# Determination of normalized extremal quasimodular forms of depth 1 with
integral Fourier coefficients

Tomoaki Nakaya  Faculty of Mathematics, Kyushu University, 744, Motooka,
Nishi-ku, Fukuoka, 819-0395, Japan<EMAIL_ADDRESS>
###### Abstract.

The main purpose of this paper is to determine all normalized extremal
quasimodular forms of depth 1 whose Fourier coefficients are integers. By
changing the local parameter at infinity from $q=e^{2\pi i\tau}$ to the
reciprocal of the elliptic modular $j$-function, we prove that all normalized
extremal quasimodular forms of depth 1 have a hypergeometric series expression
and that integrality is not affected by this change of parameters.
Furthermore, by transforming these hypergeometric series expressions into a
certain manageable form related to the Atkin(-like) polynomials and using the
lemmas that appeared in the study of $p$-adic hypergeometric series by Dwork
and Zudilin, the integrality problem can be reduced to the fact that a
polynomial vanishes modulo a prime power, which we prove. We also prove that
all extremal quasimodular forms of depth 1 with appropriate weight-dependent
leading coefficients have integral Fourier coefficients by focusing on the
hypergeometric expression of them.

###### Key words and phrases:

Extremal quasimodular forms; Fourier coefficients; hypergeometric series;
modular differential equations.

###### 2010 Mathematics Subject Classification:

11F11, 11F25, 11F30, 33C05, 34M03

##  1\. Introduction

A quasimodular form of weight $w$ on the full modular group
$\Gamma=SL_{2}(\mathbb{Z})$ is given as

$\displaystyle\sum_{\begin{subarray}{c}2\ell+4m+6n=w\\\ \ell,m,n\geq
0\end{subarray}}C_{\ell,m,n}E_{2}^{\,\ell}E_{4}^{\,m}E_{6}^{\,n}\in
QM_{*}(\Gamma)\coloneqq\mathbb{C}[E_{2},E_{4},E_{6}].$

Here $E_{k}=E_{k}(\tau)$ is the standard Eisenstein series on $\Gamma$ of
weight $k$ defined by

$\displaystyle
E_{k}(\tau)=1-\frac{2k}{B_{k}}\sum_{n=1}^{\infty}\sigma_{k-1}(n)q^{n}\;(q=e^{2\pi
i\tau}),\quad\sigma_{k}(n)=\sum_{d\mid n}d^{k},$

where $\tau$ is a variable in the complex upper half plane $\mathfrak{H}$, and
$B_{k}$ is $k$-th Bernoulli number, e.g.,
$B_{2}=\tfrac{1}{6},B_{4}=-\tfrac{1}{30},B_{6}=\tfrac{1}{42}$. For the
intrinsic definition of the quasimodular forms on a non-cocompact discrete
subgroup of $SL_{2}(\mathbb{R})$, we refer to [41, §5.3]. We denote the vector
space of modular forms and cusp forms of weight $k$ on $\Gamma$ by
$M_{k}(\Gamma)$ and $S_{k}(\Gamma)$, respectively. It is well known that
$E_{k}\in M_{k}(\Gamma)$ for even $k\geq 4$, but $E_{2}$ is not modular and is
quasimodular. Any quasimodular form $f$ of weight $w$ can be uniquely written
as

$\displaystyle f=\sum_{\ell=0}^{r}E_{2}^{\,\ell}f_{\ell},\quad f_{\ell}\in
M_{w-2\ell}(\Gamma),\quad f_{r}\not=0$

with $r\in\mathbb{Z}_{\geq 0}$ and we call it depth of $f$. Hence $E_{2}$ is
the quasimodular form with minimum weight and depth on $\Gamma$. Let
$QM_{w}^{(r)}=QM_{w}^{(r)}(\Gamma)$ denote the vector space of quasimodular
forms of weight $w$ and depth $\leq r$ on $\Gamma$. In particular,
$QM_{w}^{(0)}(\Gamma)$ is equal to $M_{w}(\Gamma)$. (In the following, we
often omit the reference to the group $\Gamma$.) From the fact
$\dim_{\mathbb{C}}QM_{w}^{(r)}=\sum_{\ell=0}^{r}\dim_{\mathbb{C}}M_{w-2\ell}$,
the generating function of the dimension of $QM_{w}^{(r)}$ is given by the
following (See [10] for the explicit formula of
$\dim_{\mathbb{C}}QM_{w}^{(r)}$.):

$\displaystyle{}\sum_{k=0}^{\infty}\dim_{\mathbb{C}}QM_{2k}^{(r)}\,T^{2k}=\frac{\sum_{\ell=0}^{r}T^{2\ell}}{(1-T^{4})(1-T^{6})}$
$\displaystyle=\frac{1-T^{2(r+1)}}{(1-T^{2})(1-T^{4})(1-T^{6})}\quad(r\in\mathbb{Z}_{\geq
0},\;|T|<1).$

In particular, since there is no quasimodular form in which the depth exceeds
half the weight, $QM_{2k}^{(k)}=QM_{2k}^{(k+1)}=QM_{2k}^{(k+2)}=\dotsb$ holds,
so by taking the limit $r\rightarrow\infty$ in the above equation, we have

$\displaystyle\sum_{k=0}^{\infty}\dim_{\mathbb{C}}QM_{2k}^{(k)}\,T^{2k}=\frac{1}{(1-T^{2})(1-T^{4})(1-T^{6})}.$

We denote by $D$ the differential operator $D=\tfrac{1}{2\pi
i}\tfrac{d}{d\tau}=q\tfrac{d}{dq}$. It is well known that the Eisenstein
series $E_{2},E_{4}$ and $E_{6}$ satisfy the following differential relation
(equation) by Ramanujan ([5, Thm. 0.21]. See also [24, §3] and [25, §4.1] for
Halphen’s contribution.):

$\displaystyle D(E_{2})=\frac{E_{2}^{2}-E_{4}}{12},\quad
D(E_{4})=\frac{E_{2}E_{4}-E_{6}}{3},\quad
D(E_{6})=\frac{E_{2}E_{6}-E_{4}^{2}}{2}.$ (1)

Therefore, the ring $QM_{*}(\Gamma)$ is closed under the derivation $D$.

For $f\in\mathbb{C}[\\![q]\\!]$, we denote
$\nu(f)=\mathrm{ord}_{q=0}(f)\in\mathbb{Z}_{\geq 0}\cup\\{\infty\\}$. The
notion of extremal quasimodular forms was introduced and studied by Kaneko and
Koike in [19]. They defined it as follows using the vanishing order $\nu(f)$
of the quasimodular form $f$.

######  Definition 1.

Let $f=\sum_{n=0}^{\infty}a_{n}q^{n}\in QM_{w}^{(r)}\backslash QM_{w}^{(r-1)}$
and $m=\dim_{\mathbb{C}}QM_{w}^{(r)}$. We call $f$ extremal if $\nu(f)=m-1$.
If, moreover, $a_{m-1}=1$, $f$ is said to be normalized. We denote by
$G_{w}^{(r)}$ the normalized extremal quasimodular form of weight $w$ and
depth $r$ on $\Gamma$ (if exists).

We set

$\displaystyle\nu_{\max}(r,w)=\max\\{n\in\mathbb{Z}_{\geq 0}\text{ such that
there exists }f\in QM_{w}^{(r)}\backslash\\{0\\}\text{ with }\nu(f)=n\\}.$

Pellarin gave an upper bound on $\nu_{\max}(r,w)$ in [30, Thm. 2.3], which
turns out to be $\nu_{\max}(r,w)=\dim_{\mathbb{C}}QM_{w}^{(r)}-1$ for the
depth $r\in\\{1,2,3,4\\}$. In a recent paper [10] Grabner constructed
inductively the normalized extremal quasimodular forms $G_{w}^{(r)}$ of weight
$w$ and depth $r\in\\{1,2,3,4\\}$. More precisely, he constructed the
quasimodular form $f_{w}^{(r)}$ such that

$\displaystyle f_{w}^{(r)}=q^{m-1}(1+O(q))\in QM_{w}^{(r)},\quad m=\dim
QM_{w}^{(r)}$

inductively by using the Serre derivative. From Definition 1 of the extremal
quasimodular forms, it is necessary to confirm that $f_{w}^{(r)}\in
QM_{w}^{(r)}\backslash QM_{w}^{(r-1)}$, which follows from
$\nu_{\max}(r,w)-\nu_{\max}(r-1,w)>0$, and hence $f_{w}^{(r)}=G_{w}^{(r)}$.
Moreover, these extremal quasimodular forms exist uniquely for
$r\in\\{1,2,3,4\\}$. This is because if the two forms $f,g=q^{m-1}+O(q^{m})\in
QM_{w}^{(r)}$ satisfy $f-g\not=0$, then the vanishing order of $f-g\in
QM_{w}^{(r)}$ is given by $\nu(f-g)\geq m>\nu_{\max}(r,w)$, which contradicts
the definition of $\nu_{\max}(r,w)$. In contrast to these cases of depth
$r\leq 4$, the existence and the uniqueness of $G_{w}^{(r)}$ for the depth
$r\geq 5$ is still open.

Let $\mathcal{E}_{r}$ be the set of weights $w$ such that the normalized
extremal quasimodular form of weight $w$ and depth $r$ on $\Gamma$ has
integral Fourier coefficients. In [15], based on Grabner’s results, Kaminaka
and Kato completely determined the sets
$\mathcal{E}_{2}=\\{4,8\\},\mathcal{E}_{3}=\\{6\\},\mathcal{E}_{4}=\emptyset$,
and showed that the set $\mathcal{E}_{1}$ is a subset of a finite set of
cardinality 22. Our main theorem asserts that their finite set coincides with
$\mathcal{E}_{1}$.

######  Theorem 1.

We have

$\displaystyle\mathcal{E}_{1}=\\{2,6,8,10,12,14,16,18,20,22,24,28,30,32,34,38,54,58,68,80,114,118\\}.$
(2)

Note that there is no quasimodular form of weight $4$ and depth $1$, since
there is no modular form of weight 2 on $\Gamma$. The minimum weight for the
normalized extremal quasimodular forms with non-integral Fourier coefficients
is 26, and the Fourier coefficients belong to $\frac{1}{5}\mathbb{Z}$:

$\displaystyle G_{26}^{(1)}$
$\displaystyle=\frac{E_{2}(E_{4}^{6}-1640E_{4}^{3}\Delta+269280\Delta^{2})-E_{4}^{2}E_{6}(E_{4}^{3}-920\Delta)}{69837768000}$
$\displaystyle=q^{4}+\frac{1176}{5}q^{5}+18816q^{6}+\cdots+316607232q^{9}+\frac{19845981568}{5}q^{10}+O(q^{11}).$

Here are some examples of the normalized extremal quasimodular forms of lower
weight and depth 1. Obviously, the following forms have integral Fourier
coefficients:

$\displaystyle G_{2}^{(1)}$
$\displaystyle=E_{2}=D(\log\Delta)=1-24\sum_{n=1}^{\infty}\sigma_{1}(n)q^{n},$
$\displaystyle G_{6}^{(1)}$
$\displaystyle=\frac{E_{2}E_{4}-E_{6}}{720}=\frac{D(E_{4})}{240}=\sum_{n=1}^{\infty}n\sigma_{3}(n)q^{n},$
$\displaystyle G_{8}^{(1)}$
$\displaystyle=\frac{E_{4}^{2}-E_{2}E_{6}}{1008}=-\frac{D(E_{6})}{504}=\sum_{n=1}^{\infty}n\sigma_{5}(n)q^{n},$
$\displaystyle G_{10}^{(1)}$
$\displaystyle=E_{4}G_{6}^{(1)}=\frac{D(E_{8})}{480}=\sum_{n=1}^{\infty}n\sigma_{7}(n)q^{n},$

where $\Delta$ is the “discriminant” cusp form of weight 12 on $\Gamma$:

$\displaystyle\Delta(\tau)=q\prod_{n=1}^{\infty}(1-q^{n})^{24}=\frac{E_{4}(\tau)^{3}-E_{6}(\tau)^{2}}{1728}\in
S_{12}(\Gamma).$

Using this infinite product expression of $\Delta(\tau)$ or the differential
relation (1), we see that $D(\log\Delta)=D(\Delta)/\Delta=E_{2}$. The first
few non-trivial examples of $G_{w}^{(1)}\in\mathbb{Z}[\\![q]\\!]$ are given
below.

$\displaystyle G_{12}^{(1)}$ $\displaystyle=\frac{E_{4}^{3}-1008\Delta-
E_{2}E_{4}E_{6}}{332640}=\frac{1}{2\cdot 3\cdot 5^{2}\cdot
7}\sum_{n=2}^{\infty}(n\sigma_{9}(n)-\tau(n))q^{n}$
$\displaystyle=q^{2}+56q^{3}+1002q^{4}+9296q^{5}+57708q^{6}+269040q^{7}+O(q^{8}),$
$\displaystyle G_{14}^{(1)}$
$\displaystyle=\frac{E_{2}(E_{4}^{3}-720\Delta)-E_{4}^{2}E_{6}}{393120}=\frac{1}{2\cdot
3\cdot 691}\sum_{n=2}^{\infty}n(\sigma_{11}(n)-\tau(n))q^{n}$
$\displaystyle=q^{2}+128q^{3}+4050q^{4}+58880q^{5}+525300q^{6}+3338496q^{7}+O(q^{8}).$

In [15, Rem. 1.2], Grabner pointed out to Kato that the integrality of these
Fourier coefficients can be proved by using a classical congruence formula
between Ramanujan’s tau-function $\tau(n)$, defined by
$\Delta=\sum_{n=1}^{\infty}\tau(n)q^{n}$, and $\sigma_{k}(n)$ (see also
Section 5 for the integrality of $G_{14}^{(1)}$).

Since $G_{16}^{(1)}=E_{4}G_{12}^{(1)}$ holds (see Proposition 2), we also see
that $G_{16}^{(1)}\in\mathbb{Z}[\\![q]\\!]$. However, it is probably difficult
to express the Fourier coefficients of, say, $G_{118}^{(1)}$ in terms of known
number-theoretic functions and to show congruence formulas that they satisfy.
The key idea that avoids this difficulty is to use the reciprocal of the
elliptic modular function
$j(\tau)^{-1}=\Delta(\tau)/E_{4}(\tau)^{3}=q-744q^{2}+356652q^{3}-\dotsb$
instead of $q=e^{2\pi i\tau}$ as the local parameter at infinity (or
equivalently, at the cusp of $\Gamma$). As will be shown later, Lemma 1
guarantees that the integrality of the coefficients is equivalent when
expanded for each local parameter. Furthermore, this idea also provides a
unified method of proof for all weights by using generalized hypergeometric
series.

Observing the examples of $G_{w}^{(1)}$ above, we find that its Fourier
coefficients are positive, except for $G_{2}^{(1)}=E_{2}$. More generally,
Kaneko and Koike conjectured in [19] that the Fourier coefficients of
$G_{w}^{(r)}$ would be positive if $w\geq 4$ and $1\leq r\leq 4$. For this
positivity conjecture, Grabner showed in [11] that the conjecture is true if
$w\leq 200$ and $1\leq r\leq 4$, by proving the asymptotic formula for the
Fourier coefficients of $G_{w}^{(r)}$. Thus, from Theorem 1, we can conclude
that the Fourier coefficients of $G_{w}^{(1)}$ are positive integers if and
only if their weight $w$ belongs to $\mathcal{E}_{1}\backslash\\{2\\}$.

The paper is organized as follows. Sections 2 to 4 are related to the main
theorem. In Section 2 we derive that the normalized extremal quasimodular
forms of depth 1 have a hypergeometric expression, and rewrite them using the
Atkin-like polynomials $A_{m,a}(X)$ and their adjoint polynomials
$B_{m,a}(X)$. Then the main theorem can be reduced to the fact that a formal
power series vanishes modulo a prime power, since the polynomial part arising
from the “factorization” of a formal power series vanishes. To “factorize” the
formal power series, we use some results on $p$-adic hypergeometric series by
Dwork and Zudilin. In Section 3, we specialize their results and prove several
propositions. In Section 4 we prove the main theorem of this paper by
combining the results of the previous sections.

In Section 5 we explicitly construct extremal quasimodular forms of depth 1
with integral Fourier coefficients by focusing on their hypergeometric
expression.

In Section 6 we summarize some previous work on the normalized extremal
quasimodular forms of depth 2 to 4 and present some supplementary results, in
particular from a hypergeometric point of view. We also present some previous
work in which the group $\Gamma$ is replaced by a low-level congruence
subgroup or Fricke group for (extremal) quasimodular forms of depth 1.

In Appendices we give: (i) A table of the integral Fourier coefficients of the
normalized extremal quasimodular forms of depth 1. (ii) Explicit formulas for
the coefficients of a certain formal power series that appears in the proof of
the main theorem.

##  2\. $G_{w}^{(1)}$ as a formal power series of $j^{-1}$

Let $p,q$ be non-negative integers and $a_{i},b_{j}\in\mathbb{C}$ with
$b_{j}\not\in\mathbb{Z}_{\leq 0}$. The generalized hypergeometric series
${}_{p}F_{q}$ is defined by

$\displaystyle{}_{p}F_{q}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};z\right)=\sum_{n=0}^{\infty}\frac{(a_{1})_{n}\dotsm(a_{p})_{n}}{(b_{1})_{n}\dotsm(b_{q})_{n}}\frac{z^{n}}{n!},$

where $(a)_{0}=1,(a)_{n}=a(a+1)\cdots(a+n-1)\;(n\geq 1)$ denotes the
Pochhammer symbol. This series is clearly invariant under the interchange of
each of the parameters $a_{i}$, and the same is true for $b_{j}$. When
$q=p-1$, the series
$F={}_{p}F_{p-1}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{p-1};z\right)$
satisfies the differential equation

$\displaystyle
z^{p-1}(1-z)\frac{d^{p}F}{dz^{p}}+\sum_{n=1}^{p-1}z^{n-1}(\alpha_{n}z+\beta_{n})\frac{d^{n}F}{dz^{n}}+\alpha_{0}\,F=0,$

where $\alpha_{n}$ and $\beta_{n}$ are some constants that depend on the
parameters $a_{i}$ and $b_{j}$. Using the Euler operator
$\Theta=z\tfrac{d}{dz}$, the above differential equation can be rewritten as
follows:

$\displaystyle\\{\Theta(\Theta+b_{1}-1)\dotsm(\Theta+b_{p-1}-1)-z(\Theta+a_{1})\dotsm(\Theta+a_{p})\\}F=0.$

Here we collect some hypergeometric series identities that we will use in
later discussions.

$\displaystyle{}_{2}F_{1}\left(\alpha,\beta;\gamma;z\right)=(1-z)^{\gamma-\alpha-\beta}{}_{2}F_{1}\left(\gamma-\alpha,\gamma-\beta;\gamma;z\right),$
(3)
$\displaystyle{}_{2}F_{1}\left(\alpha+1,\beta;\gamma;z\right)=\left(1+\tfrac{1}{\alpha}\,z\tfrac{d}{dz}\right){}_{2}F_{1}\left(\alpha,\beta;\gamma;z\right),$
(4)
$\displaystyle\begin{split}&{}_{2}F_{1}\left(\alpha,\beta;\alpha+\beta+\tfrac{1}{2};z\right)^{2}\\\
&={}_{3}F_{2}\left(2\alpha,\alpha+\beta,2\beta;2\alpha+2\beta,\alpha+\beta+\tfrac{1}{2};z\right)\end{split}\quad\text{(Clausen's
formula)},$ (5)
$\displaystyle\begin{split}&{}_{2}F_{1}\left(\alpha,\beta;\alpha+\beta-\tfrac{1}{2};z\right){}_{2}F_{1}\left(\alpha,\beta-1;\alpha+\beta-\tfrac{1}{2};z\right)\\\
&={}_{3}F_{2}\left(2\alpha,2\beta-1,\alpha+\beta-1;2\alpha+2\beta-2,\alpha+\beta-\tfrac{1}{2};z\right)\end{split}\quad\text{(Orr's
formula)}.$ (6)

See [2, pp. 85-86] for the proof and historical background of the equations
(5) and (6).

######  Proposition 1.

For sufficiently large $\Im(\tau)$, we have

$\displaystyle E_{4}(\tau)$
$\displaystyle={}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;\frac{1728}{j(\tau)}\right)^{4},\quad
E_{4}(\tau)^{1/2}={}_{3}F_{2}\left(\frac{1}{6},\frac{1}{2},\frac{5}{6};1,1;\frac{1728}{j(\tau)}\right),$
(7) $\displaystyle E_{6}(\tau)$
$\displaystyle=\left(1-\frac{1728}{j(\tau)}\right)^{1/2}{}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;\frac{1728}{j(\tau)}\right)^{6},$
(8) $\displaystyle E_{2}(\tau)$
$\displaystyle={}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;\frac{1728}{j(\tau)}\right){}_{2}F_{1}\left(-\frac{1}{12},\frac{7}{12};1;\frac{1728}{j(\tau)}\right)$
(9)
$\displaystyle=\left(1-\frac{1728}{j(\tau)}\right)^{1/2}{}_{3}F_{2}\left(\frac{1}{2},\frac{5}{6},\frac{7}{6};1,1;\frac{1728}{j(\tau)}\right).$
(10)

The hypergeometric expressions of the Eisenstein series $E_{4}$ and $E_{6}$
are classical and well known, for the proof, see [36]. In particular,
$E_{4}^{1/4}$ is one of the solutions of a hypergeometric differential
equation at $j=\infty$, and for the solution at $j=0,1728$, we refer to [1].
To obtain the second equation of (7), we use Clausen’s formula (5). See [41,
§5.4] for a more general explanation of the fact that a (holomorphic or
meromorphic) modular form satisfies a linear differential equation with a
modular function as a variable.

