# MacMahon’s sums-of-divisors and allied $q$-series

Tewodros Amdeberhan, Ken Ono and Ajit Singh  Dept. of Mathematics, Tulane
University, New Orleans, LA 70118<EMAIL_ADDRESS>Dept. of Mathematics,
University of Virginia, Charlottesville, VA 22904<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Here we investigate the $q$-series

$\begin{split}\mathcal{U}_{a}(q)=\sum_{n=0}^{\infty}MO(a;n)q^{n}&:=\sum_{0<k_{1}<k_{2}<\cdots<k_{a}}\frac{q^{k_{1}+k_{2}+\cdots+k_{a}}}{(1-q^{k_{1}})^{2}(1-q^{k_{2}})^{2}\cdots(1-q^{k_{a}})^{2}},\\\
\mathcal{U}_{a}^{\star}(q)=\sum_{n=0}^{\infty}M(a;n)q^{n}&:=\sum_{1\leq
k_{1}\leq k_{2}\leq\cdots\leq
k_{a}}\frac{q^{k_{1}+k_{2}+\cdots+k_{a}}}{(1-q^{k_{1}})^{2}(1-q^{k_{2}})^{2}\cdots(1-q^{k_{a}})^{2}}.\end{split}$

MacMahon introduced the $\mathcal{U}_{a}(q)$ in his seminal work on partitions
and divisor functions. Recent works show that these series are sums of
quasimodular forms with weights $\leq 2a.$ We make this explicit by describing
them in terms of Eisenstein series. We use these formulas to obtain explicit
and general congruences for the coefficients $MO(a;n)$ and $M(a;n).$ Notably,
we prove the conjecture of Amdeberhan-Andrews-Tauraso as the $m=0$ special
case of the infinite family of congruences

$MO(11m+10;11n+7)\equiv 0\pmod{11},$

and we prove that

$MO(17m+16;17n+15)\equiv 0\pmod{17}.$

We obtain further formulae using the limiting behavior of these series. For
$n\leq a+\binom{a+1}{2},$ we obtain a “hook length” formulae for $MO(a;n)$,
and for $n\leq 2a$, we find that
$M(a;n)=\binom{a+n-1}{n-a}+\binom{a+n-2}{n-a-1}.$

###### Key words and phrases:

divisor functions, partitions, quasimodular forms

2020 Mathematics Subject Classification. 11F03, 11A25, 11F50, 11M41, 11F11,
11F33

##  1\. Introduction and Statement of Results

At first glance, one might underestimate the value of the trivial observation
that the number of partitions of an integer $n$ into identical parts is also
the number of divisors of $n$. This fact is a glimpse of a rich theory that
relates integer partitions and divisor functions. Indeed, MacMahon’s important
paper [7] is based on the idea of connecting partitions to divisor sums:
partition of $n$ using $k_{1}$ repeated $s_{1}$ times, and $k_{2}$ repeated
$s_{2}$ times, and so on through $k_{a}$ repeated $s_{a}$ times. Using this
convention, he considered the sum of products of the _multiplicities_
$MO(a;n):=\sum s_{1}s_{2}\cdots s_{a}$ of size $n$ partitions, which has the
generating function

(1.1) $\displaystyle\mathcal{U}_{a}(q):=\sum_{n\geq
0}MO(a;n)\,q^{n}=\sum_{0<k_{1}<k_{2}<\cdots<k_{a}}\frac{q^{k_{1}+k_{2}+\cdots+k_{a}}}{(1-q^{k_{1}})^{2}(1-q^{k_{2}})^{2}\cdots(1-q^{k_{a}})^{2}}.$

His work [7] is populated with beautiful divisor function identities, where
$\sigma_{\nu}(n):=\sum_{d\mid n}d^{\nu},$ such as:

(1.2) $\mathcal{U}_{1}(q)=\sum_{n\geq 1}\sigma_{1}(n)q^{n}\ \ \ \ {\text{\rm
and}}\ \ \ \ \mathcal{U}_{2}(q)=\sum_{n\geq
1}\left(\frac{\sigma_{1}(n)}{8}-\frac{n\sigma_{1}(n)}{4}+\frac{\sigma_{3}(n)}{8}\right)q^{n}.$

To entice the reader, we offer the first few terms of
$\mathcal{U}_{1}(q),\dots,\mathcal{U}_{4}(q)$:

$\begin{split}\mathcal{U}_{1}(q)&=q+3q^{2}+4q^{3}+7q^{4}+6q^{5}+12q^{6}+8q^{7}+\dots,\\\
\mathcal{U}_{2}(q)&=q^{3}+3q^{4}+9q^{5}+15q^{6}+30q^{7}+45q^{8}+67q^{9}+\dots,\\\
\mathcal{U}_{3}(q)&=q^{6}+3q^{7}+9q^{8}+22q^{9}+42q^{10}+81q^{11}+140q^{12}+\dots,\\\
\mathcal{U}_{4}(q)&=q^{10}+3q^{11}+9q^{12}+22q^{13}+51q^{14}+97q^{15}+188q^{16}+\dots.\\\
\end{split}$

The inequalities in definition (1.1) imply that
$q^{-\frac{a(a+1)}{2}}\cdot\mathcal{U}_{a}(q)=1+3q+\dots,$ while for $a\geq
2,$ we have

$q^{-\frac{a(a+1)}{2}}\cdot\mathcal{U}_{a}(q)=1+3q+9q^{2}+\dots.$

Answering the natural question, we show that this sequence converges to a
simple infinite product, which, by the theory of Nekrasov-Okounkov [8], gives
hook length formulae for many of the $MO(a;n).$

To make this precise, recall that a partition
$\lambda=(\lambda_{1},\dots,\lambda_{\ell})$ of $n$, denoted $\lambda\vdash
n$, is a non-increasing sequence of positive integers that sum to $n$. Its
Young diagram is the left-justified array of boxes where the row lengths are
the parts. The hook $H(i,j)$ of the box in position $(i,j)$ consists of this
box, together with those below it and those to its right. Its hook length
$h(i,j):=(\lambda_{i}-i)+(\lambda_{j}^{\prime}-j)+1$ is the number of such
boxes, where $\lambda_{j}^{\prime}$ is the number of boxes in column $j$.
Denote the multiset of hook lengths of $\lambda$ by $\mathcal{H}(\lambda)$.
Finally, we recall the “exponential form” of a partition
$\lambda=(1^{m_{1}},2^{m_{2}},\dots,t^{m_{t}}),$ where $m_{i}$ is the
multiplicity of part $i.$

######  Example.

The exponential form of $\lambda=(4,4,2)$ is
$\lambda=(1^{0},2^{1},3^{0},4^{2},5^{0},6^{0},7^{0},8^{0},9^{0},10^{0})\vdash
10$. Its Young diagram is given below, and shows that
$\mathcal{H}(\lambda)=\\{6,5,5,4,3,2,2,2,1,1\\}$.

$\ytableausetup{centertableaux}\ytableau 6&532\\\ 5421\\\ 21$

We derive the following result using the work of Andrews-Rose [2] and
Nekrasov-Okounkov [8].

