0029: Part 8, Series Reversions

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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 [1] and [2] 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 [3] and [4],

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 [1] and [4] 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:

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

[2] 1/j = q - 744q² + 356652q³ - 140361152q⁴ + …. A066395

[3] 1/q = j - 744 -196884(1/j) - 167975456(1/j)² - … A178451

[4] 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,

e^π√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:

[1] r_2A(τ) = 1/q + 104 + 4372q + 96256q² + …. A007267

with reciprocal,

[2] 1/r_2A = {1, -104, 6444, -311744, 13018830, -493025760, 17411253944, -583472867840, 18770817643749, -584450497233840,…}

and series reversion coefficient lists,

[3] 1/q = {1, -104, -4372, -550944, -87663186, -15664980976, -3002554283256, -603254129022912, -125374508043698603, -26730177033272155720, -5813712718711111093252, -1284868344619718683448992,…}

[4] 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:

[1] r_3A(τ) = 1/q + 42 + 783q + 8672q² + 65367q³ + … A030197

with reciprocal and series reversions,

[2] 1/r_3A = {1, -42, 981, -16988, 244230, -3089394, 35579552, -381326952, 3860626617, -37318157820,…}

[3] 1/q = {1, -42, -783, -41558, -2788116, -210130632, -16988840521, -1439826519522, -126232024255821, -11353269833770206, -1041686623832555187, -97120395073301505966,…}

[4] 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:

[1] r₄(τ) = 1/q + 24 + 276q + 2048q² + 11202q³ + … A097340, A107080

and,

[2] 1/r₄ = {1, -24, 300, -2624, 18126, -105504, 538296, -2471424, 10400997, -40674128,…}

[3] 1/q = {1, -24, -276, -8672, -344658, -15390480, -737293560, -37026698304, -1923581395371, -102518730258488, -5573961072647172,…}

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

with similar relations between the terms of [1]-[4] as in the ones above. Example: Let τ = ½√-11, then,

where T is the tribonacci constant. Perhaps not surprisingly, [2] 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, [4] 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,

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 a_d.

B. Baby Monster

3. Cullen also gave the expression,

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

Let b = 396⁴ - 104. Then,