In contrast to $E_{4}$ and $E_{6}$, the hypergeometric expression (9) of
$E_{2}$ is less well known. To the best of the author’s knowledge, the
expression can be found in the author’s Ph.D. thesis [27, Ch. 3] and [31, Thm.
5]. There is also an equivalent expression in [26, Ch. 2.6], although it looks
a little different. For the convenience of the reader, we will briefly review
the proof in [27] here; by calculating the logarithmic derivative of
$\Delta=j^{-1}E_{4}^{3}=j^{-1}{}_{2}F_{1}(\tfrac{1}{12},\tfrac{5}{12};1;\tfrac{1728}{j})^{12}$,
we have

$\displaystyle E_{2}$
$\displaystyle=D(\log\Delta)=-\tfrac{D(j)}{j}+12\,{}_{2}F_{1}(\tfrac{1}{12},\tfrac{5}{12};1;\tfrac{1728}{j})^{-1}D\left({}_{2}F_{1}(\tfrac{1}{12},\tfrac{5}{12};1;\tfrac{1728}{j})\right)$
$\displaystyle=-\tfrac{D(j)}{j}+12\,{}_{2}F_{1}(\tfrac{1}{12},\tfrac{5}{12};1;\tfrac{1728}{j})^{-1}D\left(\tfrac{1728}{j}\right)\tfrac{d}{dz}{}_{2}F_{1}(\tfrac{1}{12},\tfrac{5}{12};1;z)\big{|}_{z=1728/j}$
$\displaystyle=-\tfrac{D(j)}{j}\,{}_{2}F_{1}(\tfrac{1}{12},\tfrac{5}{12};1;\tfrac{1728}{j})^{-1}\left\\{\left(1+12z\tfrac{d}{dz}\right){}_{2}F_{1}(\tfrac{1}{12},\tfrac{5}{12};1;z)\big{|}_{z=1728/j}\right\\}$
$\displaystyle=\tfrac{E_{6}}{E_{4}}\,{}_{2}F_{1}(\tfrac{1}{12},\tfrac{5}{12};1;\tfrac{1728}{j})^{-1}\,{}_{2}F_{1}(\tfrac{13}{12},\tfrac{5}{12};1;\tfrac{1728}{j})\quad\text{(by
\eqref{eq:dHyp})}$
$\displaystyle=\tfrac{E_{6}}{E_{4}}\,{}_{2}F_{1}(\tfrac{1}{12},\tfrac{5}{12};1;\tfrac{1728}{j})^{-1}\,\left(1-\tfrac{1728}{j}\right)^{-1/2}\,{}_{2}F_{1}(-\tfrac{1}{12},\tfrac{7}{12};1;\tfrac{1728}{j})\quad\text{(by
\eqref{eq:Euler})}$
$\displaystyle={}_{2}F_{1}(\tfrac{1}{12},\tfrac{5}{12};1;\tfrac{1728}{j})\,{}_{2}F_{1}(-\tfrac{1}{12},\tfrac{7}{12};1;\tfrac{1728}{j}).$

In the fourth equality we used the fact that $D(j)=-jE_{6}/E_{4}$, which can
be calculated with (1). Equation (10) is obtained by transforming Equation (9)
using Orr’s formula as $(\alpha,\beta)=(\tfrac{7}{12},\tfrac{11}{12})$ as
follows.

$\displaystyle E_{2}$
$\displaystyle={}_{2}F_{1}(\tfrac{1}{12},\tfrac{5}{12};1;\tfrac{1728}{j})\,{}_{2}F_{1}(-\tfrac{1}{12},\tfrac{7}{12};1;\tfrac{1728}{j})$
$\displaystyle=\left(1-\tfrac{1728}{j}\right)^{1/2}\,{}_{2}F_{1}(\tfrac{11}{12},\tfrac{7}{12};1;\tfrac{1728}{j})\,{}_{2}F_{1}(-\tfrac{1}{12},\tfrac{7}{12};1;\tfrac{1728}{j})$
$\displaystyle=\left(1-\tfrac{1728}{j}\right)^{1/2}\,{}_{3}F_{2}(\tfrac{7}{6},\tfrac{5}{6},\tfrac{1}{2};1,1;\tfrac{1728}{j}).$

Alternatively, the expressions (9) and (10) can be obtained by setting $n=0$
in equations (21) and (58), since $E_{2}=G_{2}^{(1)}=G_{2}^{(2)}$.

By setting $t=1/j(\tau)$, Proposition 1 immediately implies the following
theorem, which is a generalization of Theorem 5 in [36] for
$M_{*}(\Gamma)=\mathbb{C}[E_{4},E_{6}]$.

######  Theorem 2.

Put
$\mathcal{F}_{1}(t)={}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;1728t\right),\mathcal{F}_{2}(t)={}_{2}F_{1}\left(-\frac{1}{12},\frac{7}{12};1;1728t\right)$,
and then we have the ring isomorphism

$\displaystyle
QM_{*}(\Gamma)\simeq\mathcal{F}:=\mathbb{C}[\mathcal{F}_{1}(t)\mathcal{F}_{2}(t),\mathcal{F}_{1}(t)^{4},(1-1728t)^{1/2}\mathcal{F}_{1}(t)^{6}].$
(11)

Moreover, since $QM_{*}$ is closed under the derivation $D$, the ring
$\mathcal{F}$ is closed under the derivation
$D_{\mathcal{F}}:=(1-1728t)^{1/2}\mathcal{F}_{1}(t)^{2}\,t\frac{d}{dt}$. In
other words, the isomorphism
$(QM_{*}(\Gamma),D)\simeq(\mathcal{F},D_{\mathcal{F}})$ holds as a graded
differential algebra over $\mathbb{C}$.

Based on the ring isomorphism (11), we can consider the problem of the
integrality of the Fourier coefficients of normalized extremal quasimodular
forms as an equivalence problem in $\mathcal{F}$.

######  Example 1.

The formula in $\mathcal{F}$ equivalent to the formula
$D(E_{2})=\tfrac{1}{12}(E_{2}^{2}-E_{4})$ in $QM_{*}$ can be calculated as
follows.

$\displaystyle{}12t\frac{d}{dt}\mathcal{F}_{1}(t)\mathcal{F}_{2}(t)=\left(12t\frac{d}{dt}\mathcal{F}_{1}(t)\right)\mathcal{F}_{2}(t)+\mathcal{F}_{1}(t)\left(12t\frac{d}{dt}\mathcal{F}_{2}(t)\right)$
$\displaystyle=\mathcal{F}_{2}(t)\left(1+12t\frac{d}{dt}\right)\mathcal{F}_{1}(t)-\mathcal{F}_{1}(t)\left(1-12t\frac{d}{dt}\right)\mathcal{F}_{2}(t)$
$\displaystyle=\mathcal{F}_{2}(t)\,{}_{2}F_{1}\left(\tfrac{13}{12},\tfrac{5}{12};1;1728t\right)-\mathcal{F}_{1}(t)\,{}_{2}F_{1}\left(\tfrac{11}{12},\tfrac{7}{12};1;1728t\right)\quad(\text{by
\eqref{eq:dHyp}})$
$\displaystyle=(1-1728t)^{-1/2}\left(\mathcal{F}_{2}(t)^{2}-\mathcal{F}_{1}(t)^{2}\right)\quad(\text{by
\eqref{eq:Euler}}).$

Hence we obtain
$D_{\mathcal{F}}(\mathcal{F}_{1}(t)\mathcal{F}_{2}(t))=\tfrac{1}{12}(\mathcal{F}_{1}(t)^{2}\mathcal{F}_{2}(t)^{2}-\mathcal{F}_{1}(t)^{4})$.
The equations corresponding to the remaining equations in (1) can be
calculated in a similar way.

We introduce the Serre derivative (or Ramanujan–Serre derivative)
$\partial_{k}$ defined by

$\displaystyle\partial_{k}=D-\frac{k}{12}E_{2}.$

From this definition, it is clear that the Leibniz rule
$\partial_{k+l}(fg)=\partial_{k}(f)g+f\partial_{l}(g)$ is satisfied. According
to convention, we use the following symbols for the iterated Serre derivative:

$\displaystyle\partial_{k}^{0}(f)=f,\quad\partial_{k}^{n+1}(f)=\partial_{k+2n}\circ\partial_{k}^{n}(f)\quad(n\geq
0).$

It is well known that $\partial_{k-r}:QM_{k}^{(r)}\rightarrow QM_{k+2}^{(r)}$
for even $k$ and $r\in\mathbb{Z}_{\geq 0}$, which is a special case of
Proposition 3.3 in [19]. By rewriting (1) using the Serre derivative, we
obtain the following useful consequences.

$\displaystyle\partial_{1}(E_{2})=-\frac{1}{12}E_{4},\quad\partial_{4}(E_{4})=-\frac{1}{3}E_{6},\quad\partial_{6}(E_{6})=-\frac{1}{2}E_{4}^{2}.$
(12)

Note that the Serre derivative is not necessarily depth-preserving, as can be
seen from the first equation in (12). Therefore, it is necessary to confirm
the depth of the quasimodular form created by repeatedly applying a
differential operator including the Serre derivative as in (17).

Now we consider the following differential equation, called the Kaneko–Zagier
equation or, in a more general context, a second-order modular linear
differential equation:

$\displaystyle D^{2}(f)-\frac{w}{6}E_{2}D(f)+\frac{w(w-1)}{12}D(E_{2})f=0.$
(13)

This differential equation (with $w$ replaced by $k+1$) first appeared in the
study of the $j$-invariants of supersingular elliptic curves in [21] and was
characterized in [17, §5]. Although we consider quasimodular form solutions of
(13) in this paper, it should be emphasized that the Kaneko–Zagier equation
(13) has modular form solutions of level $1,2,3,4$ ([17]), and $5$ ([16]) and
mixed mock modular form solutions ([12]) for appropriate $w$.

It is easy to rewrite the differential equation (13) as $L_{w}(f)=0$, where

$\displaystyle L_{w}\coloneqq\partial_{w-1}^{2}-\frac{w^{2}-1}{144}E_{4}.$
(14)

We also define the differential operators $K_{w}^{\mathrm{up}}$ ([10]) and its
“adjoint” $\widehat{K_{w}^{\mathrm{up}}}$ as follows:

$\displaystyle K_{w}^{\mathrm{up}}\coloneqq
E_{4}\partial_{w-1}-\frac{w+1}{12}E_{6},\quad\widehat{K_{w}^{\mathrm{up}}}\coloneqq
E_{4}\partial_{w+3}-\frac{w+9}{12}E_{6}.$ (15)

Then we have the following identity for the composition of the differential
operators.

$\displaystyle L_{w+6}\circ
K_{w}^{\mathrm{up}}=\widehat{K_{w}^{\mathrm{up}}}\circ L_{w}.$ (16)

We will not give the proof, since it is a simple calculation by using
Ramanujan’s identity (1). However, this identity is important for
investigating the inductive structure of the solutions of the differential
equation $L_{w}(f)=0$. Indeed, if a function $f$ satisfies $L_{w}(f)=0$, we
see that $F:=K_{w}^{\mathrm{up}}(f)$ satisfies $L_{w+6}(F)=0$. Therefore, the
identity (16) gives not only the structure of the quasimodular solution of
(13), but also that of a modular solution, a logarithmic solution, and a
formal $q$-series solution.

The differential operator (16) can be considered as a third-order differential
operator, so that the given second-order differential operators $L_{w+6}$ and
$L_{w}$ are in the left and right factors, respectively. From this point of
view, the factors $K_{w}^{\mathrm{up}}$ and its adjoint are uniquely and
independently determined from the Fourier coefficients of the extremal
quasimodular forms, up to a constant multiple.

We define the sequence of power series $G_{k}^{*}$ by the following
differential recursions:

$\displaystyle{}G_{0}^{*}=1,\quad
G_{w+6}^{*}=\frac{w+6}{72(w+1)(w+5)}K_{w}^{\mathrm{up}}(G_{w}^{*}),$ (17)
$\displaystyle{}G_{w+2}^{*}=\frac{12}{w+1}\partial_{w-1}(G_{w}^{*}),\quad
G_{w+4}^{*}=E_{4}G_{w}^{*}.$ (18)

The proportionality constants appearing in the definition are chosen so that
the leading coefficients of $G_{w}^{*}$ and $G_{w+2}^{*}$ are $1$ as follows.
Since $L_{0}(G_{0}^{*})=L_{0}(1)=0$, we have $L_{w}(G_{w}^{*})=0$ for any
$w\equiv 0\pmod{6}$ with the aid of (16). Then the Frobenius ansatz (see [10,
§2.3]) gives

$\displaystyle
G_{w}^{*}=q^{w/6}\left\\{1+\frac{4w(2w-3)}{w+6}q+O(q^{2})\right\\}.$

By applying the differential operators $\partial_{w-1}$ and
$K_{w}^{\mathrm{up}}$ act on this power series, we have

$\displaystyle\partial_{w-1}(G_{w}^{*})$
$\displaystyle=q^{w/6}\left(\frac{w+1}{12}+O(q)\right),$ $\displaystyle
K_{w}^{\mathrm{up}}(G_{w}^{*})$
$\displaystyle=q^{(w+6)/6}\left\\{\frac{72(w+1)(w+5)}{w+6}+O(q)\right\\}.$

Note that $G_{w+4}^{*}$ can also be expressed as
$\frac{12}{w-1}\partial_{w+1}(G_{w+2}^{*})$.

From this construction and the property of the Serre derivative
$\partial_{w-1}:QM_{w}^{(1)}\rightarrow QM_{w+2}^{(1)}$, it is clear that
$G_{k}^{*}=q^{m-1}(1+O(q))$ for $m=\dim_{\mathbb{C}}QM_{k}^{(1)}$ and
$G_{k}^{*}\in QM_{k}^{(1)}$ for even integer $k$. Therefore, from the same
discussion in Section 1, the power series $G_{k}^{*}$ is nothing but the
normalized extremal quasimodular form $G_{k}^{(1)}$, we have
$L_{w}(G_{w}^{(1)})=0$ for $w\equiv 0\pmod{6}$.

By changing the variables $z=1728/j(\tau)$, the Kaneko–Zagier equation (13),
which is equivalent to $L_{w}(f)=0$, is transformed into

$\displaystyle
z(1-z)\frac{d^{2}g}{dz^{2}}+\left(-\frac{w-6}{6}+\frac{w-9}{6}z\right)\frac{dg}{dz}-\frac{(w-1)(w-5)}{144}g=0$

or equivalently (Recall that $\Theta=z\tfrac{d}{dz}$.)

$\displaystyle\left\\{\Theta\left(\Theta-\frac{w-6}{6}-1\right)-z\left(\Theta-\frac{w-1}{12}\right)\left(\Theta-\frac{w-5}{12}\right)\right\\}g=0,$

where $g(\tau)=E_{4}(\tau)^{-(w-1)/4}f(\tau)$. This differential equation is a
hypergeometric differential equation. Since we now assume $w\equiv 0\pmod{6}$,
the solution space of the above differential equation is spanned by the power
series solution

$\displaystyle
z^{w/6}\,{}_{2}F_{1}\left(\frac{w+1}{12},\frac{w+5}{12};\frac{w}{6}+1;z\right)$
(19)

and a logarithmic solution. Therefore, since the extremal quasimodular forms
contain no logarithmic terms, we obtain the hypergeometric expression of
$G_{w}^{(1)}$ for $w\equiv 0\pmod{6}$.

######  Proposition 2.

The normalized extremal quasimodular forms of even weight and depth $1$ on
$\Gamma$ have the following hypergeometric expressions.

$\displaystyle G_{6n}^{(1)}(\tau)$
$\displaystyle=j(\tau)^{-n}{}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;\frac{1728}{j(\tau)}\right)^{2(3n-1)}P_{n}\left(\frac{1}{j(\tau)}\right),$
$\displaystyle G_{6n+2}^{(1)}(\tau)$
$\displaystyle=j(\tau)^{-n}{}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;\frac{1728}{j(\tau)}\right)^{6n}Q_{n}\left(\frac{1}{j(\tau)}\right),$
$\displaystyle G_{6n+4}^{(1)}(\tau)$
$\displaystyle=E_{4}(\tau)G_{6n}^{(1)}(\tau),$

where

$\displaystyle P_{n}(t)$
$\displaystyle:={}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;1728t\right)\,{}_{2}F_{1}\left(\frac{6n+1}{12},\frac{6n+5}{12};n+1;1728t\right),$
(20) $\displaystyle Q_{n}(t)$
$\displaystyle:={}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;1728t\right)\,{}_{2}F_{1}\left(\frac{6n-1}{12},\frac{6n+7}{12};n+1;1728t\right).$
(21)

###### Proof.

The hypergeometric expression of $G_{6n}^{(1)}(\tau)$ can be obtained by
setting $z=\frac{1728}{j(\tau)}$ in (19) and multiplying by
$1728^{-n}E_{4}^{(6n-1)/4}$. Similar expression of $G_{6n+2}^{(1)}(\tau)$ can
be obtained by applying the Serre derivative to $G_{6n}^{(1)}(\tau)$ as
follows ($w=6n$):

$\displaystyle{}\frac{w+1}{12}G_{w+2}^{(1)}=\partial_{w-1}(G_{w}^{(1)})=D(G_{w}^{(1)})-\frac{w-1}{12}E_{2}G_{w}^{(1)}$
$\displaystyle=D(j^{-w/6}E_{4}^{(w-1)/4})\,{}_{2}F_{1}\left(\frac{w+1}{12},\frac{w+5}{12};\frac{w}{6}+1;\frac{1728}{j}\right)$
$\displaystyle{}\quad+j^{-w/6}E_{4}^{(w-1)/4}D\left(\frac{1728}{j}\right)\frac{d}{dz}{}_{2}F_{1}\left(\frac{w+1}{12},\frac{w+5}{12};\frac{w}{6}+1;z\right)\Bigg{|}_{z=1728/j}$
$\displaystyle{}\quad-\frac{w-1}{12}E_{2}\,j^{-w/6}E_{4}^{(w-1)/4}\,{}_{2}F_{1}\left(\frac{w+1}{12},\frac{w+5}{12};\frac{w}{6}+1;\frac{1728}{j}\right)$
$\displaystyle=\frac{w+1}{12}j^{-w/6}E_{4}^{(w-5)/4}E_{6}\;{}_{2}F_{1}\left(\frac{w+1}{12},\frac{w+5}{12};\frac{w}{6}+1;\frac{1728}{j}\right)$
$\displaystyle{}\quad+j^{-w/6}E_{4}^{(w-5)/4}E_{6}\;z\frac{d}{dz}{}_{2}F_{1}\left(\frac{w+1}{12},\frac{w+5}{12};\frac{w}{6}+1;z\right)\Bigg{|}_{z=1728/j}$
$\displaystyle=\frac{w+1}{12}j^{-w/6}E_{4}^{(w-5)/4}E_{6}\;{}_{2}F_{1}\left(\frac{w+13}{12},\frac{w+5}{12};\frac{w}{6}+1;\frac{1728}{j}\right)\quad(\text{by
\eqref{eq:dHyp}})$
$\displaystyle=\frac{w+1}{12}j^{-w/6}E_{4}^{(w+1)/4}\;{}_{2}F_{1}\left(\frac{w-1}{12},\frac{w+7}{12};\frac{w}{6}+1;\frac{1728}{j}\right)\quad(\text{by
\eqref{eq:Euler}}).$

∎

######  Remark 1.

When $w=0$, the Kaneko–Zagier equation (13) becomes the equation $D^{2}(f)=0$,
which has 1 and $2\pi i\tau=\log(q)$ as independent solutions. In general, for
$w=6n\,(n\in\mathbb{Z}_{\geq 0})$, the two-dimensional solution space of the
differential equation (13) has already been discussed in [17, §5], and its
basis is given by $G_{6n}^{(1)}(\tau)$ and

$\displaystyle
H_{6n}^{(1)}(\tau):=\begin{cases}G_{12m}^{(1)}(\tau)\log(q)-\dfrac{12}{N_{m,0}}E_{4}(\tau)E_{6}(\tau)\Delta(\tau)^{m-1}A_{m,0}(j(\tau))\\\
\quad\text{ if $n=2m\,(m\in\mathbb{Z}_{\geq 0})$},\\\
G_{12m+6}^{(1)}(\tau)\log(q)+\dfrac{12}{N_{m,6}}E_{4}(\tau)\Delta(\tau)^{m}A_{m,6}(j(\tau))\\\
\quad\text{ if $n=2m+1\,(m\in\mathbb{Z}_{\geq 0})$},\end{cases}$

where $A_{m,0}(X)$ and $A_{m,6}(X)$ are Atkin-like polynomials that appear
later in this section, and the numbers $N_{m,0}$ and $N_{m,6}$ are defined by
(27) and (28). The solution $H_{6n}^{(1)}$ can also be obtained by the
recurrence formula (17), where the initial value is replaced by
$H_{0}^{(1)}=\log(q)$.

It is well-known that the space $M_{k}(\Gamma)$ has a basis
$\\{E_{4}^{3m+\delta}E_{6}^{\varepsilon},E_{4}^{3m+\delta-3}E_{6}^{\varepsilon}\Delta,\dotsc,E_{6}^{\varepsilon}\Delta^{m}\\}$
for $k=12m+4\delta+6\varepsilon$ with $m\in\mathbb{Z}_{\geq
0},\delta\in\\{0,1,2\\},\varepsilon\in\\{0,1\\}$. This basis is characterized
by the fact that the leading terms of its Fourier expansion are
$1,q,\dotsc,q^{m}$. We can construct a basis for $QM_{k}^{(1)}(\Gamma)$ with
the same property using the normalized extremal quasimodular forms of depth
$1$.

######  Proposition 3.

For any even integer $k\geq 2$, a basis $\mathcal{B}_{k}^{(1)}$ of the space
$QM_{k}^{(1)}(\Gamma)$ is given by the following set: where the notation is
that in Theorem 2 and Proposition 2.