######  Theorem 1.1.

The following are true:

(i) If $a$ is a positive integer, then we have that

$q^{-\frac{a(a+1)}{2}}\cdot\mathcal{U}_{a}(q)=\prod_{n\geq
1}\frac{1}{(1-q^{n})^{3}}+O(q^{a+1}).$

(ii) If $n\leq a+\binom{a+1}{2}$, then we have that

$MO(a;n)=\sum_{\lambda\vdash
n-a}\,\prod_{h\in\mathcal{H}(\lambda)}\left(\frac{2}{h^{2}}+1\right)=\sum_{\lambda\vdash
n-a}\prod_{s=1}^{n-a}\binom{2+m_{s}}{2}.$

Inspired by the $\mathcal{U}_{a}(q),$ Amdeberhan-Andrews-Tauraso [1] initiated
the study of the $q$-series

(1.3) $\displaystyle\mathcal{U}_{a}^{\star}(q):=\sum_{n\geq
0}M(a;n)\,q^{n}=\sum_{1\leq k_{1}\leq k_{2}\leq\cdots\leq
k_{a}}\frac{q^{k_{1}+k_{2}+\cdots+k_{a}}}{(1-q^{k_{1}})^{2}(1-q^{k_{2}})^{2}\cdots(1-q^{k_{a}})^{2}},$

where the strict inequalities in (1.1) are replaced by weak inequalities. One
easily sees that

$\mathcal{U}_{a}^{*}(q)=\sum_{n\geq 0}M(a;n)q^{n}=q^{a}+(2a+1)q^{a+1}+\dots.$

To entice the reader, we offer the first few terms of
$\mathcal{U}_{1}^{*}(q),\dots,\mathcal{U}_{4}^{*}(q)$:

$\begin{split}\mathcal{U}_{1}^{\star}(q)&=q+3q^{2}+4q^{3}+7q^{4}+6q^{5}+12q^{6}+\cdots,\\\
\mathcal{U}_{2}^{\star}(q)&=q^{2}+5q^{3}+14q^{4}+29q^{5}+55q^{6}+86q^{7}+\cdots,\\\
\mathcal{U}_{3}^{\star}(q)&=q^{3}+7q^{4}+27q^{5}+77q^{6}+181q^{7}+378q^{8}+\cdots,\\\
\mathcal{U}_{4}^{\star}(q)&=q^{4}+9q^{5}+44q^{6}+156q^{7}+450q^{8}+1121q^{9}+\cdots.\end{split}$

In analogy with Theorem 1.1, we consider the limiting behavior of these
series. These series converge to specializations of the generating function
for the polynomials
$p_{0}(x):=1,p_{1}(x):=2x+1,p_{2}(x):=2x^{2}+3x,p_{3}(x):=\frac{4}{3}x^{3}+4x^{2}+\frac{5}{3}x,\dots.$
For $n\geq 1,$ these polynomials are defined by

(1.4) $p_{n}(x):=\binom{2x+n-1}{n}+\binom{2x+n-2}{n-1}.$

As a companion to Theorem 1.1, we obtain the following theorem.

######  Theorem 1.2.

The following are true:

(i) If $a$ is a positive integer, then we have that

$q^{-a}\cdot\mathcal{U}^{*}_{a}(q)=\sum_{n=0}^{a}p_{n}(a)q^{n}+O(q^{a+1}).$

(ii) If $n\leq 2a$, then we have that $M(a;n)=p_{n-a}(a).$

######  Remark.

The $\mathcal{U}_{a}(q)$ and $\mathcal{U}_{a}^{\star}(q)$ are multiple
$q$-zeta values. To make this precise, we recall the $q$-notation
$[k]_{q}:=\frac{1-q^{k}}{1-q}$ and the multiple $q$-zeta values (for example,
see [3])

$\displaystyle\zeta_{q}(m_{1},\dots,m_{a}):$
$\displaystyle=\sum_{0<k_{1}<\cdots<k_{a}}\frac{q^{(m_{1}-1)k_{1}+\cdots+(m_{a}-1)k_{a}}}{[k_{1}]_{q}^{m_{1}}\cdots[k_{a}]_{q}^{m_{a}}},$
$\displaystyle\zeta_{q}^{\star}(m_{1},\dots,m_{a}):$
$\displaystyle=\sum_{1\leq k_{1}\leq\cdots\leq
k_{a}}\frac{q^{(m_{1}-1)k_{1}+\cdots+(m_{a}-1)k_{a}}}{[k_{1}]_{q}^{m_{1}}\cdots[k_{a}]_{q}^{m_{a}}}.$

We have that $(1-q)^{2a}\cdot\mathcal{U}_{a}(q)=\zeta_{q}({2,\ldots,2})$ and
$(1-q)^{2a}\cdot\mathcal{U}_{a}^{\star}(q)=\zeta_{q}^{\star}({2,\ldots,2}).$

As divisor functions arise as the coefficients of Eisenstein series,
identities such as (1.2) suggest a strong relationship between the
$\mathcal{U}_{a}(q)$ and quasimodular forms. This speculation was confirmed by
Andrews-Rose. Indeed, they proved (see [2, Cor. 4]) and [12, Th. 1.12]) that
each $\mathcal{U}_{a}(q)$ is a linear combination of quasimodular forms on
$\mathrm{SL}_{2}(\mathbb{Z})$ with weights $\leq 2a$. Similarly, Amdeberhan-
Andrews-Tauraso [1, Th. 6.1] proved that each $\mathcal{U}_{a}^{\star}(q)$ is
a linear combination of quasimodular forms on $\mathrm{SL}_{2}(\mathbb{Z})$
with weights $\leq 2a$.

Here we make this quasimodularity explicit. In the case of
$\mathcal{U}_{a}(q),$ we employ the standard generators of the graded ring of
quasimodular forms: the quasimodular weight 2 Eisenstein series

(1.5) $E_{2}(q):=1-24\sum_{n=1}^{\infty}\sigma_{1}(n)q^{n},$

and the weight 4 and 6 modular Eisenstein series

(1.6)
$E_{4}(q):=1+240\sum_{n=1}^{\infty}\sigma_{3}(n)q^{n}\qquad\text{and}\qquad
E_{6}(q):=1-504\sum_{n=1}^{\infty}\sigma_{5}(n)q^{n}.$

It is well known [6] that the ring of quasimodular forms is
$\mathbb{C}[E_{2},E_{4},E_{6}],$ and so our goal is to obtain formulas in
terms of the monomials $E_{2}^{\alpha}(q)E_{4}^{\beta}(q)E_{6}^{\gamma}(q)$,
where $\alpha,\beta$ and $\gamma$ are non-negative integers.

