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 wellknown that the jfunction, 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,
744^{2} – 196884 = 356652 744^{3} – 2∙744∙196884 + 21493760 = 140361152
and so on. But not so wellknown are the infinite series,
Similarly, between [3] and [4],
744^{2} + 196884 = 750420 744^{3} + 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, 744^{3} + 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 + 21493760q^{2} + …. A000521
[2] 1/j = q  744q^{2} + 356652q^{3}  140361152q^{4} + …. 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} ≈ 640320^{3} + 744
it is equally valid to say it is the jfunction 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 jfunction. This gave me the idea to look at the reversion for the other moonshine functions.
II. Series Reversion for Other Moonshine Functions
The jfunction is essentially the McKayThompson 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 + c_{0}q^{2} + c_{1}q^{3} + c_{2}q^{4} + …
1/q = f + c_{0} + c_{1}(1/f) + c_{2}(1/f)^{2} + c_{3}u(1/f)^{3} + …
q = (1/f) + c_{0}(1/f)^{2} + c_{1}(1/f)^{3} + c_{3}(1/f)^{4} + …
will be given simply as a coefficient list,
{1, c_{0}, c_{1}, c_{2}, c_{3},…}
for the first ten or so coefficients (as, strangely enough, these sequences are not yet in the OEIS).
1. McKayThompson series of class 2A with a(0) = 104:
[1] r_{2A}(τ) = 1/q + 104 + 4372q + 96256q^{2} + …. 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,
104^{2} – 4372 = 6444 104^{3} – 2∙104∙4372 + 96256 = 311744
And reversions,
where,
104^{2} + 4372 = 15188 104^{3} + 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. McKayThompson series of class 3A with a(0) = 42:
[1] r_{3A}(τ) = 1/q + 42 + 783q + 8672q^{2} + 65367q^{3} + … 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,
42^{2} – 783 = 981 42^{3} – 2∙42∙783 + 8672 = 16988
And reversions,
where,
42^{2} + 783 = 2547 42^{3} + 2∙42∙783 + 41558 = 181418
3. McKayThompson series of class 4A with a(0) = 24:
[1] r_{4}(τ) = 1/q + 24 + 276q + 2048q^{2} + 11202q^{3} + … A097340, A107080
and,
[2] 1/r_{4} = {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(q^{4})/eta^{2}(q^{2})]^{24} in powers of q), and as a nonalternating series is A014103 (the expansion of [eta(q^{2})/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: PrimeGenerating 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) ≈ [(640320^{3} + 744)^{2 }– 393768]^{2 }+ 85975040(640320^{3}) – 17018389032
However, this can be simplified. It belongs to a family which is illustrated in the table below. First, recall the jfunction sequence A000521,
{1, 744, 196884, 21493760, 864299970,…}
Let a = 640320^{3} + 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  196884q^{2 }+ 103378831900730205293632q^{3} ≈ 0
which differs from zero by a mere 10^{59}. Again, it can be simplified as,
1  aq  196884q^{2} + (2∙196884a21493760)q^{3 }≈ 0
Based on the table, we can assume this can be extended to fourth powers using the next term in the sequence,
1  aq  196884q^{2} + (2∙196884a21493760)q^{3 } + (8642999702∙196884^{2})q^{4 }≈ 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 a_{427} = 5280^{3}(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) ≈ (396^{4 }– 104)^{3 }– 13116(396^{4}) – 1075296
Let b = 396^{4}  104. Then,
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 [2] and [3], it probably isn’t a surprise that, let q = exp(π√58), then we also have,
1  bq  4372q^{2} + (2∙4372b+96256)q^{3 }≈ 0
and,
1  bq  4372q^{2} + (2∙4372b+96256)q^{3 }+ (12400022∙4372^{2})q^{4 }≈ 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.