$\displaystyle\mathcal{B}_{12m}^{(1)}$
$\displaystyle=\mathcal{F}_{1}(j^{-1})^{12m-2}\left(\left\\{j^{-2\ell}P_{2\ell}(j^{-1})\right\\}_{0\leq\ell\leq
m}\cup\left\\{j^{-2\ell-1}Q_{2\ell+1}(j^{-1})\right\\}_{0\leq\ell\leq
m-1}\right)$
$\displaystyle=\left\\{E_{4}^{3m},E_{4}^{3m-2}G_{8}^{(1)},E_{4}^{3m-3}G_{12}^{(1)},E_{4}^{3m-5}G_{20}^{(1)},E_{4}^{3m-6}G_{24}^{(1)},\dotsc,G_{12m}^{(1)}\right\\},$
$\displaystyle\mathcal{B}_{12m+2}^{(1)}$
$\displaystyle=\mathcal{F}_{1}(j^{-1})^{12m}\left(\left\\{j^{-2\ell}Q_{2\ell}(j^{-1})\right\\}_{0\leq\ell\leq
m}\cup\left\\{j^{-2\ell-1}P_{2\ell+1}(j^{-1})\right\\}_{0\leq\ell\leq
m-1}\right)$
$\displaystyle=\left\\{E_{4}^{3m}G_{2}^{(1)},E_{4}^{3m-2}G_{10}^{(1)},E_{4}^{3m-3}G_{14}^{(1)},E_{4}^{3m-5}G_{22}^{(1)},E_{4}^{3m-6}G_{26}^{(1)},\dotsc,G_{12m+2}^{(1)}\right\\},$
$\displaystyle\mathcal{B}_{12m+4}^{(1)}$
$\displaystyle=E_{4}\,\mathcal{B}_{12m}^{(1)},\quad\mathcal{B}_{12m+6}^{(1)}=E_{4}\,\mathcal{B}_{12m+2}^{(1)}\cup\left\\{G_{12m+6}^{(1)}\right\\},$
$\displaystyle\mathcal{B}_{12m+8}^{(1)}$
$\displaystyle=E_{4}^{2}\,\mathcal{B}_{12m}^{(1)}\cup\left\\{G_{12m+8}^{(1)}\right\\},\quad\mathcal{B}_{12m+10}^{(1)}=E_{4}\,\mathcal{B}_{12m+6}^{(1)}.$

###### Proof.

We give a proof only for the case of $QM_{12m}^{(1)}(\Gamma)$, the remaining
cases being similar. We put
$f_{0}=E_{4}^{3m}=1+O(q),f_{1}=E_{4}^{3m-2}G_{8}^{(1)}=q+O(q^{2}),\dotsc,f_{2m}=G_{12m}^{(1)}=q^{2m}+O(q^{2m+1})$.
For any quasimodular form $f\in QM_{12m}^{(1)}(\Gamma)$, we can determine the
coefficient $a_{r}$ such that $F:=f-\sum_{r=0}^{2m}a_{r}f_{r}=O(q^{2m+1})\in
QM_{12m}^{(1)}(\Gamma)$. Since $\nu_{\max}(1,12m)=2m$, the form $F$ must be
$0$, and hence $QM_{12m}^{(1)}(\Gamma)$ is spanned by
$\\{f_{0},\dotsc,f_{2m}\\}$. For the linear independence property, by
comparing the Fourier coefficients, we have
$\sum_{r=0}^{2m}a_{r}f_{r}=0\Rightarrow a_{r}=0\,(0\leq r\leq 2m)$. ∎

######  Example 2.

Case of weight $18$: The following equation holds.

$\displaystyle\begin{pmatrix}E_{4}^{3}E_{6}\\\ E_{6}\Delta\\\
E_{2}E_{4}^{4}\\\
E_{2}E_{4}\Delta\end{pmatrix}=\mathcal{F}_{1}(j^{-1})^{17}\cdot
A\begin{pmatrix}{}_{2}F_{1}(-\tfrac{1}{12},\tfrac{7}{12};1;\tfrac{1728}{j})\\\
j^{-1}\,{}_{2}F_{1}(\tfrac{7}{12},\tfrac{11}{12};2;\tfrac{1728}{j})\\\
j^{-2}\,{}_{2}F_{1}(\tfrac{11}{12},\tfrac{19}{12};3;\tfrac{1728}{j})\\\
j^{-3}\,{}_{2}F_{1}(\tfrac{19}{12},\tfrac{23}{12};4;\tfrac{1728}{j})\end{pmatrix}=A\begin{pmatrix}E_{4}^{4}G_{2}^{(1)}\\\
E_{4}^{3}G_{6}^{(1)}\\\ E_{4}G_{14}^{(1)}\\\ G_{18}^{(1)}\end{pmatrix},$

where

$\displaystyle A=\begin{pmatrix}1&-720&0&0\\\ 0&1&-1266&269280\\\ 1&0&0&0\\\
0&1&-546&0\end{pmatrix}.$

######  Lemma 1.

The following claims hold.

  1. (1)

$f(t)\in\mathbb{Z}[\\![t]\\!]\Leftrightarrow(1-1728t)^{-1/2}f(t)\in
Z[\\![t]\\!]$.

  2. (2)

$j(\tau)^{-1}\in q(1+q\mathbb{Z}[\\![q]\\!])$.

  3. (3)

$f(\tau)\in\mathbb{Z}[\\![q]\\!]\Leftrightarrow
f(\tau)\in\mathbb{Z}[\\![j^{-1}]\\!]$.

  4. (4)

$G_{6n}^{(1)}(\tau)\in\mathbb{Z}[\\![q]\\!]\Leftrightarrow
G_{6n+4}^{(1)}(\tau)\in\mathbb{Z}[\\![q]\\!]$.

  5. (5)

$G_{6n}^{(1)}(\tau)\in\mathbb{Z}[\\![q]\\!]\Leftrightarrow
P_{n}(t)\in\mathbb{Z}[\\![t]\\!]$ and
$G_{6n+2}^{(1)}(\tau)\in\mathbb{Z}[\\![q]\\!]\Leftrightarrow
Q_{n}(t)\in\mathbb{Z}[\\![t]\\!]$.

###### Proof.

  1. (1)

It is clear that the following facts imply the claim:

$\displaystyle{}(1-4x)^{-1/2}=\sum_{r=0}^{\infty}\binom{2r}{r}x^{r}\in\mathbb{Z}[\\![x]\\!],$
$\displaystyle{}(1-4x)^{1/2}\left(=\sum_{r=0}^{\infty}\frac{1}{1-2r}\binom{2r}{r}x^{r}\right)=(1-4x)(1-4x)^{-1/2}\in\mathbb{Z}[\\![x]\\!].$

  2. (2)

Since $E_{4}\in 1+q\mathbb{Z}[\\![q]\\!]$, $1/E_{4}^{3}\in
1+q\mathbb{Z}[\\![q]\\!]$ holds, and we have $j^{-1}=\Delta/E_{4}^{3}\in
q(1+q\mathbb{Z}[\\![q]\\!])$.

  3. (3)

Using claim (2), we know that the coefficients satisfy
$a_{m}\in\mathbb{Z}[b_{0},b_{1},\dots,b_{m}]$ and
$b_{m}\in\mathbb{Z}[a_{0},a_{1},\dots,a_{m}]$ when the function $f$ is
expressed in two ways as
$f=a_{0}+a_{1}q+a_{2}q^{2}+\cdots=b_{0}+b_{1}j^{-1}+b_{2}j^{-2}+\cdots$. This
gives us the assertion.

  4. (4)

It is clear from the facts that $E_{4},1/E_{4}\in 1+q\mathbb{Z}[\\![q]\\!]$
and $G_{6n+4}^{(1)}=E_{4}G_{6n}^{(1)}$.

  5. (5)

By combining (7), (29) and (30), we have $E_{4}^{1/2}\in
1+j^{-1}\mathbb{Z}[\\![j^{-1}]\\!]$ and hence $E_{4}^{1/2}\in
1+q\mathbb{Z}[\\![q]\\!]$ and $E_{4}^{-1/2}\in 1+q\mathbb{Z}[\\![q]\\!]$. By
Proposition 2, $G_{6n}^{(1)}=j^{-n}E_{4}^{(3n-1)/2}P_{n}(j^{-1})$ holds, so if
$G_{6n}\in\mathbb{Z}[\\![q]\\!]$, then
$E_{4}^{-(3n-1)/2}G_{6n}^{(1)}\in\mathbb{Z}[\\![q]\\!]$ and hence
$j^{-n}P_{n}(j^{-1})\in\mathbb{Z}[\\![j^{-1}]\\!]$ (note (3)). And conversely,
if $P_{n}(t)\in\mathbb{Z}[\\![t]\\!]$, we have
$G_{6n}=j^{-n}E_{4}^{(3n-1)/2}P_{n}(j^{-1})\in\mathbb{Z}[\\![j^{-1}]\\!]$ and
therefore $G_{6n}^{(1)}\in\mathbb{Z}[\\![q]\\!]$ holds. The case of
$G_{6n+2}^{(1)}$ can be proved similarly.

∎

######  Remark 2.

In the above proof we used that $E_{4}^{1/2}\in 1+q\mathbb{Z}[\\![q]\\!]$, but
a stronger claim holds. The hypergeometric series ${}_{2}F_{1}$ satisfies the
following quadratic transformation of Gauss:

$\displaystyle{}_{2}F_{1}\left(\alpha,\beta;\alpha+\beta+\frac{1}{2};4z(1-z)\right)={}_{2}F_{1}\left(2\alpha,2\beta;\alpha+\beta+\frac{1}{2};z\right).$
(22)

By putting
$(\alpha,\beta,4z(1-z))=(\tfrac{1}{12},\tfrac{5}{12},\tfrac{1728}{j})$ in the
above equation, we have

$\displaystyle E_{4}^{1/4}$
$\displaystyle={}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;\frac{1728}{j}\right)={}_{2}F_{1}\left(\frac{1}{6},\frac{5}{6};1;z\right)=\sum_{n=0}^{\infty}\binom{3n}{n}\binom{6n}{3n}\left(\frac{z}{432}\right)^{n},$
$\displaystyle\frac{z}{432}$
$\displaystyle=\frac{1}{864}\left\\{1-\left(1-\frac{1728}{j}\right)^{1/2}\right\\}=\sum_{m=1}^{\infty}432^{m-1}C_{m-1}j^{-m}\in
j^{-1}\mathbb{Z}[\\![j^{-1}]\\!],$

where $C_{n}:=\tfrac{1}{n+1}\tbinom{2n}{n}=\tbinom{2n}{n}-\tbinom{2n}{n-1}$ is
the $n$-th Catalan number. (The second equality holds for $n\geq 1$.) Thus we
have $E_{4}^{1/4}\in 1+j^{-1}\mathbb{Z}[\\![j^{-1}]\\!]$ or equivalently
$E_{4}^{1/4}\in 1+q\mathbb{Z}[\\![q]\\!]$. Actually, it is known the stronger
claim $E_{4}^{1/8}\in 1+q\mathbb{Z}[\\![q]\\!]$ holds. See [13] for more
details and related results.

We classify each element of the set (2) according to the remainder of modulo
12 as follows:

$\displaystyle\mathcal{S}_{0}$
$\displaystyle=\\{12,24\\},\;\mathcal{S}_{2}=\\{2,14,38\\},\;\mathcal{S}_{4}=\mathcal{S}_{0}\oplus
4=\\{16,28\\},\;\mathcal{S}_{6}=\\{6,18,30,54,114\\},$
$\displaystyle\mathcal{S}_{8}$
$\displaystyle=\\{8,20,32,68,80\\},\;\mathcal{S}_{10}=\mathcal{S}_{6}\oplus
4=\\{10,22,34,58,118\\}.$

Here the symbol $\\{\mathrm{list}\\}\oplus n$ means that for each element in
the list the number $n$ is added. From (4) of Lemma 1, it suffices to show
that the function $G_{w}^{(1)}$ has integral Fourier coefficients when the
weight $w$ is an element of the set
$\mathcal{S}_{0}\cup\mathcal{S}_{2}\cup\mathcal{S}_{6}\cup\mathcal{S}_{8}$.

Put $P_{n}(t)=\sum_{\ell=0}^{\infty}a_{\ell}(n)t^{\ell}$ and
$Q_{n}(t)=\sum_{\ell=0}^{\infty}b_{\ell}(n)t^{\ell}$. The first few
coefficients are given by

$\displaystyle a_{1}(n)$
$\displaystyle=60+432n+\frac{60}{n+1},\;a_{2}(n)=65700+305856n+93312n^{2}+\frac{31320}{n+1}-\frac{27720}{n+2},$
$\displaystyle b_{1}(n)$
$\displaystyle=60+432n-\frac{84}{n+1},\;b_{2}(n)=3492+305856n+93312n^{2}-\frac{37800}{n+1}+\frac{32760}{n+2}.$

Consider the condition that these coefficients are integers. In such a case,
the denominator must divide the numerator of the above equations, and so
$a_{1}(n)$ and $a_{2}(n)$ are integers if $n\in\\{0,1,2,3,4,5,9,19\\}$.
Similarly, if $n\in\\{0,1,2,3,5,6,11,13\\}$, $b_{1}(n)$ and $b_{2}(n)$ are
integers. In these lists of $n$, we exclude the case of $P_{0}(t)$, because it
corresponds to the trivial case $G_{0}^{(1)}=1$. From (5) of Lemma 1, in order
to prove the main theorem, we should show that the power series $P_{n}(t)$ and
$Q_{n}(t)$ are actually have integral coefficients for these lists of $n$.
That is, we prove that

  * •

$n\in\\{1,2,3,4,5,9,19\\}\Rightarrow P_{n}(t)\in\mathbb{Z}[\\![t]\\!]$, and
then $G_{w}^{(1)}\in\mathbb{Z}[\\![q]\\!]$ for
$w\in\mathcal{S}_{0}\cup\mathcal{S}_{6}$,

  * •

$n\in\\{0,1,2,3,5,6,11,13\\}\Rightarrow Q_{n}(t)\in\mathbb{Z}[\\![t]\\!]$, and
then $G_{w}^{(1)}\in\mathbb{Z}[\\![q]\\!]$ for
$w\in\mathcal{S}_{2}\cup\mathcal{S}_{8}$.

To prove these assertions, we will now rewrite the formal power series
$P_{n}(t)$ and $Q_{n}(t)$ into a more manageable form.

Although the symbols are slightly different, in their paper [19], Kaneko and
Koike expressed the normalized extremal quasimodular forms by using the monic
polynomials $A_{m,a}(X)$ and $B_{m,a}(X)$ as follows:

$\displaystyle{}G_{12m}^{(1)}=\frac{1}{N_{m,0}}\left(-E_{2}E_{4}E_{6}\Delta^{m-1}A_{m,0}(j)+\Delta^{m}B_{m,0}(j)\right),$
(23)
$\displaystyle{}G_{12m+2}^{(1)}=\frac{1}{N_{m,2}}\left(E_{2}\Delta^{m}A_{m,2}(j)-E_{4}^{2}E_{6}\Delta^{m-1}B_{m,2}(j)\right),$
(24)
$\displaystyle{}G_{12m+6}^{(1)}=\frac{1}{N_{m,6}}\left(E_{2}E_{4}\Delta^{m}A_{m,6}(j)-E_{6}\Delta^{m}B_{m,6}(j)\right),$
(25)
$\displaystyle{}G_{12m+8}^{(1)}=\frac{1}{N_{m,8}}\left(-E_{2}E_{6}\Delta^{m}A_{m,8}(j)+E_{4}^{2}\Delta^{m}B_{m,8}(j)\right),$
(26)

where the normalizing factor $N_{m,a}$ is given below for any $m\geq 0$ except
for $N_{0,0}=N_{0,2}\coloneqq 1$.

$\displaystyle N_{m,0}$ $\displaystyle=24m\binom{6m}{2m}\binom{12m}{6m},\quad
N_{m,2}=\frac{12m+1}{12m-1}N_{m,0},$ (27) $\displaystyle N_{m,6}$
$\displaystyle=N_{m+1/2,0}=12(2m+1)\binom{6m+3}{2m+1}\binom{12m+6}{6m+3},\quad
N_{m,8}=\frac{12m+7}{12m+5}N_{m,6}.$ (28)

Here, $N_{m,2}$ and $N_{m,8}$ are integers, since
$\tfrac{1}{1-2n}\tbinom{2n}{n}\in\mathbb{Z}$ from the proof of (1) of
Corollary 1. In particular, the polynomial $A_{n,2}(X)$ is equal to the
original Atkin polynomial $A_{n}(X)$ treated in [21]. We will refer to the
polynomials $A_{m,a}(X)$ as the Atkin-like polynomials. Note that we have the
polynomial corresponding to the quasimodular part as “$A$”, which is different
between us and [19]. The reason we do not follow the symbols of [21] is to
write the best rational function approximation in a unified way, which we will
discuss shortly afterwards. For the convenience of the reader, we provide
below a comparison table of these symbols. In the table, polynomials in the
same column are equal.

Table 1. Comparison table of the Atkin-like polynomials and its adjoint polynomials Our symbols | $A_{n,0}$ | $A_{n,2}$ | $A_{n,6}$ | $A_{n,8}$ | $B_{n,0}$ | $B_{n,2}$ | $B_{n,6}$ | $B_{n,8}$
---|---|---|---|---|---|---|---|---
Symbols in [19] | $B_{n}^{*}$ | $A_{n}$ | $\widetilde{A}_{n+1}$ | $\widetilde{B}_{n+1}^{*}$ | $A_{n}^{*}$ | $B_{n}$ | $\widetilde{B}_{n+1}$ | $\widetilde{A}_{n+1}^{*}$

We also note that the polynomials $A_{m,a}(X)$ and $B_{m,a}(X)$ are the
denominator and numerator, respectively, of the best rational-function
approximation111If it is not the best approximation, it contradicts the
extremality of $G_{w}^{(1)}$. As a side note, for example, to get the rational
function $B_{m,8}(j)/A_{m,8}(j)$ by Mathematica for certain $m$, we just have
to enter PadeApproximant$[(j-1728)f_{1}/(jf_{2}),\\{j,\text{Infinity},m\\}]$,
where $f_{1}={}_{2}F_{1}\left(13/12,5/12;1;1728/j\right)$ and
$f_{2}={}_{2}F_{1}\left(1/12,5/12;1;1728/j\right)$.  of the following power
series.

$\displaystyle j(j-1728)\Phi-\frac{B_{m,0}(j)}{A_{m,0}(j)}$
$\displaystyle=-\frac{N_{m,0}\,G_{12m}^{(1)}}{\Delta^{m}A_{m,0}(j)}=-\frac{N_{m,0}}{j^{2m-1}}+O(j^{-2m}),$
$\displaystyle\Phi-\frac{B_{m,2}(j)}{A_{m,2}(j)}$
$\displaystyle=\frac{N_{m,2}\,G_{12m+2}^{(1)}}{E_{4}^{2}E_{6}\Delta^{m-1}A_{m,2}(j)}=\frac{N_{m,2}}{j^{2m+1}}+O(j^{-2m-2}),$
$\displaystyle j\Phi-\frac{B_{m,6}(j)}{A_{m,6}(j)}$
$\displaystyle=\frac{N_{m,6}\,G_{12m+6}^{(1)}}{E_{6}\Delta^{m}A_{m,6}(j)}=\frac{N_{m,6}}{j^{2m+1}}+O(j^{-2m-2}),$
$\displaystyle(j-1728)\Phi-\frac{B_{m,8}(j)}{A_{m,8}(j)}$
$\displaystyle=-\frac{N_{m,8}\,G_{12m+8}^{(1)}}{E_{4}^{2}\Delta^{m}A_{m,8}(j)}=-\frac{N_{m,8}}{j^{2m+1}}+O(j^{-2m-2}),$

where $\Phi=\Phi(j^{-1})=E_{2}E_{4}/(jE_{6})$ is the power series with
$j^{-1}$ as the variable (recall Proposition 1), and then
$(j-1728)\Phi=E_{2}E_{6}/E_{4}^{2}$. More generally, for some suitable
polynomial $\psi(x)\in\mathbb{Q}[x]$, the orthogonal polynomial that appears
when approximating the function $j\psi(j)\Phi$ has already been considered in
[3] by Basha, Getz and Nover. However, they are treated in the context of a
generalization of the Atkin orthogonal polynomials, and not from the point of
view of extremal quasimodular forms.

Here and throughout this paper, the symbols $u_{r}$, $U(t)$, and $V(t)$ are
defined as the following positive integer and formal power series,
respectively:

$\displaystyle{}u_{r}=\frac{(6r)!}{(3r)!\,r!^{3}}=\binom{2r}{r}\binom{3r}{r}\binom{6r}{3r}=\binom{4r}{r}\binom{5r}{r}\binom{6r}{r}\quad(r\in\mathbb{Z}_{\geq
0}),$ (29)
$\displaystyle{}U(t)={}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;1728t\right)^{2}={}_{3}F_{2}\left(\frac{1}{6},\frac{1}{2},\frac{5}{6};1,1;1728t\right)=\sum_{r=0}^{\infty}u_{r}t^{r},$
(30)
$\displaystyle{}V(t)={}_{3}F_{2}\left(\frac{1}{2},\frac{5}{6},\frac{7}{6};1,1;1728t\right)=\left(1+6t\frac{d}{dt}\right)U(t)=\sum_{r=0}^{\infty}(6r+1)u_{r}t^{r}.$
(31)

Note that by using Proposition 1, $U(t)=P_{1}(t)$ and
$V(t)=(1-1728t)^{-1/2}Q_{0}(t)$ hold. Therefore,
$G_{6}^{(1)}\in\mathbb{Z}[\\![q]\\!]$ and
$G_{2}^{(1)}\in\mathbb{Z}[\\![q]\\!]$ hold with the aid of Lemma 1.