Our formulas for $\mathcal{U}_{a}(q)$ use of the triple index sequence of
rational numbers defined by the recursion

$\displaystyle c(\alpha,\beta,\gamma)$
$\displaystyle:=-\frac{1}{3}(2\alpha+8\beta+12\gamma+1)\cdot
c(\alpha-1,\beta,\gamma)+\frac{2}{3}(\alpha+1)\cdot
c(\alpha+1,\beta-1,\gamma)$ (1.7) $\displaystyle\qquad\qquad\ \ \ \ \
+\frac{8}{3}(\beta+1)\cdot c(\alpha,\beta+1,\gamma-1)+4(\gamma+1)\cdot
c(\alpha,\beta-2,\gamma+1),$

where $\alpha,\beta,\gamma\geq 0.$ To seed the recursion, we let
$c(0,0,0):=1,$ and we let $c(\alpha,\beta,\gamma):=0$ if any of the arguments
are negative. Here we list the “first few” values:

$\displaystyle c(1,0,0)=-1,\ c(0,1,0)=-\frac{2}{3},\ c(0,0,1)=-\frac{16}{9},\
c(1,1,0)=\frac{14}{3},\ c(1,0,1)=\frac{64}{3},\dots.$

We also require constants for the quasimodular summands sorted by weight. For
$0\leq t\leq a$, define

(1.8) $\displaystyle
w_{t}(a):=\frac{\binom{2a}{a}}{16^{a}(2a+1)}\sum_{0\leq\ell_{1}<\cdots<\ell_{t}<a}\prod_{j=1}^{t}\frac{1}{(2\ell_{j}+1)^{2}}.$

In terms of $w_{t}(a)$ and the numbers $c(\alpha,\beta,\gamma)$, we have the
following explicit formulae for $\mathcal{U}_{a}(q).$

######  Theorem 1.3.

If $a$ is a non-negative integer, then we have that

$\mathcal{U}_{a}(q)=\sum_{t=0}^{a}w_{t}(a)\sum_{\begin{subarray}{c}\alpha,\beta,\gamma\geq
0\\\
\alpha+2\beta+3\gamma=t\end{subarray}}c(\alpha,\beta,\gamma)E_{2}(q)^{\alpha}E_{4}(q)^{\beta}E_{6}(q)^{\gamma}.$

######  Example.

For $a=3$, Theorem 1.3 gives

$\begin{split}\mathcal{U}_{3}(q)&=\frac{5}{7168}-\frac{37E_{2}(q)}{46080}+\frac{5E_{2}(q)^{2}}{27648}-\frac{E_{4}(q)}{13824}-\frac{E_{2}(q)^{3}}{82944}+\frac{E_{2}(q)E_{4}(q)}{69120}-\frac{E_{6}(q)}{181440}.\end{split}$

We turn to the $\mathcal{U}_{a}^{*}(q).$ Instead of using $E_{2}(q),E_{4}(q),$
and $E_{6}(q),$ we use all of the Eisenstein series

(1.9)
$E_{2k}(q):=1-\frac{4k}{B_{2k}}\sum_{n=1}^{\infty}\sigma_{2k-1}(n)q^{n},$

where $B_{k}$ is the usual $k$th Bernoulli number. Namely, we let
$\mathbb{E}_{0}(q):=1,$ and for positive $t$ we define

(1.10)
$\displaystyle\mathbb{E}_{2t}^{\star}(q):=\sum_{(1^{m_{1}},\dots,t^{m_{t}})\vdash
t}\,\prod_{j=1}^{t}\frac{1}{m_{j}!}\left(-\frac{B_{2j}\,E_{2j}(q)}{(2j)\cdot(2j!)}\right)^{m_{j}}.$

We require constants for the summands sorted by weight. We let
$w_{0}^{\star}(0):=1$, and for $a>0,$ we let

(1.11)
$w_{0}^{\star}(a):=\sum_{i=1}^{a}\frac{(-1)^{i-1}\binom{2i}{i}}{16^{i}(2i+1)}\,w_{0}^{\star}(a-i).$

For $1\leq t\leq a$, we define

(1.12) $w_{t}^{\star}(a):=(-1)^{a+t-1}4^{t}(2t+1)!\,w_{t-1}(a-1).$

With this notation, we obtain the following explicit expressions for
$\mathcal{U}_{a}^{*}(q)$.

######  Theorem 1.4.

If $a$ is a non-negative integer, then we have that

$\mathcal{U}_{a}^{\star}(q)=\sum_{t=0}^{a}w_{t}^{\star}(a)\cdot\mathbb{E}_{2t}^{\star}(q).$

######  Example.

For $a=5$, Theorem 1.4 gives

$\begin{split}\mathcal{U}_{5}^{*}(q)&=\frac{1295803}{12262440960}+\frac{35}{294912}\mathbb{E}_{2}^{\star}(q)-\frac{3229}{967680}\mathbb{E}_{4}^{\star}(q)+\frac{47}{1152}\mathbb{E}_{6}^{\star}(q)-\frac{7}{24}\mathbb{E}_{8}^{\star}(q)+\mathbb{E}_{10}^{\star}(q).\end{split}$

The coefficients of $\mathcal{U}_{a}(q)$ and $U_{a}^{\star}(q)$ satisfy
surprising congruences. Amdeberhan-Andrews-Tauraso [1] discovered some
congruences that are reminiscent of Ramanujan’s partition congruences, such as

$\begin{split}MO(2;5n+2)\equiv 0\pmod{5}\ \ \ \ \ {\text{\rm and}}\ \ \ \
MO(3;7n+3)\equiv MO(3;7n+5)\equiv 0\pmod{7}.\end{split}$

Moreover, they conjectured (see Conjecture 9.1 of [1]) that

(1.13) $MO(10;11n+7)\equiv 0\pmod{11}.$

######  Theorem 1.5.

For every non-negative integer $n$, we have that

$MO(10;11n+7)\equiv 0\pmod{11}.$

We offer two proofs of this result. The first proof uses the explicit
description of $\mathcal{U}_{10}(q)$ provided by Theorem 1.3, which allows us
to employ the “theory of modular forms mod $p$”. This proof illustrates an
algorithm that reduces the proof of all conjectured congruences of the form

$MO(a;pn+r)\equiv 0\pmod{p}\ \ \ \ {\text{\rm and}}\ \ \ \ M(a;pn+r)\equiv
0\pmod{p}$

to finitely many steps. Theorem 1.5 requires computing at most 20 terms of
five auxiliary $q$-series.

The second proof is a special case of one of three new infinite families of
congruences.

######  Theorem 1.6.

The following are true:

(i) For every pair of non-negative integers $n$ and $m,$ we have that

$MO(3m+2;3n+1)\equiv MO(3m+2;3n+2)\equiv 0\pmod{3}.$

(ii) For every pair of non-negative integers $n$ and $m,$ we have

$MO(11m+10;11n+7)\equiv 0\pmod{11}.$

(iii) For every pair of non-negative integers $n$ and $m,$ we have

$MO(17m+16;17n+15)\equiv 0\pmod{17}.$

Computer searches for congruences suggest that such congruences are rare,
thereby underscoring the significance of Theorem 1.5. However, it turns out
that congruences are both rare and ubiquitous.

