q at d = 163 have all terms rational, with the next being, with a difference of merely 10-46, and so on.  In “An Approximation of Pi from Monster Group Symmetries”, Cullen also discusses approximations to q = exp(π√d) using the first few terms of the series reversion for the j-function.  This gave me the idea to look at the reversion for the other moonshine functions.

II. Series Reversion for Other Moonshine Functions

The j-function is essentially the McKay-Thompson series of class pA with a(0) = 744 and p = 1.  In this section, for brevity, we will just focus on p = 2,3,4.  For convenience, the reciprocal and series reversions of a function f,

1/f  = q + c0q2 + c1q3 + c2q4 + …

1/q = f + c0 + c1(1/f) + c2(1/f)2 + c3u(1/f)3 + …

q = (1/f) + c0(1/f)2 + c1(1/f)3 + c3(1/f)4 + …

will be given simply as a coefficient list,

{1, c0, c1, c2, c3,…}

for the first ten or so coefficients (as, strangely enough, these sequences are not yet in the OEIS).

1. McKay-Thompson series of class 2A with a(0) = 104:

  r2A(τ) = 1/q + 104 + 4372q + 96256q2 + ….         A007267

with reciprocal,

  1/r2A = {1, -104, 6444, -311744, 13018830, -493025760, 17411253944, -583472867840, 18770817643749, -584450497233840,…}

and series reversion coefficient lists,

  1/q = {1, -104, -4372, -550944, -87663186, -15664980976, -3002554283256, -603254129022912, -125374508043698603, -26730177033272155720, -5813712718711111093252, -1284868344619718683448992,…}

  q = {1, 104, 15188, 2585184, 480222434, 94395247376, 19304070595976, 4064960003568320, 875470150835783183, 191947530987669939928,…}

Example:  Let τ = ½√-58, then, where,

1042 – 4372 = 6444

1043 – 2∙104∙4372 + 96256 = 311744

And reversions, where,

1042 + 4372 = 15188

1043 + 2∙104∙4372 + 550944 = 2585184

and so on.  Truncating the third at the first few terms, this gives us the nice approximation, which differs by just 10-27.

2. McKay-Thompson series of class 3A with a(0) = 42:

  r3A(τ) = 1/q + 42 + 783q + 8672q2 + 65367q3 + …  A030197

with reciprocal and series reversions,

  1/r3A = {1, -42, 981, -16988, 244230, -3089394, 35579552, -381326952, 3860626617, -37318157820,…}

  1/q = {1, -42, -783, -41558, -2788116, -210130632, -16988840521, -1439826519522, -126232024255821, -11353269833770206, -1041686623832555187, -97120395073301505966,…}

  q = {1, 42, 2547, 181418, 14147409, 1169321202, 100643939492, 8924936952648, 809787618416520, 74819208058969680,…}

Example: Let τ = (3+√-267)/6,

Then, where,

422 – 783 = 981

423 – 2∙42∙783 + 8672 = 16988

And reversions, where,

422 + 783 = 2547

423 + 2∙42∙783 + 41558 = 181418

3. McKay-Thompson series of class 4A with a(0) = 24:

  r4(τ) = 1/q + 24 + 276q + 2048q2 + 11202q3 + … A097340A107080

and,

  1/r4 = {1, -24, 300, -2624, 18126, -105504, 538296, -2471424, 10400997, -40674128,…}

  1/q = {1, -24, -276, -8672, -344658, -15390480, -737293560, -37026698304, -1923581395371, -102518730258488, -5573961072647172,…}

  q = {1, 24, 852, 35744, 1645794, 80415216, 4094489992, 214888573248, 11542515402255, 631467591949480,…}

with similar relations between the terms of - as in the ones above.  Example: Let τ = ½√-11, then, where T is the tribonacci constant.  Perhaps not surprisingly,  is A100130 (the expansion of [eta(q)eta(q4)/eta2(q2)]24 in powers of q), and as a non-alternating series is A014103 (the expansion of [eta(q2)/eta(q)]24 in powers of q).  On the other hand,  is A195130, but as an alternating series is A005149 (connected to a rapidly converging series for pi).