For a given polynomial $\alpha$, we denote its reciprocal polynomial as
$\widetilde{\alpha}$, i.e.,

$\displaystyle\alpha(j)=\sum_{k=0}^{m}c_{k}j^{k},\quad\widetilde{\alpha}(t):=t^{m}\alpha(1/t)=\sum_{k=0}^{m}c_{k}t^{m-k}.$

Comparing equations (23) to (26) with Proposition 2 and using the
hypergeometric expression of Eisenstein series in Proposition 1, we have

$\displaystyle{}N_{m,0}\,t^{2m}P_{2m}(t)=-\widetilde{A_{m,0}}(t)\,(1-1728t)V(t)+\widetilde{B_{m,0}}(t)U(t),$
(32)
$\displaystyle{}N_{m,2}\,t^{2m}(1-1728t)^{-1/2}Q_{2m}(t)=\widetilde{A_{m,2}}(t)V(t)-\widetilde{B_{m,2}}(t)U(t),$
(33)
$\displaystyle{}N_{m,6}\,t^{2m+1}(1-1728t)^{-1/2}P_{2m+1}(t)=\widetilde{A_{m,6}}(t)V(t)-\widetilde{B_{m,6}}(t)U(t),$
(34)
$\displaystyle{}N_{m,8}\,t^{2m+1}Q_{2m+1}(t)=-\widetilde{A_{m,8}}(t)\,(1-1728t)V(t)+\widetilde{B_{m,8}}(t)U(t).$
(35)

As we will see later in Section 4, the polynomials $A_{m,a}(X)$ and
$B_{m,a}(X)$ are not necessarily elements of $\mathbb{Z}[X]$. In such a case,
multiply both sides of the above equation by an appropriate factor $C$ so that
the right-hand side of (32) to (35) has integral coefficients. Under this
normalization, if all the coefficients of the formal power series on the
right-hand side are congruent to $0$ modulo $CN_{m,a}$, then we can conclude
that the corresponding formal power series $P_{n}(t)$ and $Q_{n}(t)$ have
integral coefficients. Here, for $Q_{2m}(t)$ and $P_{2m+1}(t)$, note (1) in
Lemma 1. Thus, in the next section we will investigate in detail the
congruence formulas of the formal power series $U(t)$ and $V(t)$ modulo prime
powers.

The above equations (32) to (35) are essentially the relations satisfied by a
hypergeometric series ${}_{2}F_{1}$, and can be interpreted in the following
two ways.

  1. (1)

Focusing on the left-hand side, from the definitions of the series $P_{n}(t)$
and $Q_{n}(t)$, the term depending on $m$ is a hypergeometric series with a
certain rational parameter shifted by an integer. Thus, the equations (32) to
(35) are concrete expressions of certain contiguous relations (or three term
relations) of a hypergeometric series ${}_{2}F_{1}$. For more general results
of this kind, see [8] by Ebisu.

  2. (2)

Focusing on the right-hand side, the polynomials of a certain degree or less
that are multiplied by the series $U(t)$ and $V(t)$ are chosen so that the
vanishing order for $t$ on the left-hand side is as large as possible.
Therefore, such polynomials are Hermite–Padé approximations for $U(t)$ and
$V(t)$, and the left-hand side is the corresponding remainder. For the
Hermite–Padé approximation of generalized hypergeometric series
${}_{p}F_{p-1}$, see [28] by Nesterenko.

Unfortunately, the case treated in the papers [8] and [28] is that the
corresponding hypergeometric differential equation has no logarithmic
solution, which does not include our case (recall Remark 1).

##  3\. Congruence formulas for $U(t)$ and $V(t)$ modulo prime powers

First, note that in the prime factorization of the normalizing factor
$N_{m,a}$, which appears in the proof of the main theorem in Section 4, the
exponents of all prime factors above 11 are 1. On the other hand, under the
proper normalization described at the end of the previous section, the
exponents of the prime factors 2, 3, 5, and 7 are at most 8, 5, 2, and 2,
respectively. Therefore, to prove the main theorem, we need to calculate the
specific congruence formulas for $U(t)$ and $V(t)$ modulo
$2^{8},3^{5},5^{2},7^{2}$ and the appropriate prime $p\geq 11$. (Of course,
results for the largest exponents are valid for smaller ones.) More
specifically, in this section we will prove that $U(t)/U(t^{p})$ and
$V(t)/U(t^{p})$ modulo these prime powers give some polynomials or rational
functions. Throughout this section, for the formal power series $X(t)$ and
$Y(t)$, the symbol $X(t)\equiv Y(t)\pmod{p^{s}}$ means that all the power
series coefficients of their difference $X(t)-Y(t)$ are congruent to 0 modulo
$p^{s}$.

If the prime $p$ is greater than or equal to 5, the following proposition
corresponds to the special case of Lemma 2, but here we give a simpler
alternative proof using Lucas’ theorem.

######  Proposition 4.

For any prime $p$, we have

$\displaystyle U(t)$
$\displaystyle\equiv\left(\sum_{m=0}^{[p/6]}u_{m}t^{m}\right)U(t^{p})\pmod{p},$
$\displaystyle V(t)$
$\displaystyle\equiv\left(\sum_{m=0}^{[p/6]}(6m+1)u_{m}t^{m}\right)U(t^{p})\pmod{p}.$

Thus, in particular, $u_{r}\equiv 0\pmod{p}$ for $r\geq 1$ and
$p\in\\{2,3,5\\}$.

###### Proof.

We first recall Lucas’ theorem for the binomial coefficient modulo a prime
$p$; for some nonnegative integers $n,m,a$ and $b$ such that $0\leq a,b\leq
p-1$,

$\displaystyle\binom{np+a}{mp+b}\equiv\binom{n}{m}\binom{a}{b}\pmod{p}$ (36)

holds under the usual convention $\tbinom{0}{0}=1$, and $\tbinom{x}{y}=0$ if
$x<y$. For some generalizations of Lucas’ theorem, we refer to the extensive
historical survey [22] by Meštrović.

To calculate $u_{lp+m}$ for $l\geq 0$ and $0\leq m\leq p-1$, we classify by
the value of $m$. From Lucas’ theorem, it is easy to see that if $0\leq
m<p/6$, the following congruence holds:

$\displaystyle u_{lp+m}$
$\displaystyle=\binom{2(lp+m)}{lp+m}\binom{3(lp+m)}{lp+m}\binom{6(lp+m)}{3(lp+m)}$
$\displaystyle\equiv\binom{2l}{l}\binom{3l}{l}\binom{6l}{3l}\binom{2m}{m}\binom{3m}{m}\binom{6m}{3m}$
$\displaystyle\equiv u_{l}u_{m}\pmod{p}.$

Similarly, if $p/6\leq m<p/3$, since $0\leq 6m-p<p$ and $3m<p$, so

$\displaystyle\binom{6(lp+m)}{3(lp+m)}=\binom{(6l+1)p+(6m-p)}{3lp+3m}\equiv\binom{6l+1}{3l}\binom{6m-p}{3m}\pmod{p}.$

Now we assume that $m<p/3$, $(6m-p)-3m=3m-p<0$ and thus the binomial
coefficient $\tbinom{6m-p}{3m}$ vanishes. Also, if $p/3\leq m<p/2$ and
$p/2\leq m<p$, $\binom{3(lp+m)}{(lp+m)}$ and $\binom{2(lp+m)}{lp+m}$ vanish,
respectively. Therefore, we have

$\displaystyle
U(t)=\sum_{l=0}^{\infty}\sum_{m=0}^{p-1}u_{lp+m}t^{lp+m}\equiv\sum_{l=0}^{\infty}\sum_{m=0}^{[p/6]}u_{l}u_{m}t^{lp+m}=\left(\sum_{m=0}^{[p/6]}u_{m}t^{m}\right)U(t^{p})\pmod{p}.$

The claim for $V(t)$ is obtained from a similar calculation, noting that if
$0\leq m<p/6$, then $(6(lp+m)+1)u_{lp+m}\equiv(6m+1)u_{m}u_{l}\pmod{p}$. If
$p\in\\{2,3,5\\}$ then $U(t)\equiv U(t^{p})\equiv
U(t^{p^{2}})\equiv\dotsb\pmod{p}$ holds, we see that $U(t)\equiv 1\pmod{p}$
and hence $u_{r}\equiv 0\pmod{p}$ for $r\geq 1$. ∎

######  Remark 3.

For $p=2$, we can prove that $u_{r}\equiv 0\pmod{2^{3}}$. Since $E_{4}\in
1+240q\mathbb{Z}[\\![q]\\!]$, $E_{4}^{1/2}\in 1+120q\mathbb{Z}[\\![q]\\!]$ and
so $E_{4}^{1/2}\in 1+120j^{-1}\mathbb{Z}[\\![j^{-1}]\\!]$ holds. By comparing
(7) and (30), we have $U(t)\in 1+120t\mathbb{Z}[\\![t]\\!]$ and obtain the
desired result.

######  Lemma 2.

Let $p\geq 5$ be a prime number. Let

$\displaystyle
F(x)={}_{3}F_{2}\left(\frac{1}{6},\frac{1}{2},\frac{5}{6};1,1;x\right)=\sum_{m=0}^{\infty}B(m)x^{m},\quad
F_{s}(x)=\sum_{m=0}^{p^{s}-1}B(m)x^{m}$

and $f^{(n)}(t)=\tfrac{d^{n}}{dt^{n}}f(t)$ denotes the $n$-th derivative
$(n\geq 0)$. Then the following congruence formulas hold for any
$s\in\mathbb{Z}_{\geq 0}$:

$\displaystyle\frac{F(x)}{F(x^{p})}\equiv\frac{F_{s+1}(x)}{F_{s}(x^{p})}\pmod{p^{s+1}},\quad\frac{F^{(n)}(x)}{F(x)}\equiv\frac{F^{(n)}_{s+1}(x)}{F_{s+1}(x)}\pmod{p^{s+1}}.$

###### Proof.

These assertions correspond to the special cases of Lemma 3.4 (i) and (ii) in
[7, p. 45], respectively. The lemma requires the following three assumptions:

  1. a)

$B(0)=1$.

  2. b)

$B(n+mp^{s+1})/B([n/p]+mp^{s})\equiv B(n)/B([n/p])\pmod{p^{s+1}}$ for all
$n,m,s\in\mathbb{Z}_{\geq 0}$.

  3. c)

$B(n)/B([n/p])\in\mathbb{Z}_{p}$ for all $n\in\mathbb{Z}_{\geq 0}$.

Note that since we now consider $F(x)$ to be the formal power series, a domain
$\mathfrak{O}$ in the original assumption c) can simply be the ring of
$p$-adic integers $\mathbb{Z}_{p}$.

First, the assumption a) clearly holds. Following [7, p. 30], for a given
prime number $p$ and some $\theta\in\mathbb{Z}_{p}\cap\mathbb{Q}$, the symbol
$\theta^{\prime}$ denotes the unique element $\mathbb{Z}_{p}\cap\mathbb{Q}$
such that $p\theta^{\prime}-\theta\in[0,p-1]$. Put
$(\theta_{1},\theta_{2},\theta_{3})=(\tfrac{1}{6},\tfrac{1}{2},\tfrac{5}{6})$
and then

$\displaystyle(\theta_{1}^{\prime},\theta_{2}^{\prime},\theta_{3}^{\prime})=\begin{cases}(\tfrac{1}{6},\tfrac{1}{2},\tfrac{5}{6})&\text{
if }p\equiv 1\pmod{6}\\\ (\tfrac{5}{6},\tfrac{1}{2},\tfrac{1}{6})&\text{ if
}p\equiv 5\pmod{6}\end{cases}$

holds. Therefore, this corresponds to the case of $A(n)=B(n)$ in Corollary 2
of [7, p. 36], it can be seen that assumptions b) and c) hold. ∎

We rewrite the result of this lemma for $F(x)$ into the following statement
for $U(t)$ and $V(t)$ by changing the variables.

######  Proposition 5.

The following congruence formulas are valid.

$\displaystyle U(t)$ $\displaystyle\equiv(1+20t+10t^{2})U(t^{5})\pmod{5^{2}},$
(37) $\displaystyle V(t)$
$\displaystyle\equiv(1+15t+5t^{2})U(t^{5})\pmod{5^{2}},$ (38) $\displaystyle
U(t)$
$\displaystyle\equiv\frac{1+22t+7t^{2}+21t^{3}+t^{7}+36t^{8}}{1+t^{7}}\,U(t^{7})\pmod{7^{2}},$
(39) $\displaystyle V(t)$
$\displaystyle\equiv\frac{1+7t+42t^{2}+7t^{3}+43t^{7}}{1+t^{7}}\,U(t^{7})\pmod{7^{2}}.$
(40)

###### Proof.

Assuming that $p=5$ and $s=1$ in Lemma 2, the direct calculation gives

$\displaystyle{}F_{2}(x)\equiv 1+15x+15x^{2}+15x^{5}+15x^{10}\pmod{5^{2}},$
$\displaystyle{}F_{1}(x^{5})\equiv 1+15x^{5}+15x^{10}\pmod{5^{2}},$
$\displaystyle{}(1+10x^{5}+10x^{10})F_{1}(x^{5})\equiv 1\pmod{5^{2}},$

and so

$\displaystyle\frac{F_{2}(x)}{F_{1}(x^{5})}\equiv(1+10x^{5}+10x^{10})F_{2}(x)\equiv
1+15x+15x^{2}\pmod{5^{2}}.$

On the other hand, $u_{r}\equiv 0\pmod{5}$ for $r\geq 1$ from Proposition 4,
and $1728^{4r}\equiv 1\pmod{5}$ holds, thus we have

$\displaystyle
U(t^{5})-F(1728^{5}t^{5})=\sum_{r=1}^{\infty}u_{r}\cdot(1-1728^{4r})t^{5r}\equiv
0\pmod{5^{2}}.$

Hence, with the help of Lemma 2, we obtain the congruence formula (37) for
$U(t)$:

$\displaystyle\frac{U(t)}{U(t^{5})}\equiv\frac{F(1728t)}{F(1728^{5}t^{5})}\equiv\frac{F_{2}(1728t)}{F_{1}(1728^{5}t^{5})}\equiv
1+20t+10t^{2}\pmod{5^{2}}.$ (41)

Similarly, we have

$\displaystyle{}(1+10x+10x^{2}+10x^{5}+10x^{10})F_{2}(x)\equiv 1\pmod{5^{2}},$
$\displaystyle{}\frac{d}{dx}F_{2}(x)\equiv 15+5x\pmod{5^{2}},$

and so

$\displaystyle\frac{\tfrac{d}{dx}F(x)}{F(x)}\equiv\frac{\tfrac{d}{dx}F_{2}(x)}{F_{2}(x)}\equiv
15+5x\pmod{5^{2}}.$

Note that changing the variables $x=1728t$, since
$1728\tfrac{d}{dx}F(x)=\tfrac{d}{dt}U(t)$, we have

$\displaystyle\frac{\tfrac{d}{dt}U(t)}{U(t)}\equiv 1728(15+5\cdot 1728t)\equiv
20+20t\pmod{5^{2}}.$ (42)

By combining (41) and (42), we obtain the congruence formula (38) for $V(t)$:

$\displaystyle\frac{V(t)}{U(t^{5})}$
$\displaystyle=\frac{1}{U(t^{5})}\left(1+6t\frac{d}{dt}\right)U(t)=\left(1+6t\frac{\frac{d}{dt}U(t)}{U(t)}\right)\frac{U(t)}{U(t^{5})}$
$\displaystyle\equiv(1+6t(20+20t))(1+20t+10t^{2})\equiv
1+15t+5t^{2}\pmod{5^{2}}.$

Next, assuming that $p=7$ and $s=1$ in Lemma 2, the direct calculation gives

$\displaystyle{}F_{2}(x)\equiv 1+13x+7x^{2}+28x^{3}+27x^{7}+22x^{8}$
$\displaystyle{}\qquad\qquad+7x^{14}+42x^{15}+28x^{21}+21x^{22}\pmod{7^{2}},$
$\displaystyle{}F_{1}(x^{7})\equiv
1+13x^{7}+7x^{14}+28x^{21}\equiv(1+48x^{7})(1+14x^{7}+21x^{14})\pmod{7^{2}},$
$\displaystyle{}(1+35x^{7}+28x^{14})F_{1}(x^{7})\equiv 1+48x^{7}\pmod{7^{2}},$

and so

$\displaystyle\frac{F_{2}(x)}{F_{1}(x^{7})}$
$\displaystyle\equiv\frac{(1+35x^{7}+28x^{14})F_{2}(x)}{1+48x^{7}}$
$\displaystyle\equiv\frac{1+13x+7x^{2}+28x^{3}+13x^{7}+36x^{8}}{1+48x^{7}}\pmod{7^{2}}.$

Unlike the case of $p=5$, $U(t^{7})-F(1728^{7}t^{7})$ is not congruent $0$
modulo $7^{2}$, so we first calculate the following congruence relation.
Noting that the congruence $1728^{7}\equiv-1\pmod{7^{2}}$,

$\displaystyle\frac{F(-t)}{F(1728t)}$
$\displaystyle=\frac{F(1728^{7}t^{7})}{F(1728t)}\cdot\frac{F(-t)}{F(1728^{7}t^{7})}\equiv\left(\frac{F(1728t)}{F(1728^{7}t^{7})}\right)^{-1}\frac{F(-t)}{F(-t^{7})}$
$\displaystyle\equiv\left(\frac{F_{2}(1728t)}{F_{1}(1728^{7}t^{7})}\right)^{-1}\frac{F_{2}(-t)}{F_{1}(-t^{7})}\equiv\frac{F_{2}(-t)}{F_{2}(1728t)}\equiv\frac{1+48t^{2}}{1+35t+13t^{2}}\pmod{7^{2}}.$

Replacing $t$ with $t^{7}$ in the above equation, we have

$\displaystyle\frac{F(1728^{7}t^{7})}{U(t^{7})}\equiv\frac{F(-t^{7})}{F(1728t^{7})}\equiv\frac{1+48t^{14}}{1+35t^{7}+13t^{14}}\pmod{7^{2}}$

and hence

$\displaystyle\frac{U(t)}{U(t^{7})}$
$\displaystyle=\frac{F(1728^{7}t^{7})}{U(t^{7})}\cdot\frac{F(1728t)}{F(1728^{7}t^{7})}\equiv\frac{1+48t^{14}}{1+35t^{7}+13t^{14}}\cdot\frac{F_{2}(1728t)}{F_{1}(1728^{7}t^{7})}$
$\displaystyle\equiv\frac{1+22t+7t^{2}+21t^{3}+t^{7}+36t^{8}}{1+t^{7}}\pmod{7^{2}}.$

The calculation for $V(t)$ is the same as for the case of $p=5$. By
substituting

$\displaystyle\frac{\tfrac{d}{dt}U(t)}{U(t)}$ $\displaystyle\equiv
1728\frac{\left.\tfrac{d}{dx}F_{2}(x)\middle|_{x=1728t}\right.}{F_{2}(1728t)}$
$\displaystyle\equiv\frac{22+14t+14t^{2}+7t^{6}+15t^{7}+7t^{14}+21t^{21}}{F_{2}(1728t)}\pmod{7^{2}}$

into the corresponding part, we obtain

$\displaystyle\frac{V(t)}{U(t^{7})}$
$\displaystyle=\frac{1}{U(t^{7})}\left(1+6t\frac{d}{dt}\right)U(t)=\left(1+6t\frac{\frac{d}{dt}U(t)}{U(t)}\right)\frac{U(t)}{U(t^{7})}$
$\displaystyle\equiv\frac{1+7t+42t^{2}+7t^{3}+15t^{7}+7t^{14}+21t^{21}}{F_{2}(1728t)}\cdot\frac{U(t)}{U(t^{7})}$
$\displaystyle\equiv\frac{1+7t+42t^{2}+7t^{3}+43t^{7}}{1+t^{7}}\pmod{7^{2}}.$

This completes the proof of the proposition. ∎

Unfortunately, since $\tfrac{1}{2}\not\in\mathbb{Z}_{2}\cap\mathbb{Q}$ and
$\tfrac{1}{6},\tfrac{5}{6}\not\in\mathbb{Z}_{3}\cap\mathbb{Q}$, Lemma 2 cannot
be applied directly to $p=2$ and $3$. Therefore, we consider the following
formal power series to which Lemma 2 can be applied:

$\displaystyle F(z)$
$\displaystyle={}_{2}F_{1}\left(\frac{1}{6},\frac{5}{6};1;432z\right)=\sum_{m=0}^{\infty}A(m)z^{m},\quad
F_{s}(z)=\sum_{m=0}^{p^{s}-1}A(m)z^{m},$ (43) $\displaystyle A(m)$
$\displaystyle=A_{6}(m)=\frac{(6m)!}{(3m)!(2m)!m!}=\binom{3m}{m}\binom{6m}{3m}\quad(m\in\mathbb{Z}_{\geq
0}).$ (44)

######  Lemma 3.

Suppose that the sequence of integers $A(m)$ defined by (44) and let
$p\in\\{2,3\\}$. Then for all nonnegative integers $a,b,s$, one has

$\displaystyle\frac{A(a+bp^{s+1})}{A([a/p]+bp^{s})}\equiv\frac{A(a)}{A([a/p])}\pmod{p^{s+1}},\quad\frac{A(a)}{A([a/p])}\in\mathbb{Z}_{p}.$
(45)

Although it looks a slightly different, this lemma corresponds (partially) to
the case of $N=6$ of Lemmas 11 and 12 in [42] by Zudilin. We will not prove
the above lemma again here, but will explain below how to rewrite the
calculation in [42]. From the equation that appeared in the proof of Lemma 12
in [42], for all nonnegative integers $u,v,n,s$ such that $0\leq u<p^{s}$ and
$0\leq v<p$, we have

$\displaystyle\frac{A(v+up+np^{s+1})}{A(u+np^{s})}=\frac{A(v+up)}{A(u)}\left(1+O(p^{s+1})\right).$

By putting $n=n_{1}+n_{2},\,a=v+up+n_{1}p^{s+1}\in\mathbb{Z}_{\geq 0}$ and
$b=n_{2}\in\mathbb{Z}_{\geq 0}$ in this equation, we obtain

$\displaystyle\frac{A(a+bp^{s+1})}{A([a/p]+bp^{s})}\equiv\frac{A(v+up)}{A(u)}\pmod{p^{s+1}}.$

Since this congruence formula holds even when $b=0$, we obtain the first
assertion in (45). Next, by combining [42, Eq. (34)] and the definition of a
constant $k$ [42, Eq. (1)] depending on $N$, we have

$\displaystyle\mathrm{ord}_{p}\frac{A(v+mp)}{A(mp)}=\frac{kv}{p-1}\in\mathbb{Z}_{\geq
0}$

for all nonnegative integers $v,m$ such that $0\leq v<p$. The second assertion
in (45) can be obtained by rewriting $v+mp$ as $a$ in this equation.
Consequently, Lemma 3 guarantees assumptions b) and c) of Lemma 3.4 in [7, p.
45], and hence the following lemma holds.