######  Theorem 1.7.

For positive integers $a$ and $m$, the following are true:

(i) There are infinitely many non-nested arithmetic progressions $tn+r$ (resp.
$t^{*}n+r^{*}$) for which

$\begin{split}M(a;tn+r)&\equiv 0\pmod{m},\\\ MO(a;t^{*}n+r^{*})&\equiv
0\pmod{m}.\end{split}$

(ii) There are infinitely many non-nested arithmetic progressions $tn+r$ for
which

$M(a;tn+r)\equiv MO(a;tn+r)\equiv 0\pmod{m}.$

(iii) There exists a positive real number $\alpha(a,m)>0$ for which

$\begin{split}\\#\\{n\leq X\ :\ M(a;n)\not\equiv
0\pmod{m}\\}&=O\left(X/\log^{\alpha(a,m)}X\right)\\\ \\#\\{n\leq X\ :\
MO(a;n)\not\equiv
0\pmod{m}\\}&=O\left(X/\log^{\alpha(a,m)X}\right).\end{split}$

In other words, the values $M(a;n)$ and $MO(a;n)$ are almost always multiples
of any integer $m.$

To conclude, we offer infinite families of congruences when
$a\in\\{2,3,4,5\\}$. For convenience, we let

$\displaystyle N_{a}:=\begin{cases}2^{3}&\mbox{if $a=2$},\\\ 2^{7}3\cdot
5&\mbox{if $a=3$},\\\ 2^{10}3^{3}\cdot 5\cdot 7&\mbox{if $a=4$},\\\
2^{15}3^{3}5^{2}\cdot 7&\mbox{if $a=5$}.\end{cases}$

######  Corollary 1.8.

If $a\in\\{2,3,4,5\\},$ then the following are true:

(i) If $\ell\in\\{2,3,5,7\\}$ and
$p\equiv-1\pmod{\ell^{\mathrm{o}rd_{\ell}(N_{a})+1}}$ is prime, then for every
$n$ coprime to $p$ we have

$MO(a;pn)\equiv 0\pmod{\ell}.$

(ii) If $\ell\geq 11$ is prime and $p\equiv-1\pmod{\ell}$, then for every
integer $n$ coprime to $p$ we have

$MO(a;pn)\equiv 0\pmod{\ell}.$

######  Example.

The following congruence is an example of Corollary 1.8 $(i):$

$\begin{split}MO(2,19^{2}n+19)&\equiv MO(2,19^{2}n+38)\equiv
MO(2,19^{2}n+57)\equiv MO(2,19^{2}n+76)\equiv 0\pmod{5},\\\ \end{split}$

As an example of Corollary 1.8 $(ii)$, for $1\leq t\leq 18$, we have

$\begin{split}MO(a,37^{2}n+37t)&\equiv 0\pmod{19}.\\\ \end{split}$

######  Remark.

Most of the congruences in Theorem 1.7 do not belong to infinite families such
as those in Corollary 1.8. For instance, if $p\in\\{67,101,271,373\\}$, then
for every non-negative integer $n$ we have

$M(6;pn)\equiv MO(6;pn)\equiv 0\pmod{17}.$

The coefficients of the expansion of $\mathcal{U}_{6}(q)$ provided by Theorem
1.3 are units modulo 17, and so these congruences follow from the fact that
all of the monomials $E_{2}(q)^{\alpha}E_{4}(q)^{\beta}E_{6}(q)^{\gamma},$
with $\alpha,\beta,\gamma\geq 0$ and $\alpha+2\beta+3\gamma\leq 6$, are
annihilated modulo 17 by the Hecke operators $T_{p}$ for
$p\in\\{67,101,271,373\\}$.

This paper is organized as follows. In Section 2, we recall the Nekrasov-
Okounkov hook formulae and relevant results of Andrews-Rose and Amdeberhan-
Andrews-Tauraso, which we then employ to prove Theorems 1.1 and 1.2 on the
limiting behavior of $U_{a}(q)$ and $U_{a}^{\star}(q)$. In Section 3 we recall
pertinent facts about symmetric functions, as well as results on the
quasimodularity of $U_{a}(q)$, which we then use to prove Theorems 1.3 and
1.4. Finally, in Section 4 we prove Theorems 1.5 and 1.6, and in Section 5 we
prove Theorem 1.7 using modularity.

## Acknowledgements

The second author thanks the Thomas Jefferson Fund and the NSF (DMS-2002265
and DMS-2055118). The third author is grateful for the support of a Fulbright
Nehru Postdoctoral Fellowship.

##  2\. Proofs of Theorems 1.1 and 1.2

Here we prove Theorems 1.1 and 1.2 using earlier work of Nekrasov-Okounkov and
Andrews-Rose.

###  2.1. Proof of Theorem 1.1

We require a beautiful identity of Andrews-Rose for $\mathcal{U}_{a}(q)$.

######  Lemma 2.1.

[2, Cor. 2] If $a$ is a positive integer, then as formal power series we have
that

$\mathcal{U}_{a}(q)\cdot\prod_{n\geq
1}(1-q^{n})^{3}=\frac{(-1)^{a}}{(2a+1)!}\,\sum_{n\geq
0}(-1)^{n}(2n+1)\frac{(n+a)!}{(n-a)!}\,q^{\frac{n(n+1)}{2}}.$

We also require the celebrated Nekrasov-Okounkov hook length identity (see
(6.12) on page 569 of [8]; see also Th. 1.3 of [5]).

######  Theorem 2.2.

As a formal power series, we have

$\prod_{j\geq 1}\frac{1}{(1-q^{j})^{z+1}}=\sum_{m\geq
0}q^{m}\,\sum_{\lambda\vdash
m}\prod_{h\in\mathcal{H}(\lambda)}\left(\frac{z}{h^{2}}+1\right).$

###### Proof of Theorem 1.1.

Thanks to Lemma 2.1 for $\mathcal{U}_{a}(q),$ we find that

$\displaystyle q^{-\binom{a+1}{2}}\cdot\mathcal{U}_{a}(q)\cdot\prod_{n\geq
1}(1-q^{n})^{3}$ $\displaystyle=\sum_{j\geq
0}(-1)^{j}\cdot\frac{2j+2a+1}{2a+1}\cdot\binom{j+2a}{j}\,q^{aj+\binom{j+1}{2}}$
$\displaystyle=1-(2a+3)q^{a+1}+(a+1)(2a+5)q^{2a+3}+\cdots\,\,.$

Claim $(i)$ follows immediately.

The first formula in $(ii)$ follows by letting $z=2$ in Theorem 2.2, giving

$\prod_{n\geq 1}\frac{1}{(1-q^{n})^{3}}=\sum_{m\geq
0}q^{m}\,\sum_{\lambda\vdash
m}\,\prod_{h\in\mathcal{H}(\lambda)}\left(\frac{2}{h^{2}}+1\right),$

while the other claim arises from the interpretation of the $q$-product in
terms of 3-colored partitions. ∎

###  2.2. Proof of Theorem 1.2

Amdeberhan-Andrews-Tauraso express $\mathcal{U}_{a}^{\star}(q)$ as a single
sum.