Note:  It is quite easy to find these coefficients using certain eta quotients evaluated at a large enough d.  One interesting thing about these moonshine functions is that, at certain arguments τ, they evaluate as integers, hence the series for q and 1/q will have rational terms. (A full list of such τ is given in Table 3 of Part 2: Prime-Generating Polynomials, as the roots of these polynomials.

III. Cullen’s and Gosper’s Approximations

In Cullen’s article cited in the Introduction, in addition to approximations to q = exp(±π√d), he also considered powers of q.  Similarly, Gosper found a nice approximation that was a polynomial in q.  It turns out there are common integers that appear in their work.

A. Monster Group

1.  Cullen pointed out that,

exp(4π√163) ≈ [(6403203 + 744)2 – 393768]2 + 85975040(6403203) – 17018389032

However, this can be simplified.  It belongs to a family which is illustrated in the table below.  First, recall the j-function sequence A000521,

{1, 744, 196884, 21493760, 864299970,…}

Let a = 6403203 + 744, then,

 k Error Approximation 1 10-13 exp(π√163) ≈ a 2 10-10 exp(2π√163) ≈ a2 – 2∙196884 3 10-9 exp(3π√163) ≈ a3 – 3(196884a + 21493760) 4 10-7 exp(4π√163) ≈ a4 – 4(196884a2 - 21493760a + 864299970) + 2∙1968842

So it is the case k = 4, and can be neatly expressed by the first few terms of A000521. But to continue the table to k = 5 using the next term of the sequence apparently does not yield a polynomial in “a” as neat and simple as the four above.

2.  Let q = exp(-π√163).  Gosper gave (see Almost Integer, eq. 55),

1 - 262537412640768744q - 196884q2 + 103378831900730205293632q3  ≈ 0

which differs from zero by a mere 10-59.  Again, it can be simplified as,

1 - aq - 196884q2 + (2∙196884a-21493760)q3  ≈ 0

Based on the table, we can assume this can be extended to fourth powers using the next term in the sequence,

1 - aq - 196884q2 + (2∙196884a-21493760)q3  + (864299970-2∙1968842)q4  ≈ 0

which reduces the error to only 10-75.  But to go to 5th powers and find a similarly simple expression is not as easy.

Note:  The results are not unique to d = 163.  If we used d = 427 and a427 = 52803(236674+30303√61)3 + 744, then we will find that the quartic polynomial will differ from zero by a mere 10-129.  And so on for other d and their ad.

B. Baby Monster

3. Cullen also gave the expression,

exp(3π√58) ≈ (3964 – 104)3 – 13116(3964) – 1075296

Let b = 3964 - 104.   Then,

 k Error Approximation 1 10-7 exp(π√58) ≈ b 2 10-6 exp(2π√58) ≈ b2 - 2∙4372 3 10-4 exp(3π√58) ≈ b3 - 3(4372b + 96256) 4 10-3 exp(4π√58) ≈ b4 - 4(4372b2 + 96256b + 1240002) + 2∙43722

Compare to A007267: {1, 104, 4372, 96256, 1240002, …} as well as to the degrees of the irreducible representations of the baby monster, A001378,

{1, 4371, 96255, 1139374,…}

(Note that 4371 + 96255 + 1139374 + 2 = 1240002.)

4. Based on  and , it probably isn’t a surprise that, let q = exp(-π√58), then we also have,

1 - bq - 4372q2 + (2∙4372b+96256)q3  ≈ 0

and,

1 - bq - 4372q2 + (2∙4372b+96256)q3  + (1240002-2∙43722)q4  ≈ 0

which differs from zero by 10-34 and 10-44, respectively.  Similar results presumably can be found for the other moonshine functions.

-- End --

© Tito Piezas III, Sept 2011

You can email author at tpiezas@gmail.com.