######  Lemma 4.

Let $F(z)$ and $F_{s}(z)$ be the formal power series and polynomials defined
by (43) respectively. For $p\in\\{2,3\\}$, the following congruence formulas
hold for any $s\in\mathbb{Z}_{\geq 0}$:

$\displaystyle\frac{F(z)}{F(z^{p})}\equiv\frac{F_{s+1}(z)}{F_{s}(z^{p})}\pmod{p^{s+1}},\quad\frac{F^{(n)}(z)}{F(z)}\equiv\frac{F^{(n)}_{s+1}(z)}{F_{s+1}(z)}\pmod{p^{s+1}}.$

Here, $F^{(n)}(z)=\tfrac{d^{n}}{dz^{n}}F(z)$ denotes the $n$-th derivative
$(n\geq 0)$.

######  Proposition 6.

The following congruence formulas are valid.

$\displaystyle U(t)$
$\displaystyle\equiv(1+120t+96t^{2}+128t^{3})U(t^{2})\pmod{2^{8}},$ (46)
$\displaystyle V(t)$
$\displaystyle\equiv(1+72t+128t^{2}+128t^{3}+64t^{4}+128t^{8})U(t^{2})\pmod{2^{8}},$
(47)
$\displaystyle\begin{split}U(t)&\equiv(1+120t+54t^{2}+189t^{3}+135t^{4}+81t^{5}+162t^{6}+81t^{7}+162t^{10})\\\
&{}\quad\times U(t^{3})\pmod{3^{5}},\end{split}$ (48)
$\displaystyle\begin{split}V(t)&\equiv(1+111t+216t^{2}+162t^{3}+135t^{4}+81t^{5}+81t^{7}+162t^{9}+162t^{10})\\\
&{}\quad\times U(t^{3})\pmod{3^{5}}.\end{split}$ (49)

###### Proof.

By putting $t=z(1-432z)$ and using the quadratic transformation (22), we have

$\displaystyle
U(t)={}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;1728t\right)^{2}={}_{2}F_{1}\left(\frac{1}{6},\frac{5}{6};1;432z\right)^{2}=F(z)^{2}.$

and then

$\displaystyle F(z^{2})^{2}$
$\displaystyle={}_{2}F_{1}\left(\frac{1}{6},\frac{5}{6};1;432z^{2}\right)^{2}={}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;1728z^{2}(1-432z^{2})\right)^{2}$
$\displaystyle={}_{3}F_{2}\left(\frac{1}{6},\frac{1}{2},\frac{5}{6};1,1;1728z^{2}(1-432z^{2})\right)=\sum_{r=0}^{\infty}u_{r}z^{2r}(1-432z^{2})^{r}$
$\displaystyle\equiv\sum_{r=0}^{\infty}u_{r}z^{2r}(1+80rz^{2})\pmod{2^{8}},$
$\displaystyle U(t^{2})$
$\displaystyle\equiv{}_{3}F_{2}\left(\frac{1}{6},\frac{1}{2},\frac{5}{6};1,1;1728z^{2}(1+160z)\right)\equiv\sum_{r=0}^{\infty}u_{r}z^{2r}(1+160z)^{r}$
$\displaystyle\equiv\sum_{r=0}^{\infty}u_{r}z^{2r}(1+160rz)\pmod{2^{8}}.$

To explicitly calculate the difference between these formal power series, we
focus on the following congruence formula. By changing the variable $z^{2}=x$,
we have

$\displaystyle\frac{2z\tfrac{d}{dz}F(z^{2})}{F(z^{2})}$
$\displaystyle=4x\frac{\tfrac{d}{dx}F(x)}{F(x)}\equiv
4x\frac{\tfrac{d}{dx}F_{8}(x)}{F_{8}(x)}\equiv
240x+224x^{2}+192x^{4}+128x^{8}$ $\displaystyle\equiv
240z^{2}+224z^{4}+192z^{8}+128z^{16}\pmod{2^{8}}$

and then

$\displaystyle{}z\frac{d}{dz}F(z^{2})^{2}=F(z^{2})^{2}\cdot\frac{2z\tfrac{d}{dz}F(z^{2})}{F(z^{2})}$
$\displaystyle\equiv(240z^{2}+224z^{4}+192z^{8}+128z^{16})F(z^{2})^{2}\pmod{2^{8}}.$

On the other hand, from the power series expansion of $F(z^{2})^{2}$
calculated above,

$\displaystyle z\frac{d}{dz}F(z^{2})^{2}$ $\displaystyle\equiv
z\frac{d}{dz}\sum_{r=0}^{\infty}u_{r}z^{2r}(1+80rz^{2})$
$\displaystyle\equiv\sum_{r=0}^{\infty}2ru_{r}z^{2r}+160\sum_{r=0}^{\infty}r(r+1)u_{r}z^{2r+2}\pmod{2^{8}}.$

Since we know that $u_{r}\equiv 0\pmod{2^{3}}$ as mentioned in Remark 3,
$160u_{r}\equiv 0\pmod{2^{8}}$ holds for $r\geq 0$, that is, the second term
of the above equation is congruent to $0$ modulo $2^{8}$. Therefore,

$\displaystyle F(z^{2})^{2}-U(t^{2})$ $\displaystyle\equiv
40z(z-2)\sum_{r=0}^{\infty}2ru_{r}z^{r}$ $\displaystyle\equiv 40z(z-2)\cdot
z\frac{d}{dz}F(z^{2})^{2}\equiv 128z^{4}F(z^{2})^{2}\pmod{2^{8}}.$

Hence we have $(1-128z^{4})F(z^{2})^{2}\equiv U(t^{2})\pmod{2^{8}}$ and so
$F(z^{2})^{2}\equiv(1+128z^{4})U(t^{2})\pmod{2^{8}}$. By combining these
congruences, we obtain the congruence formula (46) for $U(t)$:

$\displaystyle\frac{U(t)}{U(t^{2})}$
$\displaystyle=\frac{F(z^{2})^{2}}{U(t^{2})}\cdot\left(\frac{F(z)}{F(z^{2})}\right)^{2}\equiv(1+128z^{4})\left(\frac{F_{8}(z)}{F_{7}(z^{2})}\right)^{2}$
$\displaystyle\equiv 1+120z+224z^{2}+128z^{3}\equiv
1+120t+96t^{2}+128t^{3}\pmod{2^{8}}.$

Note that the last equality uses the congruence relation $z\equiv
t(1+176t)\pmod{2^{8}}$. To obtain the formula (47), we calculate the change of
the variable $t=z(1-432z)$ with the following:

$\displaystyle\frac{\tfrac{d}{dt}U(t)}{U(t)}$
$\displaystyle\equiv\frac{(1+96z)\tfrac{d}{dz}F(z)^{2}}{F(z)^{2}}\equiv
2(1+96z)\frac{\tfrac{d}{dz}F(z)}{F(z)}\equiv
2(1+96z)\frac{\tfrac{d}{dz}F_{8}(z)}{F_{8}(z)}$ $\displaystyle\equiv
120+112z+128z^{2}+224z^{3}+192z^{7}+128z^{15}$ $\displaystyle\equiv
120+112t+128t^{2}+224t^{3}+192t^{7}+128t^{15}\pmod{2^{8}}.$

Hence we have

$\displaystyle\frac{V(t)}{U(t^{2})}$
$\displaystyle=\frac{1}{U(t^{2})}\left(1+6t\frac{d}{dt}\right)U(t)=\left(1+6t\frac{\frac{d}{dt}U(t)}{U(t)}\right)\frac{U(t)}{U(t^{2})}$
$\displaystyle\equiv 1+72t+128t^{2}+128t^{3}+64t^{4}+128t^{8}\pmod{2^{8}}.$

Since equations (48) and (49) can be obtained by the similar calculation with
$p=3$ and $s=4$ in Lemma 4, we omit the proof. ∎

From Propositions 5 and 6, we obtain the following corollary.

######  Corollary 1.

The following formal infinite product expressions hold.

$\displaystyle U(t)$
$\displaystyle\equiv\prod_{k=0}^{\infty}(1+120t^{2^{k}}+96t^{2^{k+1}}+128t^{3\cdot
2^{k}})\pmod{2^{8}},$ $\displaystyle U(t)$
$\displaystyle\equiv\prod_{k=0}^{\infty}\genfrac{(}{)}{0.0pt}{}{1+120t^{3^{k}}+54t^{2\cdot
3^{k}}+189t^{3^{k+1}}+135t^{4\cdot 3^{k}}}{+81t^{5\cdot 3^{k}}+162t^{6\cdot
3^{k}}+81t^{7\cdot 3^{k}}+162t^{10\cdot 3^{k}}}\pmod{3^{5}},$ $\displaystyle
U(t)$ $\displaystyle\equiv\prod_{k=0}^{\infty}(1+20t^{5^{k}}+10t^{2\cdot
5^{k}})\pmod{5^{2}},$ $\displaystyle U(t)$
$\displaystyle\equiv\prod_{k=0}^{\infty}\frac{(1+22t^{7^{k}}+7t^{2\cdot
7^{k}}+21t^{3\cdot 7^{k}}+t^{7^{k+1}}+36t^{8\cdot
7^{k}})}{(1+t^{7^{k+1}})}\pmod{7^{2}}.$

######  Question 1.

Recall that $U(j(\tau)^{-1})=E_{4}(\tau)^{1/2}$ and
$V(j(\tau)^{-1})=E_{2}(\tau)E_{4}(\tau)^{3/2}E_{6}(\tau)^{-1}$. What are the
counterparts of the above congruence formulas in the (quasi) modular form
side?

We still cannot answer this vague question, but we can prove the following
assertion as a closely related result. (Recall that
$U(t)=\mathcal{F}_{1}(t)^{2}$.)

######  Proposition 7.

Let $\mu(n)$ be the Möbius function and $(\cdot,\cdot)$ be the Atkin inner
product defined as

$\displaystyle(f,g)=\text{constant term of $fgE_{2}$ as a Laurent series in
$q$.}\quad(f(\tau),g(\tau)\in\mathbb{C}[j(\tau)]).$

Then the formal power series $\mathcal{F}_{1}(t)$ can be expressed formally as
follows:

$\displaystyle\mathcal{F}_{1}(t)$
$\displaystyle={}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;1728t\right)=\prod_{n=1}^{\infty}(1-t^{n})^{-c(n)},\quad
c(n)=\frac{1}{12n}\sum_{d|n}\mu(\tfrac{n}{d})\,(j^{d},1),$ (50)
$\displaystyle\mathcal{F}_{1}(t)$
$\displaystyle=\exp\left(\sum_{m=1}^{\infty}\frac{1}{12}(j^{m},1)\,\frac{t^{m}}{m}\right).$
(51)

Here are some examples of exponents $c(n)$.

$\\{c(n)\\}_{n\geq
1}=\\{60,37950,36139180,40792523310,50608476466548,\cdots\\}.$

###### Proof.

As already mentioned in [21, §5], the moment-generating function of the Atkin
inner product is given by

$\displaystyle\frac{E_{2}(\tau)E_{4}(\tau)}{j(\tau)E_{6}(\tau)}=\sum_{m=0}^{\infty}\frac{(j^{m},1)}{j(\tau)^{m+1}}=\frac{1}{j(\tau)}+\frac{720}{j(\tau)^{2}}+\frac{911520}{j(\tau)^{3}}+\frac{1301011200}{j(\tau)^{4}}+\cdots.$
(52)

By transforming the left-hand side of the above equation using the
hypergeometric expressions of the Eisenstein series in Proposition 1, we have

$\displaystyle\sum_{m=0}^{\infty}(j^{m},1)\,t^{m}$
$\displaystyle=(1-1728t)^{-1/2}\cdot\frac{{}_{2}F_{1}\left(-\frac{1}{12},\frac{7}{12};1;1728t\right)}{{}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;1728t\right)}=\frac{{}_{2}F_{1}\left(\frac{13}{12},\frac{5}{12};1;1728t\right)}{{}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;1728t\right)}$
$\displaystyle=\left\\{\left(1+12t\frac{d}{dt}\right){}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;1728t\right)\right\\}{}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;1728t\right)^{-1}$
$\displaystyle=1+12t\frac{d}{dt}\log{}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;1728t\right)$
$\displaystyle=1+12t\frac{d}{dt}\log\prod_{n=1}^{\infty}(1-t^{n})^{-c(n)}$
$\displaystyle=1+12\sum_{n=1}^{\infty}c(n)\frac{nt^{n}}{1-t^{n}}.$

Therefore, we have $(j^{m},1)=12\sum_{d|m}d\cdot c(d)$ by comparing the
coefficients of $t^{m}$ on both sides of the above equation, and obtain the
desired expression (50) of $c(n)$ by using the Möbius inversion formula.
Equation (51) is obtained by dividing both sides of the equation

$\displaystyle
t\frac{d}{dt}\log{}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;1728t\right)=\sum_{m=1}^{\infty}\frac{1}{12}(j^{m},1)\,t^{m}$

by $t$ and then integrating with respect to $t$. ∎

######  Remark 4.

Equation (52), and therefore equation (51), is equivalent to:

$\displaystyle\Delta(\tau)=j(\tau)^{-1}\exp\left(\sum_{m=1}^{\infty}(j^{m},1)\,\frac{j(\tau)^{-m}}{m}\right).$

##  4\. Proof of the main theorem

First, we summarize the normalizing factors $N_{m,a}$ and the polynomials
$A_{m,a}(X)$ and $B_{m,a}(X)$ that appear in the main theorem. From the
definition of the normalizing factor in Section 1, we see that its prime
factors do not exceed the weight of the corresponding normalized extremal
quasimodular form.

$\bullet$ Case of weight $w\in\mathcal{S}_{0}=\\{12,24\\}$.
Prime factorization of normalizing factors.

$\displaystyle N_{1,0}$ $\displaystyle=2^{5}\cdot 3^{3}\cdot 5\cdot 7\cdot
11,$ $\displaystyle N_{2,0}$ $\displaystyle=2^{6}\cdot 3^{3}\cdot 5\cdot
7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23.$

Atkin-like polynomials.

$\displaystyle A_{1,0}(X)=1,\;A_{2,0}(X)=X-824.$

Adjoint polynomials.

$\displaystyle B_{1,0}(X)=X-1008,\;B_{2,0}(X)=X^{2}-1832X+497952.$

$\bullet$ Case of weight $w\in\mathcal{S}_{2}=\\{2,14,38\\}$.
Prime factorization of normalizing factors.

$\displaystyle N_{0,2}$ $\displaystyle=1,$ $\displaystyle N_{1,2}$
$\displaystyle=2^{5}\cdot 3^{3}\cdot 5\cdot 7\cdot 13,$ $\displaystyle
5N_{3,2}$ $\displaystyle=2^{7}\cdot 3^{4}\cdot 5^{2}\cdot 7\cdot 11\cdot
13\cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 37.$

(Original) Atkin polynomials.

$\displaystyle A_{0,2}(X)$ $\displaystyle=1,\;A_{1,2}(X)=X-720,$
$\displaystyle A_{3,2}(X)$
$\displaystyle=X^{3}-\frac{12576}{5}X^{2}+1526958X-107765856.$

Adjoint polynomials.

$\displaystyle
B_{0,2}(X)=0,\;B_{1,2}(X)=1,\;B_{3,2}(X)=X^{2}-\frac{8976}{5}X+627534.$

$\bullet$ Case of weight $w\in\mathcal{S}_{6}=\\{6,18,30,54,114\\}$.
Prime factorization of normalizing factors.

$\displaystyle N_{0,6}$ $\displaystyle=2^{4}\cdot 3^{2}\cdot 5,$
$\displaystyle N_{1,6}$ $\displaystyle=2^{6}\cdot 3^{3}\cdot 5\cdot 7\cdot
11\cdot 13\cdot 17,$ $\displaystyle N_{2,6}$ $\displaystyle=2^{6}\cdot
3^{4}\cdot 5^{2}\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23\cdot 29,$
$\displaystyle 2N_{4,6}$ $\displaystyle=2^{7}\cdot 3^{4}\cdot 5^{2}\cdot
7^{2}\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 37\cdot
41\cdot 43\cdot 47\cdot 53,$ $\displaystyle 51N_{9,6}$
$\displaystyle=2^{8}\cdot 3^{5}\cdot 5^{2}\cdot 7^{2}\cdot 11\cdot 13\cdot
17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 37\cdot 41\cdot 43\cdot 47\cdot
53\cdot 59$ $\displaystyle{}\quad\cdot 61\cdot 67\cdot 71\cdot 73\cdot 79\cdot
83\cdot 89\cdot 97\cdot 101\cdot 103\cdot 107\cdot 109\cdot 113.$

Atkin-like polynomials.

$\displaystyle A_{0,6}(X)$ $\displaystyle=1,\quad A_{1,6}(X)=X-1266,\quad
A_{2,6}(X)=X^{2}-2115X+870630,$ $\displaystyle A_{4,6}(X)$
$\displaystyle=X^{4}-\frac{7671}{2}X^{3}+4871313X^{2}-2260803660X+273189722310,$
$\displaystyle A_{9,6}(X)$
$\displaystyle=X^{9}-\frac{24454}{3}X^{8}+\frac{474979296}{17}X^{7}-\frac{888804457205}{17}X^{6}$
$\displaystyle{}\quad+58002865348421X^{5}-38759471954111394X^{4}$
$\displaystyle{}\quad+15135088185868167792X^{3}-3173598010686486090312X^{2}$
$\displaystyle{}\quad+297473555337690122052390X-7840346480159903987708940.$

Adjoint polynomials.

$\displaystyle B_{0,6}(X)$ $\displaystyle=1,\quad B_{1,6}(X)=X-546,\quad
B_{2,6}(X)=X^{2}-1395X+259350,$ $\displaystyle B_{4,6}(X)$
$\displaystyle=X^{4}-\frac{6231}{2}X^{3}+3021273X^{2}-948582060X+53723885670,$
$\displaystyle B_{9,6}(X)$
$\displaystyle=X^{9}-\frac{22294}{3}X^{8}+\frac{390702816}{17}X^{7}-\frac{651013930805}{17}X^{6}$
$\displaystyle{}\quad+37180279576181X^{5}-21228003877921074X^{4}$
$\displaystyle{}\quad+6835398004395374832X^{3}-1114698418843177975752X^{2}$
$\displaystyle{}\quad+72322444486635699257190X-919318930586739576036780.$

$\bullet$ Case of weight $w\in\mathcal{S}_{8}=\\{8,20,32,68,80\\}$.
Prime factorization of normalizing factors.

$\displaystyle N_{0,8}$ $\displaystyle=2^{4}\cdot 3^{2}\cdot 7,$
$\displaystyle N_{1,8}$ $\displaystyle=2^{6}\cdot 3^{3}\cdot 5\cdot 7\cdot
11\cdot 13\cdot 19,$ $\displaystyle N_{2,8}$ $\displaystyle=2^{6}\cdot
3^{4}\cdot 5^{2}\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23\cdot 31,$
$\displaystyle 5N_{5,8}$ $\displaystyle=2^{8}\cdot 3^{4}\cdot 5^{2}\cdot
7^{2}\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 37\cdot
41\cdot 43\cdot 47\cdot 53\cdot 59$ $\displaystyle{}\quad\cdot 61\cdot 67,$
$\displaystyle 11N_{6,8}$ $\displaystyle=2^{8}\cdot 3^{4}\cdot 5^{2}\cdot
7^{2}\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 37\cdot
41\cdot 43\cdot 47\cdot 53\cdot 59$ $\displaystyle{}\quad\cdot 61\cdot 67\cdot
71\cdot 73\cdot 79.$

Atkin-like polynomials.

$\displaystyle A_{0,8}(X)$ $\displaystyle=1,\quad A_{1,8}(X)=X-330,\quad
A_{2,8}(X)=X^{2}-1215X+129030,$ $\displaystyle A_{5,8}(X)$
$\displaystyle=X^{5}-\frac{19098}{5}X^{4}+\frac{25015408}{5}X^{3}-\frac{12959037322}{5}X^{2}+441761976414X$
$\displaystyle{}\quad-9018997829292,$ $\displaystyle A_{6,8}(X)$
$\displaystyle=X^{6}-4685X^{5}+\frac{89349390}{11}X^{4}-6372443376X^{3}+2195718854056X^{2}$
$\displaystyle{}\quad-261120476348550X+3783879543834780.$

Adjoint polynomials.

$\displaystyle B_{0,8}(X)$ $\displaystyle=1,\quad B_{1,8}(X)=X-1338,\quad
B_{2,8}(X)=X^{2}-2223X+1021110,$ $\displaystyle B_{5,8}(X)$
$\displaystyle=X^{5}-\frac{24138}{5}X^{4}+\frac{42602992}{5}X^{3}-\frac{33192286666}{5}X^{2}+\frac{10734540754806}{5}X$
$\displaystyle{}\quad-202399435400844,$ $\displaystyle B_{6,8}(X)$
$\displaystyle=X^{6}-5693X^{5}+\frac{137637630}{11}X^{4}-\frac{146033508816}{11}X^{3}+6911247661864X^{2}$
$\displaystyle{}\quad-1568906774156358X+105994437115386300.$

###### Proof of Theorem 1.

We prove only the non-trivial and highest weight case
$w=114\in\mathcal{S}_{6}$. The remaining cases can be proved in a similar way.