######  Lemma 2.3.

[1, Prop. 4.1] We have the identity

$\mathcal{U}_{a}^{*}(q)=\sum_{k\geq
1}(-1)^{k-1}\,\frac{(1+q^{k})\,q^{\binom{k}{2}+ak}}{(1-q^{k})^{2a}}.$

###### Proof of Theorem 1.2.

The expansion $(1-q^{k})^{-2a}=\sum_{m\geq 0}\binom{2a+m-1}{m}\,q^{km}$ and
Lemma 2.3 imply that

$\displaystyle q^{-a}\cdot\mathcal{U}_{a}^{\star}(q)$
$\displaystyle=\sum_{k\geq
1}(-1)^{k-1}\,\frac{q^{\binom{k}{2}+a(k-1)}}{(1-q^{k})^{2a}}+\sum_{k\geq
1}(-1)^{k-1}\,\frac{q^{\binom{k+1}{2}+a(k-1)}}{(1-q^{k})^{2a}}$
$\displaystyle=\sum_{k\geq 1}\sum_{m\geq
0}(-1)^{k-1}\binom{2a+m-1}{m}\,q^{km+\binom{k}{2}+a(k-1)}+\sum_{k\geq
1}\sum_{m\geq 0}\binom{2a+m-1}{m}\,q^{km+\binom{k+1}{2}+a(k-1)}.$

We compare coefficients of $q^{n}$ for $n\leq a$. Namely, in the double sums
we require $km+\binom{k}{2}+a(k-1)\leq a$ and $km+\binom{k+1}{2}+a(k-1)\leq
a$. The former results in $k=1,m=n$ and the latter forces $k=1,m=n-1$.
Consequently, if we let $q^{-a}\cdot\mathcal{U}_{a}^{\star}(q)=:\sum_{n\geq
0}p_{n}(a)\,q^{n}$, then we find that

$p_{n}(a)=\binom{2a+n-1}{n}+\binom{2a+n-2}{n-1}.$

∎

##  3\. Proof of Theorems 1.3 and 1.4

Here we prove the explicit descriptions of $\mathcal{U}_{a}(q)$ and
$\mathcal{U}_{a}^{*}(q)$ in terms of Eisenstein series.

###  3.1. Nuts and Bolts

We make use of the differential operator $\Theta:=q\frac{d}{dq}$, which acts
by

(3.1) $\Theta\left(\sum a(n)q^{n}\right):=\sum na(n)q^{n}.$

Ramanujan famously obtained the following formulas [11, p. 181] for the action
of $\Theta$:

(3.2) $\displaystyle\Theta(E_{2}(q))$
$\displaystyle=\frac{E_{2}^{2}(q)-E_{4}(q)}{12},\qquad\Theta(E_{4}(q))=\frac{E_{2}(q)E_{4}(q)-E_{6}(q)}{3},$
$\displaystyle\Theta(E_{6}(q))$
$\displaystyle=\frac{E_{2}(q)E_{6}(q)-E_{4}^{2}(q)}{2}.$

The $q$-series $\mathcal{U}_{a}(q)$ and $\mathcal{U}_{a}^{\star}(q)$ satisfy
the following convenient convolution (see [1, p. 13]).

######  Lemma 3.1.

If $a$ is a positive integer, then we have that

$\sum_{i=0}^{a}(-1)^{i}\cdot\mathcal{U}_{i}(q)\cdot\mathcal{U}_{a-i}^{\star}(q)=0.$

Recall the Dedekind eta-function
$\eta(q)=q^{\frac{1}{24}}\prod_{m=1}^{\infty}(1-q^{m}).$ The following result
of Rose [12, Th. 1.12] describes the structural framework of
$\mathcal{U}_{a}(q)$ in terms of iterated derivatives of $\eta(q)^{3}.$

######  Theorem 3.2.

Each $\mathcal{U}_{a}(q)$ is a finite sum of quasimodular forms with weight
$\leq 2a$ on $\mathrm{SL}_{2}(\mathbb{Z})$. Moreover, the weight $2t$ summand
is a (possibly zero) scalar multiple of

$2^{t}\cdot\frac{\Theta^{t}\left(\eta(q)^{3}\right)}{\eta(q)^{3}}.$

Our next result expresses these $q$-series as a linear combination of
monomials $E_{2}(q)^{\alpha}E_{4}(q)^{\beta}E_{6}(q)^{\gamma}$.

######  Lemma 3.3.

If $t$ is a positive integer, then we have that

$(-8)^{t}\cdot\frac{\Theta^{t}\left(\eta(q)^{3}\right)}{\eta(q)^{3}}=\sum_{\begin{subarray}{c}\alpha,\beta,\gamma\geq
0\\\ \alpha+2\beta+3\gamma=t\end{subarray}}c(\alpha,\beta,\gamma)\cdot
E_{2}(q)^{\alpha}E_{4}(q)^{\beta}E_{6}(q)^{\gamma}$

where the coefficients $c(\alpha.\beta,\gamma)$ are defined by (1).

###### Proof.

For convenience, we let $\psi(q):=\eta(q)^{3}.$ We calculate
$\frac{\Theta^{t}(\psi(q))}{\psi(q)}$ by inducting on $t$. First, it is easy
to check $\Theta(\psi(q))=\frac{1}{8}\psi(q)E_{2}(q)$. Theorem 3.2 implies the
existence of numbers $\widetilde{c}(\alpha,\beta,\gamma)$ for which

$\frac{\Theta^{t}(\psi(q))}{\psi(q)}=\sum_{\begin{subarray}{c}\alpha,\beta,\gamma\geq
0\\\
\alpha+2\beta+3\gamma=t\end{subarray}}\widetilde{c}(\alpha,\beta,\gamma)\cdot
E_{2}^{\alpha}(q)E_{4}^{\beta}(q)E_{6}^{\gamma}(q).$

This comprises of all weight $2t$ quasimodular summands in
$\mathcal{U}_{a}(q)$. One more derivative $\Theta=q\frac{d}{dq}$ turns the
last equation into (for brevity, we write $\widetilde{c}$ in place of
$\widetilde{c}(\alpha,\beta,\gamma)$)

$\displaystyle\Theta^{t+1}(\psi(q))$
$\displaystyle=\Theta(\psi(q))\cdot\left(\sum_{\alpha,\beta,\gamma}\widetilde{c}\cdot
E_{2}^{\alpha}(q)E_{4}^{\beta}(q)E_{6}^{\gamma}(q)\right)+\psi(q)\cdot\sum_{\alpha,\beta,\gamma}\widetilde{c}\cdot\Theta(E_{2}^{\alpha}(q)E_{4}^{\beta}(q)E_{6}^{\gamma}(q)).$

On the other hand, Ramanujan’s identities (3.2) imply that

$\displaystyle\Theta(E_{2}^{\alpha}E_{4}^{\beta}E_{6}^{\gamma})$
$\displaystyle=\left(\frac{\alpha}{12}+\frac{\beta}{3}+\frac{\gamma}{2}\right)E_{2}^{\alpha+1}E_{4}^{\beta}E_{6}^{\gamma}-\frac{\alpha}{12}E_{2}^{\alpha-1}E_{4}^{\beta+1}E_{6}^{\gamma}-\frac{\beta}{3}E_{2}^{\alpha}E_{4}^{\beta-1}E_{6}^{\gamma+1}-\frac{\gamma}{2}E_{2}^{\alpha}E_{4}^{\beta+2}E_{6}^{\gamma-1}.$