Since $G_{114}^{(1)}(\tau)\in\mathbb{Z}[\\![q]\\!]\Leftrightarrow
P_{19}(t)\in\mathbb{Z}[\\![t]\\!]$ holds from (5) of Lemma 1, we transform the
formal power series $P_{19}(t)$ according to (34) as follows:

$\displaystyle
N_{9,6}\,t^{19}(1-1728t)^{-1/2}P_{19}(t)=\widetilde{A_{9,6}}(t)V(t)-\widetilde{B_{9,6}}(t)U(t).$

Since the polynomials $A_{9,6}(X)$ and $B_{9,6}(X)$ belong to
$\frac{1}{51}\mathbb{Z}[X]$, we multiply the both sides of the above equation
by $51$, so that the right-hand side is the power series with integral
coefficients. Using Proposition 4, we obtain the following congruence formula
for prime numbers $p\geq 11$:

$\displaystyle{}51\left(\widetilde{A_{9,6}}(t)V(t)-\widetilde{B_{9,6}}(t)U(t)\right)$
$\displaystyle\equiv\left\\{51\widetilde{A_{9,6}}(t)\sum_{m=0}^{[p/6]}(6m+1)u_{m}t^{m}-51\widetilde{B_{9,6}}(t)\sum_{m=0}^{[p/6]}u_{m}t^{m}\right\\}U(t^{p})\pmod{p}.$

By direct calculation using Mathematica, we can see that the polynomial part
of the right-hand side of the above equation is congruent to 0 modulo
$p\;(11\leq p\leq 113)$. To perform similar calculations for modulo
$2^{8},3^{5},5^{2}$, and $7^{2}$, we use Propositions 5 and 6. For example,
from (46) and (47), we have

$\displaystyle{}51\left(\widetilde{A_{9,6}}(t)V(t)-\widetilde{B_{9,6}}(t)U(t)\right)$
$\displaystyle\equiv\left\\{51\widetilde{A_{9,6}}(t)(1+72t+128t^{2}+128t^{3}+64t^{4}+128t^{8})\right.$
$\displaystyle{}\quad\left.-51\widetilde{B_{9,6}}(t)(1+120t+96t^{2}+128t^{3})\right\\}U(t^{2})\pmod{2^{8}}$

and then the polynomial part of the right-hand side is congruent to 0 modulo
$2^{8}$. The same assertion holds for the remaining cases. Combining the
results for each of these primes, we have the desired congruence

$\displaystyle{}51N_{9,6}\,t^{19}(1-1728t)^{-1/2}P_{19}(t)$
$\displaystyle=51\left(\widetilde{A_{9,6}}(t)V(t)-\widetilde{B_{9,6}}(t)U(t)\right)\equiv
0\pmod{51N_{9,6}}$

and hence $t^{19}(1-1728t)^{-1/2}P_{19}(t)\in\mathbb{Z}[\\![t]\\!]$, and so
$P_{19}(t)\in\mathbb{Z}[\\![t]\\!]$ holds from (1) in Lemma 1. ∎

######  Question 2.

The proof of Theorem 1 in this paper is a “hypergeometric” proof. Can this
theorem be proved using only the theory of modular forms?

##  5\. Other choice of the leading coefficients

Why did we choose the leading coefficient as 1 in Definition 1 of the
normalized extremal quasimodular form? Was it really a natural choice? Of
course, the set of weights $w$ of $c\,G_{w}^{(r)}$ with integral Fourier
coefficients changes if we choose a number $c$ other than 1 as the leading
coefficient. In this section we show that $c\,G_{w}^{(1)}$ has integral
Fourier coefficients if we choose a constant $c$ based on the hypergeometric
expressions of $G_{w}^{(1)}$ in Proposition 2.

The reason we often focus on (positive) integers is that we can expect them to
count “something”, such as the dimension or degree or order of some
mathematical objects, the number of curves or points with certain arithmetic
or geometric properties, and so on. Interestingly, Nebe proved the integrality
of the Fourier coefficients of $G_{14}^{(1)}$ by using the properties of the
automorphism group of the Leech lattice $\Lambda_{24}$ in the appendix of
[30]222The author speculates that Nebe probably uses $q$ to mean $e^{\pi
i\tau}$ in the proof of Theorem A.1. Hence the correct expression of
$f_{1,14}$ for $q=e^{2\pi i\tau}$ is given by
$A^{-1}\sum_{a=1}^{\infty}\tfrac{a}{2}|L_{a}|q^{a/2}$.:

$\displaystyle\theta_{\Lambda_{24}}$
$\displaystyle=\sum_{\lambda\in\Lambda_{24}}e^{\pi
i\tau\|\lambda\|^{2}}=\sum_{n=0}^{\infty}|\\{\lambda\in\Lambda_{24}\mid\|\lambda\|^{2}=n\\}|\,q^{n/2}$
$\displaystyle=E_{4}^{3}-720\Delta=E_{12}-\frac{65520}{691}\Delta$
$\displaystyle=1+196560q^{2}+16773120q^{3}+398034000q^{4}+4629381120q^{5}+O(q^{6}),$
$\displaystyle G_{14}^{(1)}$
$\displaystyle=\frac{E_{2}(E_{4}^{3}-720\Delta)-E_{4}^{2}E_{6}}{393120}=\frac{D(\theta_{\Lambda_{24}})}{393120}$
$\displaystyle=q^{2}+128q^{3}+4050q^{4}+58880q^{5}+525300q^{6}+O(q^{7}).$

Therefore, the number 393120 is twice the number of lattice vectors of squared
norm 4 in $\Lambda_{24}$, and $N_{1,2}$ is exactly equal to this number. It is
known that the Eisenstein series $E_{4}$ is the theta series of the
$E_{8}$-lattice. Since $G_{6}^{(1)}=D(E_{4})/240=D(\theta_{E_{8}})/240$, as
with $G_{14}^{(1)}$, the Fourier coefficients of $G_{6}^{(1)}$ are related to
the number of lattice vectors in the $E_{8}$-lattice. Note that although
$G_{6}^{(1)}$ and $G_{14}^{(1)}$ were obtained as derivatives of a certain
modular form, the only such $G_{w}^{(1)}$ that can be obtained this way are
for $w=6,8,10,14$. This fact is easily seen by comparing the dimension
formulas of $M_{w}$ and $QM_{w+2}^{(1)}$.

Inspired by these coincidences333The modular solutions of the Kaneko–Zagier
equation (13) with small weights are also closely related to the theta series
of the ADE-type root lattice. For more details, see [17, p. 158].
$\tfrac{1}{3}N_{0,6}\,G_{6}^{(1)}=D(\theta_{E_{8}})\in\mathbb{Z}[\\![q]\\!]$
and $N_{1,2}\,G_{14}^{(1)}=D(\theta_{\Lambda_{24}})\in\mathbb{Z}[\\![q]\\!]$,
although we still do not know what the Fourier coefficients of these forms
counting up, we have arrived at the following theorem.

######  Theorem 3.

Let $N_{m,a}$ be the normalizing factor defined by (27) and (28). Then, for
any $m\geq 0$, the Fourier coefficients of the following extremal quasimodular
forms are all integers:

$\displaystyle\frac{N_{m,0}}{24m}\,G_{12m}^{(1)},\;\frac{N_{m,2}}{24m}\,G_{12m+2}^{(1)},\;\frac{N_{m,6}}{12(2m+1)}\,G_{12m+6}^{(1)},\;\frac{N_{m,8}}{12(2m+1)}\,G_{12m+8}^{(1)}.$

Furthermore, since $G_{6n+4}^{(1)}=E_{4}G_{6n}^{(1)}$, the forms
$\tfrac{1}{24m}N_{m,0}\,G_{12m+4}^{(1)}$ and
$\tfrac{1}{12(2m+1)}N_{m,6}\,G_{12m+10}^{(1)}$ have the same properties.
(Note: Since the factors $N_{m,0}$ and $N_{m,2}$ are defined by the product of
binomial coefficients (27), we substitute $m=0$ after dividing by $m$ and
reducing.)

In the proof of this theorem, we use the following power series instead of $q$
as the local parameter at infinity, based on the method of Remark 2:

$\displaystyle{}\frac{z(q)}{432}=\frac{1}{864}\left\\{1-\left(1-\frac{1728}{j(q)}\right)^{1/2}\right\\}=\frac{1}{864}\left(1-E_{4}(q)^{-3/2}E_{6}(q)\right)$
(53)
$\displaystyle=q-312q^{2}+87084q^{3}-23067968q^{4}+5930898126q^{5}+O(q^{6}).$

###### Proof.

Throughout this proof, we consider $z$ to be
$z=\tfrac{1}{2}\\{1-(1-1728t)^{1/2}\\}\in 432t\mathbb{Z}[\\![t]\\!]$ and
$t=j^{-1}$. From Proposition 2, the normalized extremal quasimodular forms
$G_{w}^{(1)}$ are the product of the power of
$\mathcal{F}_{1}(t)={}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;1728t\right)$
and a hypergeometric series. Furthermore, since $\mathcal{F}_{1}(t)\in
1+t\mathbb{Z}[\\![t]\\!]$ holds from Remark 2, we have
$\mathcal{F}_{1}(t)^{w-1}\in 1+t\mathbb{Z}[\\![t]\\!]$. Therefore, as already
mentioned in Lemma 1 (3), it suffices to show that
$C_{w}\mathcal{F}_{1}(t)^{1-w}G_{w}^{(1)}\in\mathbb{Z}[\\![t]\\!]$ to prove
the theorem, where the constant $C_{w}$ is an appropriate normalization
constant.

First we calculate the case of $w=12m$. From the quadratic transformation
formula (22) we have

$\displaystyle{}\frac{N_{m,0}}{24m}\,{}_{2}F_{1}\left(m+\frac{1}{12},m+\frac{5}{12};2m+1;1728t\right)$
$\displaystyle=\frac{(12m)!}{(2m)!(4m)!(6m)!}\,{}_{2}F_{1}\left(2m+\frac{1}{6},2m+\frac{5}{6};2m+1;z\right)$
$\displaystyle=\frac{(12m)!}{(2m)!(4m)!(6m)!}\cdot\frac{(1)_{2m}}{(\tfrac{1}{6})_{2m}(\tfrac{5}{6})_{2m}}\cdot\frac{d^{2m}}{dz^{2m}}{}_{2}F_{1}\left(\frac{1}{6},\frac{5}{6};1;z\right)$
$\displaystyle=\sum_{n=0}^{\infty}\binom{2m+n}{n}\binom{6m+3n}{2m+n}\binom{12m+6n}{6m+3n}\left(\frac{z}{432}\right)^{n}\in\mathbb{Z}[\\![t]\\!].$
(54)

It is convenient to use the following formulas to calculate the last equality:

$\displaystyle
432^{k}\left(\frac{1}{6}\right)_{k}\left(\frac{5}{6}\right)_{k}=\frac{\Gamma(k)\Gamma(6k)}{\Gamma(2k)\Gamma(3k)}=\frac{\Gamma(k+1)\Gamma(6k+1)}{\Gamma(2k+1)\Gamma(3k+1)},\quad(\alpha+k)_{\ell}=\frac{(\alpha)_{k+\ell}}{(\alpha)_{k}}.$

A similar calculation for $w=12m+6$ yields

$\displaystyle{}\frac{N_{m,6}}{12(2m+1)}\,{}_{2}F_{1}\left(m+\frac{7}{12},m+\frac{11}{12};2m+2;1728t\right)$
$\displaystyle=\sum_{n=0}^{\infty}\binom{2m+n+1}{n}\binom{6m+3n+3}{2m+n+1}\binom{12m+6n+6}{6m+3n+3}\left(\frac{z}{432}\right)^{n}\in\mathbb{Z}[\\![t]\\!].$
(55)

To prove the assertions about $G_{12m+2}^{(1)}$ and $G_{12m+8}^{(1)}$, we
recall $\tfrac{1}{1-2n}\tbinom{2n}{n}\in\mathbb{Z}$. Then, we can see that the
coefficient of $(z/432)^{n}$ of the following formal power series is an
integer:

$\displaystyle{}_{2}F_{1}\left(-\frac{1}{6},\frac{7}{6};1;z\right)=\sum_{n=0}^{\infty}(6n+1)\binom{3n}{n}\cdot\frac{1}{1-6n}\binom{6n}{3n}\cdot\left(\frac{z}{432}\right)^{n}\in\mathbb{Z}[\\![t]\\!].$

Hence we have

$\displaystyle{}\frac{N_{m,2}}{24m}\,{}_{2}F_{1}\left(m-\frac{1}{12},m+\frac{7}{12};2m+1;1728t\right)$
$\displaystyle=\frac{N_{m,2}}{24m}{}_{2}F_{1}\left(2m-\frac{1}{6},2m+\frac{7}{6};2m+1;z\right)$
$\displaystyle=\frac{N_{m,2}}{24m}\cdot\frac{(1)_{2m}}{(-\tfrac{1}{6})_{2m}(\tfrac{7}{6})_{2m}}\cdot\frac{d^{2m}}{dz^{2m}}{}_{2}F_{1}\left(-\frac{1}{6},\frac{7}{6};1;z\right)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{12m+6n+1}{12m+6n-1}\binom{2m+n}{n}\binom{6m+3n}{2m+n}\binom{12m+6n}{6m+3n}\left(\frac{z}{432}\right)^{n}\in\mathbb{Z}[\\![t]\\!]$

and also have

$\displaystyle{}\frac{N_{m,8}}{12(2m+1)}\,{}_{2}F_{1}\left(m+\frac{5}{12},m+\frac{13}{12};2m+2;1728t\right)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{12m+6n+7}{12m+6n+5}\binom{2m+n+1}{n}\binom{6m+3n+3}{2m+n+1}\binom{12m+6n+6}{6m+3n+3}\left(\frac{z}{432}\right)^{n}\in\mathbb{Z}[\\![t]\\!].$

This completes the proof of Theorem 3. ∎

Considering the divisors of the normalizing factors $N_{m,a}$, Theorem 3
implies the following claim.

######  Corollary 2.

The denominators of the Fourier coefficients of the normalized extremal
quasimodular forms $G_{w}^{(1)}$ are only divisible by prime numbers $<w$.

This claim was originally stated as
$G_{w}^{(r)}\in\mathbb{Z}\bigl{[}\tfrac{1}{p}:p<w\bigr{]}[\\![q]\\!]\;(1\leq
r\leq 4)$ in [19]. When $(w,r)=(6n,1)$ and $(w,r)=(6n+2,1),(6n+4,1)$, they are
proved in [30, Thm. 3.3] and [23, Thm. 1.8, Thm. 1.9], respectively, and the
original claim is proved in [10]. Note that our proof is mainly based on
hypergeometric expressions of $G_{w}^{(1)}$, which is different from their
proof.

According to Theorem 3, there exists a constant $c_{w}$ such that
$c_{w}\,G_{w}^{(1)}\in\mathbb{Z}[\\![q]\\!]$. However, this constant is
probably too large, and integrality holds for smaller numbers. For example,
although
$\tfrac{1}{48}N_{2,2}\,G_{26}^{(1)}=1454953500\,G_{26}^{(1)}\in\mathbb{Z}[\\![q]\\!]$
holds, we can show that the stronger claim
$5G_{26}^{(1)}\in\mathbb{Z}[\\![q]\\!]$ holds in the similar way as in the
proof of Theorem 1. As a more general setting, we define the number
$c_{w}^{(r)}$ as
$c_{w}^{(r)}:=\min\\{\,c\,|\,c\,G_{w}^{(r)}\in\mathbb{Z}[\\![q]\\!]\\}$. How
do such numbers depend on weight and depth?

######  Remark 5.

In the study of the denominator of Atkin polynomials [21, §9], the formal
power series $\Phi_{n}(t)$ and $\Psi_{n}(t)$, which are defined as follows,
play a central role:

$\displaystyle\Phi_{n+1}(t)=\Phi_{n}(t)-\lambda_{n}^{-}\,t\,\Phi_{n-1}(t),\quad\Psi_{n+1}(t)=\Psi_{n}(t)-\lambda_{n}^{+}\,t\,\Psi_{n-1}(t)\quad(n\geq
1),$

where $\lambda_{1}^{-}=84,\;\lambda_{1}^{+}=-60$ and for $n>1$,

$\displaystyle\lambda_{n}^{\mp}=12\left(6\mp\dfrac{(-1)^{n}}{n-1}\right)\left(6\mp\dfrac{(-1)^{n}}{n}\right).$

The initial power series are given by the following.

$\displaystyle\Phi_{0}(t)$
$\displaystyle={}_{2}F_{1}\left(\tfrac{1}{12},\tfrac{5}{12};1;1728t\right)^{2},$
$\displaystyle\Phi_{1}(t)$
$\displaystyle=84t\,{}_{2}F_{1}\left(\tfrac{1}{12},\tfrac{5}{12};1;1728t\right){}_{2}F_{1}\left(\tfrac{5}{12},\tfrac{13}{12};2;1728t\right),$
$\displaystyle\Psi_{0}(t)$
$\displaystyle={}_{2}F_{1}\left(\tfrac{1}{12},\tfrac{5}{12};1;1728t\right){}_{2}F_{1}\left(-\tfrac{1}{12},\tfrac{7}{12};1;1728t\right),$
$\displaystyle\Psi_{1}(t)$
$\displaystyle=-60t\,{}_{2}F_{1}\left(\tfrac{1}{12},\tfrac{5}{12};1;1728t\right){}_{2}F_{1}\left(\tfrac{7}{12},\tfrac{11}{12};2;1728t\right).$

Although the definition for the power series $\Psi_{n}(t)$ is not given in
[21], the details are omitted here because it can be derived from the same
consideration as $\Phi_{n}(t)$. In fact, these power series and the power
series $P_{n}(t)$ and $Q_{n}(t)$ defined by (20) and (21) have the following
correspondences:

$\displaystyle\Phi_{2m}(t)$
$\displaystyle=\tfrac{1}{12}N_{m,0}\,t^{2m}P_{2m}(t),\quad\Phi_{2m+1}(t)=\tfrac{1}{12}N_{m,8}\,t^{2m+1}Q_{2m+1}(t),$
$\displaystyle\Psi_{2m}(t)$
$\displaystyle=-\tfrac{1}{12}N_{m,2}\,t^{2m}Q_{2m}(t),\quad\Psi_{2m+1}(t)=-\tfrac{1}{12}N_{m,6}\,t^{2m+1}P_{2m+1}(t).$

These correspondences can be proved by using a contiguous relation of a
certain hypergeometric series. In [21, p. 120], it is stated without proof
that “the power series $\Phi_{n}(t)$ has integral coefficients and is
divisible by $t^{n}$ for all $n$”, which is essentially equivalent to a weaker
version of our Theorem 3. However, it should be emphasized that essentially
the equivalent forms appeared in [21] before the introduction of the
normalized extremal quasimodular forms in [19].

##  6\. Further directions

Up to the previous section we have discussed the extremal quasimodular forms
of depth $1$ on $\Gamma=SL_{2}(\mathbb{Z})$. It is therefore natural to
consider a generalization that changes these factors. The case of depth $2$ on
$\Gamma$ has already been considered in [19, 10, 15], but we note that in
these papers there is no perspective of using a generalized hypergeometric
series. Therefore, we focus mainly on these hypergeometric aspects and present
in this section some complementary results of previous studies without
detailed proofs.

To describe the case of depth $\geq 2$, we introduce the Rankin–Cohen brackets
([41, §5.2], see also [40, 4]), which is defined for integers $k,\ell,n\geq 0$
and functions $f,g$ on $\mathfrak{H}$ by

$\displaystyle[f,g]_{n}^{(k,\ell)}:=\sum_{i=0}^{n}(-1)^{i}\binom{n+k-1}{n-i}\binom{n+\ell-1}{i}D^{i}(f)D^{n-i}(g).$

We then define the $(r+1)$-th order differential operator $\theta_{k}^{(r)}$
as follows.

$\displaystyle\theta_{k}^{(r)}(f)$
$\displaystyle:=D^{r+1}(f)-\frac{k+r}{12}[E_{2},f]_{r}^{(2,k)}$
$\displaystyle=D^{r+1}(f)-\frac{k+r}{12}\sum_{i=0}^{r}(-1)^{i}\binom{r+1}{i+1}\binom{k+r-1}{i}D^{i}(E_{2})D^{r-i}(f).$

In particular, when $r=0,1$, this operator gives the Serre derivative
$\partial_{k}$ and the differential operator $L_{k+1}$ defined by (14),
respectively. Also, by specializing Proposition 3.3 in [19], we can see that
$f\in QM_{k}^{(r)}\Rightarrow\theta_{k-r}^{(r)}(f)\in QM_{k+2(r+1)}^{(r)}$.

The following identity holds for the integer $r\in\\{2,3,4\\}$:

$\displaystyle\theta_{k}^{(r)}=\frac{(k+r-1)(k+2r)}{2(r-1)(k+r)}\theta_{k+2}^{(r-1)}\circ\partial_{k}-\frac{k(k+r+1)}{2(r-1)(k+r)}\partial_{k+2r}\circ\theta_{k}^{(r-1)}.$
(56)

The proof of this identity by direct calculation is long and tedious, but by
using (56) repeatedly, we can easily rewrite the differential operator
$\theta_{k}^{(r)}$ into a form using the Serre derivative $\partial_{k}$ for
$r\in\\{2,3,4\\}$. As will be described later, some extremal quasimodular
forms are annihilated by the differential operator $\theta_{k}^{(r)}$.
Therefore, such rewriting is useful to derive an inductive structure of
extremal quasimodular forms according to Grabner’s method. Incidentally, the
identity (56) does not seem to hold for $r\geq 5$, for example, we have the
following identities for $r=5$ and $6$:

$\displaystyle{}\theta_{k}^{(5)}-\left\\{\frac{(k+4)(k+10)}{8(k+5)}\theta_{k+2}^{(4)}\circ\partial_{k}-\frac{k(k+6)}{8(k+5)}\partial_{k+10}\circ\theta_{k}^{(4)}\right\\}$
$\displaystyle=\frac{k(k+4)(k+6)(k+10)(k^{2}+10k+36)}{1440}\Delta,$
$\displaystyle{}\theta_{k}^{(6)}-\left\\{\frac{(k+5)(k+12)}{10(k+6)}\theta_{k+2}^{(5)}\circ\partial_{k}-\frac{k(k+7)}{10(k+6)}\partial_{k+12}\circ\theta_{k}^{(5)}\right\\}$
$\displaystyle=\frac{k(k+5)(k+7)(k+12)(k^{2}+12k+47)}{300}\Delta\partial_{k}.$

In [19] Kaneko and Koike showed that
$\theta_{w-1}^{(1)}(G_{w}^{(1)})=0\;(w\equiv 0\pmod{6})$ (Recall the
Kaneko–Zagier equation (13).) and $\theta_{w-2}^{(2)}(G_{w}^{(2)})=0\;(w\equiv
0\pmod{4})$, and conjectured that $\theta_{w-3}^{(3)}(G_{w}^{(3)})=0\;(w\equiv
0\pmod{6})$ and $\theta_{w-4}^{(4)}(G_{w}^{(4)})=0\;(w\equiv 0\pmod{12})$.
Hence, for $r\in\\{1,2,3,4\\}$, we also call the differential equation
$\theta_{w-r}^{(r)}(f)=0$ the Kaneko–Zagier equation. Recently, Grabner [10]
gave an affirmative answer for their conjecture, more precisely, he gave a
concrete form of the differential equation satisfied by the balanced
quasimodular forms. By specializing his results, we can find the concrete form
of the differential equation satisfied by the normalized extremal quasimodular
forms $G_{w}^{(r)}$ for any even integer $w\geq 0$ and $r\in\\{1,2,3\\}$ and
$G_{12n}^{(4)}$. Of course, for $w\equiv 0\pmod{12}$ and
$a\in\\{2,4,6,8,10\\}$, there is a fifth-order differential operator
$L_{w,a}^{(4)}$ such that $L_{w,a}^{(4)}(G_{w+a}^{(4)})=0$. The specific form
of $L_{w,a}^{(4)}$ is not described here, but is of the form shown below:

$\displaystyle L_{w,a}^{(4)}=f_{a}\,\partial_{w+a-4}^{5}+(\text{lower-order
terms on $\partial_{*}$})\in M_{*}(\Gamma)[\partial_{*}],$

where $(a,f_{a})=(2,E_{10}),(4,E_{8}),(6,E_{6}),(8,E_{4}),(10,E_{14})$.

###  6.1. Case of depth $r\geq 2$

Depth 2. The Kaneko–Zagier equation $\theta_{w-2}^{(2)}(f)=0$ is transformed
into

$\displaystyle{}z^{2}(1-z)\frac{d^{3}g}{dz^{3}}+z\left(-\frac{w-12}{4}+\frac{w-18}{4}z\right)\frac{d^{2}g}{dz^{2}}$
$\displaystyle{}\quad+\left(-\frac{w-4}{4}-\frac{3w^{2}-72w+452}{144}z\right)\frac{dg}{dz}+\frac{(w-2)(w-6)(w-10)}{1728}g=0$

or equivalently

$\displaystyle\left\\{\Theta^{2}\left(\Theta-\frac{w-4}{4}-1\right)-t\left(\Theta-\frac{w-2}{12}\right)\left(\Theta-\frac{w-6}{12}\right)\left(\Theta-\frac{w-10}{12}\right)\right\\}g=0,$

where $g(\tau)=E_{4}(\tau)^{-(w-2)/4}f(\tau)$ and $z=1728/j(\tau)$. Since this
differential equation is a hypergeometric differential equation, a similar
statement as in Proposition 2 holds for depth $2$.

######  Proposition 8.

The normalized extremal quasimodular forms of even weight and depth $2$ on
$\Gamma$ have the following hypergeometric expressions.

$\displaystyle\begin{split}G_{4n}^{(2)}(\tau)&=j(\tau)^{-n}E_{4}(\tau)^{(2n-1)/2}\\\
&{}\quad\times{}_{3}F_{2}\left(\frac{4n+1}{6},\frac{4n+3}{6},\frac{4n+5}{6};n+1,n+1;\frac{1728}{j(\tau)}\right),\end{split}$
(57)
$\displaystyle\begin{split}G_{4n+2}^{(2)}(\tau)&=j(\tau)^{-n}E_{4}(\tau)^{(2n-3)/2}E_{6}(\tau)\\\
&{}\quad\times{}_{3}F_{2}\left(\frac{4n+3}{6},\frac{4n+5}{6},\frac{4n+7}{6};n+1,n+1;\frac{1728}{j(\tau)}\right).\end{split}$
(58)

From Grabner’s result [10], the relation
$G_{4n+2}^{(2)}=\frac{6}{4n+1}\partial_{4n-2}(G_{4n}^{(2)})$ holds, so the
latter expression can be obtained from the former after a little calculation.
We omit the proof because it is similar to Proposition 2. It is known from
[15] that $\mathcal{E}_{2}=\\{4,8\\}$, and the Fourier coefficients of the
corresponding normalized extremal quasimodular forms are given below:

$\displaystyle G_{4}^{(2)}$
$\displaystyle=\frac{E_{4}-E_{2}^{2}}{288}=-\frac{D(E_{2})}{24}=\sum_{n=1}^{\infty}n\sigma_{1}(n)q^{n}$
$\displaystyle=q+6q^{2}+12q^{3}+28q^{4}+30q^{5}+72q^{6}+56q^{7}+120q^{8}+O(q^{9}),$
$\displaystyle G_{8}^{(2)}$
$\displaystyle=\frac{5E_{4}^{2}+2E_{2}E_{6}-7E_{2}^{2}E_{4}}{362880}=\frac{1}{30}\sum_{n=2}^{\infty}n(\sigma_{5}(n)-n\sigma_{3}(n))q^{n}$
$\displaystyle=q^{2}+16q^{3}+102q^{4}+416q^{5}+1308q^{6}+3360q^{7}+7772q^{8}+O(q^{9}).$

The positivity of the Fourier coefficients of $G_{8}^{(2)}$ can also be seen
from the fact that $D(G_{8}^{(2)})=2G_{6}^{(1)}G_{4}^{(2)}$. In these cases,
it is easier to show the integrality by calculating the Fourier coefficients
directly, but here we introduce a proof using a hypergeometric series as in
the case of depth 1. We define

$\displaystyle
R_{n}(t)={}_{3}F_{2}\left(\frac{4n+1}{6},\frac{4n+3}{6},\frac{4n+5}{6};n+1,n+1;1728t\right),$

and then $R_{1}(t)\in\mathbb{Z}[\\![t]\\!]\Leftrightarrow
G_{4}^{(2)}\in\mathbb{Z}[\\![q]\\!]$ and
$R_{2}(t)\in\mathbb{Z}[\\![t]\\!]\Leftrightarrow
G_{8}^{(2)}\in\mathbb{Z}[\\![q]\\!]$ hold. Since

$\displaystyle R_{1}(t)$
$\displaystyle=\sum_{r=0}^{\infty}\binom{2r+1}{r}\binom{3r+1}{r+1}\binom{6r+1}{3r}\,t^{r},$
$\displaystyle R_{2}(t)$
$\displaystyle=\sum_{r=0}^{\infty}\frac{1}{210}\binom{2r+2}{r}\binom{3r+4}{r+2}\binom{6r+7}{3r+3}\,t^{r},$

the integrality of the former is obvious, while that of the latter is not.
Noticing that the exponents of the prime factors of $210=2\cdot 3\cdot 5\cdot
7$ are all $1$, we can prove the latter integrality using the Lucas’ theorem
in (36). For $p=5,7$ we set $r=\ell p+m\;(\ell\in\mathbb{Z}_{\geq 0},0\leq
m\leq p-1)$ and use Lucas’ theorem only once, but for $p=2,3$, we have to use
it several times. For example, the calculation for $p=2$ and $r=8\ell+2$ is
performed as follows.

$\displaystyle{}\binom{16\ell+6}{8\ell+2}\binom{24\ell+10}{8\ell+4}\binom{48\ell+19}{24\ell+9}\equiv\binom{0}{0}^{2}\binom{1}{1}\binom{8\ell+3}{4\ell+1}\binom{12\ell+5}{4\ell+2}\binom{24\ell+9}{12\ell+4}$
$\displaystyle\equiv\binom{1}{1}\binom{1}{0}^{2}\binom{4\ell+1}{2\ell}\binom{6\ell+2}{2\ell+1}\binom{12\ell+4}{6\ell+2}\equiv\binom{1}{0}\binom{0}{1}\binom{0}{0}\binom{2\ell}{\ell}\binom{3\ell+1}{\ell}\binom{6\ell+2}{3\ell+1}$
$\displaystyle\equiv 0\pmod{2}\;\left(\text{note }\tbinom{0}{1}=0\right).$

The following assertions are the depth $2$ counterparts of Proposition 3. The
proof is the same, so it is omitted.

######  Proposition 9.

For each even integer $k\geq 2$, a basis $\mathcal{B}_{k}^{(2)}$ of the space
$QM_{k}^{(2)}(\Gamma)$ is given by the following set:

$\displaystyle\mathcal{B}_{4n}^{(2)}$
$\displaystyle=\left\\{E_{4}^{n},E_{4}^{n-1}G_{4}^{(2)},\cdots,E_{4}G_{4n-4}^{(2)},G_{4n}^{(2)}\right\\},$
$\displaystyle\mathcal{B}_{4n+2}^{(2)}$
$\displaystyle=\left\\{E_{4}^{n}G_{2}^{(2)},E_{4}^{n-1}G_{6}^{(2)},\cdots,E_{4}G_{4n-2}^{(2)},G_{4n+2}^{(2)}\right\\}.$

Depth 3. The Kaneko–Zagier equation $\theta_{w-3}^{(3)}(f)=0$ is transformed
into

$\displaystyle{}z^{3}(1-z)^{2}\frac{d^{4}g}{dz^{4}}+z^{2}(1-z)\left(-\frac{w-18}{3}+\frac{w-27}{3}z\right)\frac{d^{3}g}{dz^{3}}$
$\displaystyle+z\left(-w+7-\frac{3w^{2}-198w+1825}{72}z+\frac{3w^{2}-126w+1375}{72}z^{2}\right)\frac{d^{2}g}{dz^{2}}$
$\displaystyle+\left\\{-\frac{w-3}{3}+\frac{w^{3}-36w^{2}+781w-3540}{432}z-\frac{(w-15)(w^{2}-30w+241)}{432}z^{2}\right\\}\frac{dg}{dz}$
$\displaystyle+\frac{(w-3)\left\\{-4(w^{3}-3w^{2}+62w-180)+(w-7)(w-11)(w-15)z\right\\}}{20736}g=0,$

where $g(\tau)=E_{4}(\tau)^{-\frac{w-3}{4}}f(\tau)$ and $z=1728/j(\tau)$. This
differential equation is not a hypergeometric differential equation. However,
noting the “accidental relation”
$G_{6}^{(3)}=\tfrac{1}{30}(D(G_{4}^{(2)})-E_{2}G_{4}^{(2)})$, we obtain the
following expression of $G_{6}^{(3)}$ using a hypergeometric series:

$\displaystyle G_{6}^{(3)}$
$\displaystyle=j^{-2}\left(1-\frac{1728}{j}\right)^{1/2}E_{4}^{3/4}$
$\displaystyle{}\quad\times\left\\{21\,{}_{2}F_{1}\left(\frac{1}{12},\frac{5}{12};1;\frac{1728}{j}\right){}_{3}F_{2}\left(\frac{11}{6},\frac{13}{6},\frac{5}{2};3,3;\frac{1728}{j}\right)\right.$
$\displaystyle{}\qquad\left.-20\,{}_{2}F_{1}\left(\frac{13}{12},\frac{17}{12};2;\frac{1728}{j}\right){}_{3}F_{2}\left(\frac{5}{6},\frac{7}{6},\frac{3}{2};2,2;\frac{1728}{j}\right)\right\\}$
$\displaystyle=t^{2}+1496t^{3}+2072262t^{4}+2893548528t^{5}+O(t^{6})\quad(t=1/j).$

It is known from [15] that $\mathcal{E}_{3}=\\{6\\}$, more specifically, we
have

$\displaystyle G_{6}^{(3)}$
$\displaystyle=\frac{5E_{2}^{3}-3E_{2}E_{4}-2E_{6}}{51840}=\frac{1}{6}\sum_{n=2}^{\infty}n(\sigma_{3}(n)-n\sigma_{1}(n))q^{n}$
$\displaystyle=q^{2}+8q^{3}+30q^{4}+80q^{5}+180q^{6}+336q^{7}+620q^{8}+960q^{9}+O(q^{10})\in\mathbb{Z}[\\![q]\\!].$

The positivity of the Fourier coefficients of $G_{6}^{(3)}$ can also be seen
from the fact that $D(G_{6}^{(3)})=2(G_{4}^{(2)})^{2}$. In considering the
meaning of integrality of Fourier coefficients, it is worth noting that the
$d$-th Fourier coefficient of $G_{6}^{(3)}$ gives the number of simply
ramified coverings of genus $2$ and degree $d$ of an elliptic curve over
$\mathbb{C}$. See [6, 20] for more details on this claim.

######  Proposition 10.

For any even integer $k\geq 2$, a basis $\mathcal{B}_{k}^{(3)}$ of the space
$QM_{k}^{(3)}(\Gamma)$ is given by the following set:

$\displaystyle\mathcal{B}_{12m}^{(3)}$
$\displaystyle=\left\\{E_{4}^{3(m-\ell)}G_{12\ell}^{(3)}\right\\}_{0\leq\ell\leq
m}\cup\left\\{E_{4}^{3(m-\ell)-1}G_{12\ell+4}^{(3)}\right\\}_{0\leq\ell\leq
m-1}$
$\displaystyle{}\quad\cup\left\\{E_{4}^{3(m-\ell)-2}G_{12\ell+8}^{(3)}\right\\}_{0\leq\ell\leq
m-1}\cup\left\\{E_{4}^{3(m-\ell)-4}E_{6}G_{12\ell+10}^{(3)}\right\\}_{0\leq\ell\leq
m-2}$ $\displaystyle{}\quad\cup\left\\{E_{6}G_{12m-6}^{(3)}\right\\},$
$\displaystyle\mathcal{B}_{12m+2}^{(3)}$
$\displaystyle=\left\\{E_{4}^{3(m-\ell)}G_{12\ell+2}^{(3)}\right\\}_{0\leq\ell\leq
m}\cup\left\\{E_{4}^{3(m-\ell)-2}E_{6}G_{12\ell+4}^{(3)}\right\\}_{0\leq\ell\leq
m-1}$
$\displaystyle{}\quad\cup\left\\{E_{4}^{3(m-\ell)-1}G_{12\ell+6}^{(3)}\right\\}_{0\leq\ell\leq
m-1}\cup\left\\{E_{4}^{3(m-\ell)-2}G_{12\ell+10}^{(3)}\right\\}_{0\leq\ell\leq
m-1},$ $\displaystyle\mathcal{B}_{12m+4}^{(3)}$
$\displaystyle=E_{4}\mathcal{B}_{12m}^{(3)}\cup\left\\{G_{12m+4}^{(3)}\right\\},\quad\mathcal{B}_{12m+6}^{(3)}=E_{4}\mathcal{B}_{12m+2}^{(3)}\cup\left\\{E_{6}G_{12m}^{(3)},G_{12m+6}^{(3)}\right\\},$
$\displaystyle\mathcal{B}_{12m+8}^{(3)}$
$\displaystyle=E_{4}\mathcal{B}_{12m+4}^{(3)}\cup\left\\{G_{12m+8}^{(3)}\right\\},\quad\mathcal{B}_{12m+10}^{(3)}=E_{4}\mathcal{B}_{12m+6}^{(3)}\cup\left\\{G_{12m+10}^{(3)}\right\\}.$

Note that the Fourier expansion of each element of the basis
$\mathcal{B}_{12m+6}^{(3)}$ is slightly different from that of depths 1 and 2.
More specifically, there are two forms $E_{4}G_{12m+2}^{(3)}$ and
$E_{6}G_{12m}^{(3)}$ whose Fourier expansion is $q^{4m}(1+O(q))$. Therefore,
the form $f\in QM_{12m+6}^{(3)}$ such that $f=q^{4m+1}(1+O(q))$ is expressed
as a linear combination of $E_{4}G_{12m+2}^{(3)},E_{6}G_{12m}^{(3)}$, and
$G_{12m+6}^{(3)}=q^{4m+2}(1+O(q))$.

Depth 4. The Kaneko–Zagier equation $\theta_{w-4}^{(4)}(f)=0$ is transformed
into

$\displaystyle{}z^{4}(1-z)^{2}\frac{d^{5}g}{dz^{5}}+5z^{3}(1-z)\left(-\frac{w-24}{12}+\frac{w-36}{12}z\right)\frac{d^{4}g}{dz^{4}}$
$\displaystyle{}+5z^{2}\left(-\frac{w-10}{2}-\frac{w^{2}-96w+1234}{72}z+\frac{w^{2}-60w+928}{72}z^{2}\right)\frac{d^{3}g}{dz^{3}}$
$\displaystyle{}-5z\left\\{\frac{7w-36}{12}-\frac{w^{3}-54w^{2}+2118w-14796}{864}z+\frac{(w-24)(w^{2}-48w+624)}{864}z^{2}\right\\}\frac{d^{2}g}{dz^{2}}$
$\displaystyle{}+\left\\{-\frac{5w-12}{12}-\frac{5(w-5)(w^{3}-7w^{2}+216w-3600)}{5184}z\right.$
$\displaystyle{}\left.\qquad+\frac{5(w-16)(w-20)(w^{2}-36w+352)}{20736}z^{2}\right\\}\frac{dg}{dz}$
$\displaystyle{}-\frac{(w-4)(w-8)\left\\{-12(w^{3}+2w^{2}+27w-90)+(w-12)(w-16)(w-20)z\right\\}}{248832}g=0,$

where $g(\tau)=E_{4}(\tau)^{-(w-4)/4}f(\tau)$ and $z=1728/j(\tau)$. This
differential equation is not a hypergeometric differential equation. It is
known from [15] that $\mathcal{E}_{4}=\emptyset$, that is, there are no
normalized extremal quasimodular forms with integral Fourier coefficients.
However, we note that $D(G_{8}^{(4)})=3G_{4}^{(2)}G_{6}^{(3)}$ holds as for
depths 2 and 3.

######  Remark 6.

The fourth- and fifth-order differential equations that appear for depths $3$
and $4$, respectively, are “rigid” in the sense of Katz (see [9, Rem. 17]).

######  Proposition 11.

For any even integer $k\geq 2$, a basis $\mathcal{B}_{k}^{(4)}$ of the space
$QM_{k}^{(4)}(\Gamma)$ is given by the following set:

$\displaystyle\mathcal{B}_{12m}^{(4)}$
$\displaystyle=\left\\{E_{4}^{3(m-\ell)}G_{12\ell}^{(4)}\right\\}_{0\leq\ell\leq
m}\cup\left\\{E_{4}^{3(m-\ell)-1}G_{12\ell+4}^{(4)}\right\\}_{0\leq\ell\leq
m-1}$
$\displaystyle{}\quad\cup\left\\{E_{4}^{3(m-\ell-1)}E_{6}G_{12\ell+6}^{(4)}\right\\}_{0\leq\ell\leq
m-1}\cup\left\\{E_{4}^{3(m-\ell)-2}G_{12\ell+8}^{(4)}\right\\}_{0\leq\ell\leq
m-1}$
$\displaystyle{}\quad\cup\left\\{E_{4}^{3(m-\ell)-4}E_{6}G_{12\ell+10}^{(4)}\right\\}_{0\leq\ell\leq
m-2}\cup\left\\{E_{4}E_{6}G_{12m-10}^{(4)}\right\\},$
$\displaystyle\mathcal{B}_{12m+2}^{(4)}$
$\displaystyle=\left\\{E_{4}^{3(m-\ell)}G_{12\ell+2}^{(4)}\right\\}_{0\leq\ell\leq
m}\cup\left\\{E_{4}^{3(m-\ell)-2}E_{6}G_{12\ell+4}^{(4)}\right\\}_{0\leq\ell\leq
m-1}$
$\displaystyle{}\quad\cup\left\\{E_{4}^{3(m-\ell)-1}G_{12\ell+6}^{(4)}\right\\}_{0\leq\ell\leq
m-1}\cup\left\\{E_{4}^{3(m-\ell-1)}E_{6}G_{12\ell+8}^{(4)}\right\\}_{0\leq\ell\leq
m-1}$
$\displaystyle{}\quad\cup\left\\{E_{4}^{3(m-\ell)-2}G_{12\ell+10}^{(4)}\right\\}_{0\leq\ell\leq
m-1},$ $\displaystyle\mathcal{B}_{12m+4}^{(4)}$
$\displaystyle=E_{4}\mathcal{B}_{12m}^{(4)}\cup\left\\{G_{12m+4}^{(4)}\right\\},$
$\displaystyle\mathcal{B}_{12m+6}^{(4)}$
$\displaystyle=E_{4}\mathcal{B}_{12m+2}^{(4)}\cup\left\\{E_{6}G_{12m}^{(4)},G_{12m+6}^{(4)}\right\\},$
$\displaystyle\mathcal{B}_{12m+8}^{(4)}$
$\displaystyle=E_{4}\mathcal{B}_{12m+4}^{(4)}\cup\left\\{E_{6}G_{12m+2}^{(4)},G_{12m+8}^{(4)}\right\\},$
$\displaystyle\mathcal{B}_{12m+10}^{(4)}$
$\displaystyle=E_{4}\mathcal{B}_{12m+6}^{(4)}\cup\left\\{G_{12m+10}^{(4)}\right\\}.$

The same (but slightly more complicated) phenomenon as in the case of depth
$3$ occurs for the Fourier expansion of each element of these bases. For
instance, there is no element in the set $\mathcal{B}_{12m}^{(4)}$ for which
$q^{5\ell+4}(1+O(q))\;(0\leq\ell\leq m-1)$, but there are two elements for
which $q^{5\ell+3}(1+O(q))\;(0\leq\ell\leq m-2)$. However, there is only one
element for which $q^{5m-2}(1+O(q))$, and there are two elements for which
$q^{5m-5}(1+O(q))$.