We find that the homogeneous weight $2t+2$ form satisfies

$\displaystyle\frac{\Theta^{t+1}(\psi(q))}{\psi(q)}$
$\displaystyle=\sum_{\begin{subarray}{c}\alpha,\beta,\gamma\geq 0\\\
\alpha+2\beta+3\gamma=t\end{subarray}}\left(\frac{\alpha}{12}+\frac{\beta}{3}+\frac{\gamma}{2}+\frac{1}{8}\right)\widetilde{c}\cdot
E_{2}^{\alpha+1}E_{4}^{\beta}E_{6}^{\gamma}-\sum_{\alpha,\beta,\gamma}\frac{\alpha}{12}\,\widetilde{c}\cdot
E_{2}^{\alpha-1}E_{4}^{\beta+1}E_{6}^{\gamma}$
$\displaystyle\qquad-\sum_{\alpha,\beta,\gamma}\frac{\beta}{3}\,\widetilde{c}\cdot
E_{2}^{\alpha}E_{4}^{\beta-1}E_{6}^{\gamma+1}-\sum_{\alpha,\beta,\gamma}\frac{\gamma}{2}\,\widetilde{c}\cdot
E_{2}^{\alpha}E_{4}^{\beta+2}E_{6}^{\gamma-1}.$

By comparing the coefficients of $E_{2}^{\alpha}E_{4}^{\beta}E_{6}^{\gamma}$
on both sides of the equation above, we obtain the recursion (with
$\widetilde{c}(\alpha,\beta,\gamma)=\delta_{(0,0,0)}(\alpha,\beta,\gamma)$, a
Dirac delta boundary conditions)

$\displaystyle\widetilde{c}(\alpha,\beta,\gamma)$
$\displaystyle=\left(\frac{\alpha}{12}+\frac{\beta}{3}+\frac{\gamma}{2}+\frac{1}{24}\right)\widetilde{c}(\alpha-1,\beta,\gamma)-\frac{\alpha+1}{12}\cdot\widetilde{c}(\alpha+1,\beta-1,\gamma)$
$\displaystyle\qquad-\frac{\beta+1}{3}\cdot\widetilde{c}(\alpha,\beta+1,\gamma-1)-\frac{\gamma+1}{2}\cdot\widetilde{c}(\alpha,\beta-2,\gamma+1).$

To determine the exact weight $2t$ term (independent of $a$), we take into
account the factor of $(-8)^{\alpha+2\beta+3\gamma}$ to determine
$c(\alpha,\beta,\gamma):=(-8)^{\alpha+2\beta+3\gamma}\cdot\widetilde{c}(\alpha,\beta,\gamma)$.
As a result, we obtain the desired

$\displaystyle c(\alpha,\beta,\gamma)$
$\displaystyle=-\frac{1}{3}\left(2\alpha+8\beta+12\gamma+1\right)\cdot
c(\alpha-1,\beta,\gamma)+\frac{2}{3}(\alpha+1)\cdot
c(\alpha+1,\beta-1,\gamma)$ $\displaystyle\qquad+\frac{8}{3}(\beta+1)\cdot
c(\alpha,\beta+1,\gamma-1)+4(\gamma+1)\cdot c(\alpha,\beta-2,\gamma+1).$

∎

###  3.2. Proof of Theorem 1.3

We let $\mathcal{E}_{t}(q):=(-8)^{t}\cdot\frac{\Theta^{t}(\psi)}{\psi},$ and
we define

$\mathbb{E}_{2t}(q):=\sum_{(1^{m_{1}},\dots,t^{m_{t}})\vdash
t}\,\prod_{j=1}^{t}\frac{1}{m_{j}!}\left(\frac{B_{2j}\,E_{2j}(q)}{(2j)\cdot(2j)!}\right)^{m_{j}}.$

By inspection, we see that $\mathbb{E}_{2t}(q)$ has weight $2t.$ We claim that

(3.3)
$\displaystyle\mathbb{E}_{2t}(q)=\frac{(-1)^{t}}{4^{t}(2t+1)!}\cdot\mathcal{E}_{t}(q).$

Let $\mathbf{S}_{r}(q):=\sum_{m\geq 1}\frac{m^{r}q^{m}}{1-q^{m}}=\sum_{n\geq
1}\sigma_{r}(n)q^{n}$. By expanding $\sum_{j,k\geq
1}\frac{q^{kj}\cos(2kx)}{k}$ in two different ways, we find that it equals
both of these

(3.4) $\displaystyle\prod_{j\geq
1}\left[1+\frac{4(\sin^{2}x)q^{j}}{(1-q^{j})^{2}}\right]$
$\displaystyle=\exp\left(-2\sum_{r\geq
1}\frac{\mathbf{S}_{2r-1}(q)}{(2r)!}(-4x^{2})^{r}\right).$

Using the identity [1, p. 13], we obtain

(3.5) $\prod_{k\geq
1}\left(1+\frac{4q^{k}\sin^{2}x}{(1-q^{k})^{2}}\right)=\sum_{a\geq
0}4^{a}\mathcal{U}_{a}(q)(\sin x)^{2a},$

and the Jacobi Triple Product then implies that

$\displaystyle\sin x\prod_{k\geq
1}\left(1+\frac{4q^{k}\sin^{2}x}{(1-q^{k})^{2}}\right)$
$\displaystyle=\frac{e^{ix}-e^{-ix}}{2i}\prod_{j\geq
1}\frac{(1-q^{j}e^{2ix})(1-q^{j}e^{-2ix})}{(1-q^{j})^{2}}$
$\displaystyle=\frac{1}{2i\cdot\psi(q)}\sum_{j\in\mathbb{Z}}(-1)^{j}q^{\binom{j+1}{2}}e^{(2n+1)ix}$
$\displaystyle=\frac{1}{\psi(q)}\sum_{t\geq
0}(-1)^{t}\frac{x^{2t+1}}{(2t+1)!}\sum_{n\geq
0}(-1)^{n}(2n+1)^{2t+1}q^{\binom{n+1}{2}}$ (3.6) $\displaystyle=\sum_{t\geq
0}\mathcal{E}_{t}(q)\frac{x^{2t+1}}{(2t+1)!}.$

Using (1.9) and the generating function for Pólya’s _cycle index formula_ [10,
(1,5)], we obtain

(3.7) $\displaystyle\sin x\cdot\exp\left(-2\sum_{r\geq
1}\frac{\mathbf{S}_{2r-1}(q)}{(2r)!}\,(-4x^{2})^{2r}\right)$
$\displaystyle=\sin x\cdot\frac{x}{\sin x}\cdot\sum_{t\geq
0}\left(\sum_{\lambda\vdash
t}\prod_{j=1}^{t}\frac{1}{m_{j}!}\left(\frac{B_{2j}\cdot
E_{2s}(q)}{(2j)\cdot(2j!}\right)^{m_{j}}\right)(-4x^{2})^{t}.$

Combining (3.4), (3.2), (3.7) and then comparing the coefficients of
$x^{2t+1},$ we confirm (3.3).

To the complete the proof, it suffices to determine the constants $b_{t}(a)$
for which

(3.8) $\displaystyle\mathcal{U}_{a}(q)$
$\displaystyle=\sum_{t=0}^{a}b_{t}(a)\cdot\mathbb{E}_{2t}(q).$

It is convenient to recall the Andrews-Rose recursion [2, Cor. 3]

(3.9) $\displaystyle\mathcal{U}_{a}(q)$
$\displaystyle=\frac{1}{2a(2a+1)}\left[(6\mathcal{U}_{1}(q)+a(a-1))\mathcal{U}_{a-1}(q)-2\Theta(\mathcal{U}_{a-1}(q))\right].$

The structure of equation (3.8) is preserved by (3.9) because of the identity

$\Theta(\mathbb{E}_{2t-2})=t(2t+1)\mathbb{E}_{2t}-3\mathbb{E}_{2}\mathbb{E}_{2t-2}.$

It is straightforward to see that

$b_{t}(a)=\frac{1}{8a(2a+1)}\left[(2a-1)^{2}\cdot b_{t}(a-1)-8t(2t+1)\cdot
b_{t-1}(a-1)\right],$

with initial boundary conditions $b_{0}(0)=1$ and $b_{t}(a)=0$ when $t<0$ or
$t>a$. Finally, one checks that $(-4)^{t}(2t+1)!\,w_{t}(a)$ satisfies this
recurrence, thereby completing the proof of the theorem.

###  3.3. Proof of Theorem 1.4

By reciprocating (3.5), we have

$\sum_{n\geq 0}(-4)^{a}\,\mathcal{U}_{a}^{\star}(q)(\sin x)^{2a}=\prod_{k\geq
1}\frac{1}{1+\frac{4q^{k}\sin^{2}x}{(1-q^{k})^{2}}}.$

In analogy with the previous formula for $\mathbb{E}_{2t}(q)$ involving the
$\mathcal{U}_{a}(q)$, we use (3.3) to obtain an identity for
$\mathcal{U}_{a}^{\star}(q)$ with $\mathbb{E}_{2t}^{*}(q)$ (see (1.10)).
Arguing as in the proof of Theorem 1.3 with Lemma 3.1, we get

$\mathcal{U}_{a}^{\star}(q)=\sum_{t=0}^{a}w_{t}^{\star}(a)\cdot\mathbb{E}_{2t}^{\star}(q).$

##  4\. Proof of Theorems 1.5 and 1.6

Here we prove Theorem 1.5 using Serre’s theory of modular forms modulo primes
$p$ (see [9, Section 2.8], or [17]) and a well-known criterion of Sturm that
determines congruences between modular forms. In the sequel, we tacitly assume
that $q:=e^{2\pi iz},$ the uniformizer for the point at infinity. We also
prove Theorem 1.6 by combining work of Andrews-Rose with a classical result of
Gordon, together with other allied observations.

###  4.1. Modular forms modulo $p$

We recall some facts from the theory of modular forms mod $p$. The key tool in
the proof of Theorem 1.5 is the following theorem of Sturm (see [16] or p. 40
of [9]).

######  Theorem 4.1.

Let $p$ be a prime. If $f(z)=\sum_{n\geq 0}a(n)q^{n}$ and $g(z)=\sum_{n\geq
0}b(n)q^{n}$ are modular forms of weight $k$ on $\mathrm{SL}_{2}(\mathbb{Z})$
with integer coefficients, then $f(z)\equiv g(z)\pmod{p}$ if and only if
$a(n)\equiv b(n)\pmod{p}$ for all $n\leq k/12$.

We shall make use of derivatives of modular forms. Although differentiation
does not preserve modularity, it does preserve modular forms modulo $p$ (for
example, see [13]).

######  Lemma 4.2.

If $f(z)=\sum_{n\geq 0}a(n)q^{n}\in M_{k}\cap\mathbb{Z}[[q]]$, then there is a
modular form $g(z)=\sum_{n\geq 0}b(n)q^{n}\in M_{k+p+1}\cap\mathbb{Z}[[q]]$
for which

$g\equiv\Theta(f):=\sum_{n\geq 0}na(n)q^{n}\pmod{p}.$

###  4.2. Proof of Theorem 1.5

We let $\mathcal{U}_{10}(q)=F_{0}(q)+F_{2}(q)+F_{4}(q)+F_{6}(q)+F_{8}(q),$
where each $F_{2i}(q)$ is a sum of $E_{2}^{\alpha}E_{4}^{\beta}E_{6}^{\gamma}$
(suppressing the $q$), where $2\alpha+4\beta+6\gamma\equiv 2i\pmod{10}.$
Theorem 1.3 then gives

$\begin{split}F_{0}(q)=&\frac{46189}{5772436045824}-\frac{2008213E_{4}E_{6}}{4271802792542208000}+\dots+\frac{E_{2}^{10}}{230078188847156428800},\\\
F_{2}(q)=&-\frac{25587296781661E_{2}}{2645567198945303592960}-\frac{604841E_{6}^{2}}{48057781416099840000}+\dots+\frac{7862933E_{2}^{6}}{63910608013099008000},\\\
F_{4}(q)=&-\frac{79923511502753E_{4}}{67133754108574433280000}+\frac{79923511502753E_{2}^{2}}{26853501643429773312000}+\dots-\frac{16333E_{2}^{7}}{4473742560916930560},\\\
F_{6}(q)=&-\frac{70726885883E_{6}}{333200617818292224000}+\frac{70726885883E_{2}E_{4}}{126933568692682752000}-\dots+\frac{1819E_{2}^{8}}{25564243205239603200},\\\
F_{8}(q)=&-\frac{316100258731E_{4}^{2}}{20732482886471516160000}+\frac{316100258731E_{2}E_{6}}{3887340541213409280000}-\dots-\frac{19E_{2}^{9}}{23007818884715642880}.\end{split}$

Each of these $q$-series is 11-integral, and so they may be reduced modulo 11
to obtain

$\begin{split}&\widehat{F}_{0}(q):=F_{0}(q)\\!\\!\\!\\!\pmod{11}\equiv
2q^{3}+6q^{4}+7q^{5}+8q^{6}+5q^{7}+2q^{8}+2q^{9}+\dots\,\,\,\,\,\,\pmod{11},\\\
&\widehat{F}_{2}(q):=F_{2}(q)\\!\\!\\!\\!\pmod{11}\equiv
6q^{3}+7q^{4}+10q^{5}+7q^{6}+8q^{7}+7q^{8}+6q^{9}+\dots\,\,\,\pmod{11},\\\
&\widehat{F}_{4}(q):=F_{4}(q)\\!\\!\\!\\!\pmod{11}\equiv
7q^{3}+10q^{4}+8q^{5}+2q^{6}+4q^{8}+7q^{9}+\dots\,\,\,\,\,\,\,\ \ \ \ \ \
\pmod{11},\\\ &\widehat{F}_{6}(q):=F_{6}(q)\\!\\!\\!\\!\pmod{11}\equiv
10q^{3}+8q^{4}+9q^{5}+10q^{6}+q^{7}+4q^{8}+10q^{9}+\dots\pmod{11},\\\
&\widehat{F}_{8}(q):=F_{8}(q)\\!\\!\\!\\!\pmod{11}\equiv
8q^{3}+2q^{4}+6q^{5}+6q^{6}+8q^{7}+5q^{8}+8q^{9}+\dots\,\,\,\,\,\,\pmod{11}.\end{split}$

Using the congruences $E_{2}(q)\equiv E_{12}(q)\pmod{11}$ and $E_{10}(q)\equiv
1\pmod{11}$, we observe that
$\widehat{F}_{0}(q),\widehat{F}_{2}(q),\widehat{F}_{4}(q),\widehat{F}_{6}(q),$
and $\widehat{F_{8}}(q)$ are modular forms modulo $11$ of weight
$120,72,84,96,$ and $108$, respectively, on $\mathrm{SL}_{2}(\mathbb{Z}).$