For a given even integer $12n+a\;(n\in\mathbb{Z}_{\geq 0},a\in
2\mathbb{Z}_{\geq 0})$ and depth $r\geq 1$, let
$12n+a-2r=4s+6t+12u\;(s\in\\{0,1,2\\},t\in\\{0,1\\},u\in\mathbb{Z}_{\geq 0})$
as before, and express the normalized extremal quasimodular form
$G_{12n+a}^{(r)}$ as follows:

$\displaystyle
G_{12n+a}^{(r)}=\frac{1}{N_{n,a}^{(r)}}E_{2}^{r}E_{4}^{s}E_{6}^{t}\Delta^{u}A_{u,a}^{(r)}(j)+(\text{lower
order terms on $E_{2}$}),$

where the number $N_{n,a}^{(r)}$ is chosen so that the polynomial
$A_{u,a}^{(r)}(X)$ of degree $u$ is the monic polynomial. We call this
polynomial the generalized Atkin polynomial. Although it is possible to
calculate the polynomial $A_{u,a}^{(r)}(X)$ separately for a given $n,a$ and
$r$, as already mentioned in Section 1, the existence and uniqueness of
$G_{w}^{(r)}$ for the depth $r\geq 5$ are still open, it is not yet clear
whether the polynomial $A_{u,a}^{(r)}(X)$ determined by the above equation
always exists for $r\geq 5$.

The following conjecture is suggested by numerical experiments .

######  Conjecture 1.

Let $p\geq 5$ be a prime number and put
$p-1=12m+4\delta+6\varepsilon\;(m\in\mathbb{Z}_{\geq
0},\delta,\varepsilon\in\\{0,1\\})$. Then we have
$A_{m+\delta+\varepsilon,2}^{(1)}(X)\equiv
A_{m+\delta+\varepsilon,2r}^{(r)}(X)\pmod{p}$ for $r\geq 2$.

######  Example 3.

The case of $m+\delta+\varepsilon=2$, that is, the case of
$p\in\\{11,17,19\\}$:

$\displaystyle A_{2,2}^{(1)}(X)$
$\displaystyle=X^{2}-1640X+269280\equiv\begin{cases}X(X+10)\pmod{11}\\\
X(X+9)\pmod{17}\\\ (X+1)(X+12)\pmod{19},\end{cases}$ $\displaystyle
A_{2,4}^{(2)}(X)$
$\displaystyle=X^{2}+\frac{5367564}{4847}X+\frac{97748640}{4847},$
$\displaystyle A_{2,6}^{(3)}(X)$
$\displaystyle=X^{2}-\frac{100925285400}{6736603}X+\frac{184720492440000}{6736603},$
$\displaystyle A_{2,8}^{(4)}(X)$
$\displaystyle=X^{2}+\frac{13326301537125}{303744733}X+\frac{8760324756150000}{303744733},$
$\displaystyle A_{2,10}^{(5)}(X)$
$\displaystyle=X^{2}-\frac{567274769925055704588000}{7988288882724700441}X$
$\displaystyle{}\quad+\frac{302601299124728270224800000}{7988288882724700441},$
$\displaystyle A_{2,12}^{(6)}(X)$
$\displaystyle=X^{2}+\frac{67508245504783855161034500000}{433955868750758754759533}X$
$\displaystyle{}\quad-\frac{163976620145430859347886034400000}{433955868750758754759533}.$

For the prime $p\in\\{11,17,19\\}$, these polynomials satisfy the above
conjecture.

For the depths greater than 5, almost nothing is known. For example, we do not
even know what differential equation the extremal quasimodular form satisfies.
Unlike the case of depth $r\leq 4$, as already pointed out in [19], any
extremal modular form of weight $w$ and depth $r\geq 5$ cannot satisfy the
differential equation $\theta_{w-r}^{(r)}(f)=0$. Of course, it is possible to
calculate specific examples separately, e.g., the normalized extremal
quasimodular form of weight $10$ and depth $5$

$\displaystyle G_{10}^{(5)}$
$\displaystyle=\frac{140E_{2}^{5}-35E_{2}^{3}E_{4}-65E_{2}^{2}E_{6}-33E_{2}E_{4}^{2}-7E_{4}E_{6}}{1437004800}$
$\displaystyle=q^{4}+\frac{144}{11}q^{5}+\frac{936}{11}q^{6}+\frac{4160}{11}q^{7}+\frac{14490}{11}q^{8}+\frac{42432}{11}q^{9}+O(q^{10})$

is the solution of the following modular linear differential equation:

$\displaystyle{}\left(E_{4}^{3}-\frac{731087}{4380623}E_{6}^{2}\right)\partial_{5}^{6}(f)+\frac{3649536}{4380623}E_{4}^{2}E_{6}\partial_{5}^{5}(f)$
$\displaystyle{}-\frac{5E_{4}(845736619E_{4}^{3}-170572459E_{6}^{2})}{630809712}\partial_{5}^{4}(f)$
$\displaystyle{}-\frac{5E_{6}(2032753837E_{4}^{3}-164191405E_{6}^{2})}{946214568}\partial_{5}^{3}(f)$
$\displaystyle{}+\frac{E_{4}^{2}(262935868013E_{4}^{3}-746094289517E_{6}^{2})}{90836598528}\partial_{5}^{2}(f)$
$\displaystyle{}+\frac{E_{4}E_{6}(80592093937E_{4}^{3}-122767956721E_{6}^{2})}{45418299264}\partial_{5}(f)$
$\displaystyle{}+\frac{55(3672965829E_{4}^{6}-7414522789E_{4}^{3}E_{6}^{2}-5174923040E_{6}^{4})}{13080470188032}f=0.$

As can be expected from this example, as the weights or depths increases, the
coefficients of the differential equation become more complex and the
calculations rapidly become unmanageable. From the observation of some
specific examples, it seems that the coefficients cannot be a simple
polynomial with $w$ as a variable. We also note that the denominator 11 of the
Fourier coefficients of $G_{10}^{(5)}$ is greater than the weight 10. (cf.
Corollary 2)

At the end of Section 2, we pointed out that the normalized extremal
quasimodular forms of depth 1 are essentially the remainder of some
Hermite–Padé approximation. In fact, with the help of the ring isomorphism in
Theorem 2, the extremal quasimodular forms of any depth can be regarded as the
remainder of some Hermite–Padé approximation (of the first kind) for a
suitable function. On the other hand, there is an approximation called the
Hermite–Padé approximation of the second kind (or simultaneous Padé
approximation) [28]. For the depth $r=1$, these two approximations are
essentially the same, but not for the depth $\geq 2$. What are the properties
of the quasimodular forms derived from the simultaneous Padé approximation?

###  6.2. Case of other groups

In this subsection we summarize previous works on Atkin orthogonal polynomials
and extremal quasimodular forms of depth 1 on the congruence subgroup
$\Gamma_{0}(N)$ and the Fricke group $\Gamma_{0}^{*}(N)$ of low-levels. If the
level is less than or equal to 4, the corresponding quasimodular Eisenstein
series of weight 2 can be expressed using the hypergeometric series
${}_{2}F_{1}$ (see [27, Ch. 3]), so it is probably possible to treat the
normalized extremal quasimodular forms in the same way as in this paper. On
the other hand, for the Fricke groups of levels 5 and 7, the local Heun
function appears instead of the hypergeometric series ${}_{2}F_{1}$. Because
of this difference, it may be difficult to obtain various concrete expressions
for these levels as in this paper. Note also that in the case of level 1, the
Atkin polynomials and the normalized extremal quasimodular forms are directly
connected as in (24), but this is not necessarily the case for level $\geq 2$.

The next table summarizes the previous works by Tsutsumi, Kaneko, Koike,
Sakai, and Shimizu. Note that some extremal quasimodular forms also appear in
[33], and the Kaneko–Zagier equation for $\Gamma_{0}^{*}(2)$ is treated in
[18], but its quasimodular solution is the quasimodular form on
$\Gamma_{0}(2)$.

Table 2. The previous works related to this paper (with levels). | Atkin inner product | Modular linear differential equation
---|---|---
| Atkin polynomials | (Extremal) quasimodular forms
$\Gamma_{0}(N)$ | Tsutsumi [37, 38] $(N=2,3,4)$ | Sakai–Tsutsumi [35] $(N=2,3,4)$
Sakai–Shimizu [34] $(N=2,3,4)$
$\Gamma_{0}^{*}(N)$ | Sakai [33] $(N=5,7)$ | Kaneko–Koike [18] $(N=2)$
Sakai [32] $(N=2,3)$

Here we mention a few problems.

  * •

In the case of level 1, three differential equations appeared in [19, Thm.
2.1], but since there are relations (17) and (18), it is essentially
sufficient to consider only the differential equation (13). Extend this result
to level $\geq 2$ and clarify the relationship between quasimodular solutions
of some differential equations using Grabner’s method.

  * •

Extend Theorem 1 to level $\geq 2$ and depth $\geq 1$.

Although of little relevance to quasimodular forms, it is worth noting that a
certain basis of the vector space of modular forms of weight
$2n/7\;(n\in\mathbb{Z}_{\geq 1})$ on $\Gamma(7)$ is expressed by a generalized
hypergeometric series ${}_{3}F_{2}$. For more details on this fact, see [14,
9, 39] and its references.

## Acknowledgement

The author thanks to Professor Masanobu Kaneko for many valuable comments on
reading an early draft of this paper.

##  Appendix A Appendices

###  A.1. Table of the integral Fourier coefficients of $G_{w}^{(1)}$

Put $G_{w}^{(1)}=q^{[w/6]}\left(1+\sum_{n=1}^{\infty}a_{w}(n)\,q^{n}\right)$
and then the first few terms of the integral Fourier coefficients of
$G_{w}^{(1)}$ are given in the following table.

Table 3. The first few integral Fourier coefficients $a_{w}(n)$ $w\backslash n$ | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
2 | $-24$ | $-72$ | $-96$ | $-168$ | $-144$
6 | 18 | 84 | 292 | 630 | 1512
8 | 66 | 732 | 4228 | 15630 | 48312
10 | 258 | 6564 | 66052 | 390630 | 1693512
12 | 56 | 1002 | 9296 | 57708 | 269040
14 | 128 | 4050 | 58880 | 525300 | 3338496
16 | 296 | 16602 | 377456 | 4846908 | 41943120
18 | 99 | 3510 | 64944 | 764874 | 6478758
20 | 183 | 10134 | 269832 | 4326546 | 47862918
22 | 339 | 29430 | 1127904 | 24615834 | 355679478
24 | 144 | 7944 | 235840 | 4451130 | 59405952
28 | 384 | 44664 | 2460160 | 79196970 | 1693028352
30 | 190 | 14460 | 608570 | 16463120 | 314562708
32 | 286 | 29988 | 1652834 | 56608952 | 1335336084
34 | 430 | 62220 | 4496090 | 195047840 | 5680752948
38 | 336 | 43587 | 3065648 | 136437750 | 4219436160
54 | 378 | 62532 | 6109740 | 401161950 | 19083824856
58 | 618 | 155412 | 21940620 | 2005126350 | 128986599096
68 | 581 | 147042 | 21956168 | 2203554570 | 160242315903
80 | 678 | 204756 | 37135249 | 4592036697 | 416237464122
114 | 855 | 341886 | 85507600 | 15092041050 | 2010698806050
118 | 1095 | 549246 | 169413760 | 36358101930 | 5819797557810

###  A.2. Explicit formulas for the coefficients of $P_{n}(t)$ and $Q_{n}(t)$

We give explicit formulas for the coefficients of the power series $P_{n}(t)$
and $Q_{n}(t)$ that appear in the proof of Theorem 1. However, as we have
already seen in equations (33) and (34), we multiply the power series
$P_{\text{odd}}(t)$ and $Q_{\text{even}}(t)$ by the factor $(1-1728t)^{-1/2}$.
Our main theorem is equivalent to the fact that the following power series
belong to $\mathbb{Z}[\\![t]\\!]$.

$\bullet$
$G_{w}^{(1)}\in\mathbb{Z}[\\![q]\\!],\,w\in\mathcal{S}_{0}=\\{12,24\\}$. The
coefficients of the power series $P_{0}(t)$ and $P_{2}(t)$ correspond to the
sequences A001421 and A145493 in the Online Encyclopedia of Integer Sequences
(OEIS) [29], respectively.

$\displaystyle P_{0}(t)$
$\displaystyle=\sum_{r=0}^{\infty}\frac{(6r)!}{(3r)!\,r!^{3}}\,t^{r}={}_{3}F_{2}\left(\frac{1}{6},\frac{1}{2},\frac{5}{6};1,1;1728t\right)\quad(\Leftrightarrow
G_{0}^{(1)}=1)$
$\displaystyle=1+120t+83160t^{2}+81681600t^{3}+93699005400t^{4}+O(t^{5}),$
$\displaystyle P_{2}(t)$
$\displaystyle=\sum_{r=0}^{\infty}\frac{(41r+77)\,(6r+6)!}{2310\,(3r+3)!\,r!\,(r+2)!^{2}}\,t^{r}$
$\displaystyle=1+944t+1054170t^{2}+1297994880t^{3}+1700941165560t^{4}+O(t^{5}),$
$\displaystyle P_{4}(t)$
$\displaystyle=\sum_{r=0}^{\infty}\frac{(17377r^{2}+117219r+193154)\,(6r+12)!}{223092870\,(3r+6)!\,r!\,(r+4)!^{2}}\,t^{r}$
$\displaystyle=1+1800t+2783760t^{2}+4183182720t^{3}+6274984354650t^{4}+O(t^{5}).$

$\bullet$
$G_{w}^{(1)}\in\mathbb{Z}[\\![q]\\!],\,w\in\mathcal{S}_{6}=\\{6,18,30,54,114\\}$.

$\displaystyle{}(1-1728t)^{-1/2}P_{1}(t)=\sum_{r=0}^{\infty}\frac{(6r+6)!}{120\,(3r+3)!\,r!\,(r+1)!^{2}}\,t^{r}$
$\displaystyle{}\quad={}_{3}F_{2}\left(\frac{7}{6},\frac{3}{2},\frac{11}{6};2,2;1728t\right)$
$\displaystyle{}\quad=1+1386t+2042040t^{2}+3123300180t^{3}+4891088081880t^{4}+O(t^{5}),$
$\displaystyle{}(1-1728t)^{-1/2}P_{3}(t)=\sum_{r=0}^{\infty}\frac{(77r+221)\,(6r+12)!}{4084080\,(3r+6)!\,r!\,(r+3)!^{2}}\,t^{r}$
$\displaystyle{}\quad=1+2235t+4129650t^{2}+7217526960t^{3}+12344776903800t^{4}+O(t^{5}),$
$\displaystyle{}(1-1728t)^{-1/2}P_{5}(t)=\sum_{r=0}^{\infty}\frac{(33649r^{2}+294051r+633650)\,(6r+18)!}{776363187600\,(3r+9)!\,r!\,(r+5)!^{2}}\,t^{r}$
$\displaystyle{}\quad=1+3094t+6975504t^{2}+13953546090t^{3}+26319290241530t^{4}+O(t^{5}),$
$\displaystyle{}(1-1728t)^{-1/2}P_{9}(t)=\sum_{r=0}^{\infty}\frac{p_{9}(r)\,(6r+30)!}{(3r+15)!\,r!\,(r+9)!^{2}}\,t^{r}$
$\displaystyle{}\quad=1+4818t+14913288t^{2}+37889152860t^{3}+86182007602320t^{4}+O(t^{5}),$
$\displaystyle{}(1-1728t)^{-1/2}P_{19}(t)=\sum_{r=0}^{\infty}\frac{p_{19}(r)\,(6r+60)!}{(3r+30)!\,r!\,(r+19)!^{2}}\,t^{r}$
$\displaystyle{}\quad=1+9135t+47828730t^{2}+188818914000t^{3}+625280243661000t^{4}+O(t^{5}),$

where

$\displaystyle p_{9}(r)$ $\displaystyle=\frac{1}{1303566339087601789200}$
$\displaystyle{}\times(301163357r^{4}+8876894690r^{3}+97346883895r^{2}$
$\displaystyle{}\quad+470641033450r+846250112568),$ $\displaystyle p_{19}(r)$
$\displaystyle=\frac{1}{7586413113700225869154849509970478998385924877600}$
$\displaystyle{}\times(116055861444395385601913r^{9}$
$\displaystyle{}\quad+15530138946748752922984725r^{8}$
$\displaystyle{}\quad+920111315629981006299003510r^{7}$
$\displaystyle{}\quad+31676880792353832401375777850r^{6}$
$\displaystyle{}\quad+698329420677409164956468289249r^{5}$
$\displaystyle{}\quad+10222801871323855615909703388405r^{4}$
$\displaystyle{}\quad+99369498641304011775924341700640r^{3}$
$\displaystyle{}\quad+618440343527755839046417085216700r^{2}$
$\displaystyle{}\quad+2236089229125717720580535903583888r$
$\displaystyle{}\quad+3578581860690243122001381266421120)$

and these polynomials are irreducible over $\mathbb{Q}$.

$\bullet$
$G_{w}^{(1)}\in\mathbb{Z}[\\![q]\\!],\,w\in\mathcal{S}_{2}=\\{2,14,38\\}$.

$\displaystyle{}(1-1728t)^{-1/2}Q_{0}(t)=\sum_{r=0}^{\infty}\frac{(6r+1)!}{(3r)!\,r!^{3}}\,t^{r}={}_{3}F_{2}\left(\frac{1}{2},\frac{5}{6},\frac{7}{6};1,1;1728t\right)$
$\displaystyle{}\quad=1+840t+1081080t^{2}+1551950400t^{3}+2342475135000t^{4}+O(t^{5}),$
$\displaystyle{}(1-1728t)^{-1/2}Q_{2}(t)=\sum_{r=0}^{\infty}\frac{(7r+13)\,(6r+7)!}{2730\,(3r+3)!\,r!\,(r+2)!^{2}}\,t^{r}$
$\displaystyle{}\quad=1+1760t+2877930t^{2}+4667789280t^{3}+7590443164920t^{4}+O(t^{5}),$
$\displaystyle{}(1-1728t)^{-1/2}Q_{6}(t)$
$\displaystyle{}\quad=\sum_{r=0}^{\infty}\frac{(1043119r^{3}+15220608r^{2}+72947639r+114757350)\,(6r+19)!}{74207381348100\,(3r+9)!\,r!\,(r+6)!^{2}}\,t^{r}$
$\displaystyle{}\quad=1+3504t+8597259t^{2}+18287498240t^{3}+36144224452050t^{4}+O(t^{5}).$

$\bullet$
$G_{w}^{(1)}\in\mathbb{Z}[\\![q]\\!],\,w\in\mathcal{S}_{8}=\\{8,20,32,68,80\\}$.
The coefficients of the power series $Q_{1}(t)$ and $Q_{3}(t)$ correspond to
the sequences A145492 and A145494 in OEIS, respectively.

$\displaystyle Q_{1}(t)$
$\displaystyle=\sum_{r=0}^{\infty}\frac{(8r+7)\,(6r+1)!}{7(3r)!\,r!\,(r+1)!^{2}}\,t^{r}$
$\displaystyle=1+450t+394680t^{2}+429557700t^{3}+522037315800t^{4}+O(t^{5}),$
$\displaystyle Q_{3}(t)$
$\displaystyle=\sum_{r=0}^{\infty}\frac{(1528r^{2}+7231r+8151)\,(6r+7)!}{190190\,(3r+3)!\,r!\,(r+3)!^{2}}\,t^{r}$
$\displaystyle=1+1335t+1757970t^{2}+2386445040t^{3}+3336565609080t^{4}+O(t^{5}),$
$\displaystyle Q_{5}(t)$
$\displaystyle=\sum_{r=0}^{\infty}\frac{(1070744r^{3}+12418991r^{2}+46901365r+57574750)\,(6r+13)!}{34579394850\,(3r+6)!\,r!\,(r+5)!^{2}}\,t^{r}$
$\displaystyle=1+2206t+3863952t^{2}+6319180098t^{3}+10079991804410t^{4}+O(t^{5}),$
$\displaystyle Q_{11}(t)$
$\displaystyle=\sum_{r=0}^{\infty}\frac{q_{11}(r)\,(6r+31)!}{(3r+15)!\,r!\,(r+11)!^{2}}\,t^{r}$
$\displaystyle=1+4805t+14658030t^{2}+36441948000t^{3}+80761720666320t^{4}+O(t^{5}),$
$\displaystyle Q_{13}(t)$
$\displaystyle=\sum_{r=0}^{\infty}\frac{q_{13}(r)\,(6r+37)!}{(3r+18)!\,r!\,(r+13)!^{2}}\,t^{r}$
$\displaystyle=1+5670t+19748832t^{2}+54741797937t^{3}+132878837538099t^{4}+O(t^{5}),$

where

$\displaystyle q_{11}(r)$ $\displaystyle=\frac{1}{15716643102160534111758180}$
$\displaystyle{}\times(13252649705176r^{6}+665298552506263r^{5}$
$\displaystyle{}\quad+13797873461407945r^{4}+151287554887490515r^{3}$
$\displaystyle{}\quad+924734694751472239r^{2}+2986992686186751022r$
$\displaystyle{}\quad+3982438425105968520),$ $\displaystyle q_{13}(r)$
$\displaystyle=\frac{1}{32176447673406729078990845541300}$
$\displaystyle{}\times(74198322973160504r^{7}+5124808625350611463r^{6}$
$\displaystyle{}\quad+150642927750066254963r^{5}+2442571823969345600665r^{4}$
$\displaystyle{}\quad+23590276457107577780801r^{3}+135688184492311416306712r^{2}$
$\displaystyle{}\quad+430315970858396108150652r+580367220881648001413040)$

and these polynomials are irreducible over $\mathbb{Q}$.

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