We proceed to isolate the arithmetic progression of coefficients that is
relevant for the theorem. We apply the differential operators to
$\mathcal{U}_{10}(q)$ to eliminate terms with exponents $n\equiv
0,1,3,4,5,9\pmod{11}.$ The non-zero classes are the quadratic residues modulo
11. Using Fermat’s Little Theorem and Euler’s Criterion, this is achieved by

$\begin{split}G_{1}(q):&\equiv\sum_{n\equiv
2,6,7,8,10\pmod{11}}MO(10;n)q^{n}\equiv\sum_{i=0}^{4}-5[\Theta^{10}(\widehat{F}_{2i}(q))-\Theta^{5}(\widehat{F}_{2i}(q))]\pmod{11}.\end{split}$

Next, we proceed to remove the terms with exponents that are quadratic non-
residues apart from those with $n\equiv 7\pmod{11}$. For instance, to
eliminate $n\equiv 2\pmod{11}$ from $G_{1}(q)$ compute

$\begin{split}G_{2}(q):=\Theta(G_{1}(q))-2G_{1}(q)&\equiv\sum_{n\equiv
6,7,8,10\pmod{11}}MO(10;n)q^{n}\pmod{11}.\end{split}$

We repeat this process to remove terms with exponents $n\equiv
6,8,10\pmod{11},$ and we get

$\begin{split}\sum_{n\equiv 7\pmod{11}}&MO(10;n)q^{n}\pmod{11}\\\
&\equiv-5(\Theta^{4}(\widehat{F}_{2})-\Theta^{9}(\widehat{F}_{2}))+9(\Theta^{3}(\widehat{F}_{4})-\Theta^{8}(\widehat{F}_{4}))-3(\Theta^{2}(\widehat{F}_{6})-\Theta^{7}(\widehat{F}_{6}))+\Theta(\widehat{F}_{8})\\\
&\ \ \ \
-\Theta^{6}(\widehat{F}_{8})-4(\widehat{F}_{0}-\Theta^{5}(\widehat{F}_{0}))-5(\Theta^{4}(\widehat{F}_{4})-\Theta^{9}(\widehat{F}_{4}))+9(\Theta^{3}(\widehat{F}_{6})-\Theta^{8}(\widehat{F}_{6}))\\\
&\ \ \ \
-3(\Theta^{2}(\widehat{F}_{8})-\Theta^{7}(\widehat{F}_{8}))+\Theta(\widehat{F}_{0})-\Theta^{6}(\widehat{F}_{0})-4(\widehat{F}_{2}-\Theta^{5}(\widehat{F}_{2}))-5(\Theta^{4}(\widehat{F}_{6})-\Theta^{9}(\widehat{F}_{6}))\\\
&\ \ \ \
+9(\Theta^{3}(\widehat{F}_{8})-\Theta^{8}(\widehat{F}_{8}))-3(\Theta^{2}(\widehat{F}_{0})-\Theta^{7}(\widehat{F}_{0}))+\Theta(\widehat{F}_{2})-\Theta^{6}(\widehat{F}_{2})-4(\widehat{F}_{4}-\Theta^{5}(\widehat{F}_{4}))\\\
&\ \ \ \
-5(\Theta^{4}(\widehat{F}_{8})-\Theta^{9}(\widehat{F}_{8}))+9(\Theta^{3}(\widehat{F}_{0})-\Theta^{8}(\widehat{F}_{0}))-3(\Theta^{2}(\widehat{F}_{2})-\Theta^{7}(\widehat{F}_{2}))+\Theta(\widehat{F}_{4})\\\
&\ \ \ \
-\Theta^{6}(\widehat{F}_{4})4(\widehat{F}_{6}-\Theta^{5}(\widehat{F}_{6}))-5(\Theta^{4}(\widehat{F}_{0})-\Theta^{9}(\widehat{F}_{0}))+9(\Theta^{3}(\widehat{F}_{2})-\Theta^{8}(\widehat{F}_{2}))\\\
&\ \ \ \
-3(\Theta^{2}(\widehat{F}_{4})-\Theta^{7}(\widehat{F}_{4}))+\Theta(\widehat{F}_{6})-\Theta^{6}(\widehat{F}_{6})-4(\widehat{F}_{8}-\Theta^{5}(\widehat{F}_{8})).\end{split}$

Now we collect these terms so that

$\sum_{n\equiv
7\pmod{11}}MO(10;n)q^{n}\pmod{11}=Y_{0}(q)+Y_{2}(q)+Y_{4}(q)+Y_{6}(q)+Y_{8}(q),$

where $Y_{2i}(q)$ consists of those $\Theta^{t}(\widehat{F}_{j}(q))$ with
weight congruent to $i$ modulo $10$. By Lemma 4.2, we find that
$Y_{0}(q),Y_{2}(q),Y_{4}(q),Y_{6}(q)$, and $Y_{8}(q)$ are modular forms modulo
11 with weights $228,180,192,204$, and $216$, respectively on
$\mathrm{SL}_{2}(\mathbb{Z})$. Finally, by Sturm’s Theorem 4.1 (i.e. checking
at most $20$ terms), we find that each of these modular forms vanishes modulo
11, which implies the theorem.

###  4.3. Proof of Theorem 1.6

The generating function for the 3-colored partition function satisfies

$P_{3}(q)=\sum_{n\geq 0}c_{3}(n)q^{n}=\prod_{n\geq
1}\frac{1}{(1-q^{n})^{3}}\equiv\sum_{n\geq 0}p(n)q^{3n}\pmod{3}.$

Therefore, we have that $3\mid c_{3}(n)$ whenever $3\nmid n.$ Furthermore,
Gordon [4] proved that

$c_{3}(11n+7)\equiv 0\pmod{11}.$

We further claim that $c_{3}(17n+15)\equiv 0\pmod{17}$. To prove this
congruence, we employ Ramanujan’s weight 12 cusp form
$\Delta(z):=\eta(z)^{24}$ through the following observation:

$q^{2}\sum_{n\geq 0}c_{3}(n)q^{n}\cdot\prod_{n\geq 1}(1-q^{17n})^{3}\equiv
q^{2}\prod_{n\geq 1}(1-q^{n})^{48}\pmod{17}=\Delta(z)^{2}.$

One easily checks that $\Delta^{2}\ |\ T_{17}\equiv 0\pmod{17},$ which means
that every seventeenth coefficient of $\Delta(z)^{2}$ vanishes modulo 17. This
congruence follows immediately from the fact that

$\prod_{n\geq 1}(1-q^{17n})^{3}\in(\mathbb{Z}/17\mathbb{Z})[q^{17}].$

Now suppose that $\ell\in\\{3,11,17\\}$ and $1\leq a\equiv\ell-1\pmod{\ell}.$
If $\ell\nmid n(n+1)/2,$ then we have

$\frac{(2n+1)(n+a)!}{(2a+1)!(n-a)!}\equiv 0\pmod{\ell}.$

Therefore, the identity in Lemma 2.1 collapses modulo $\ell$ and gives

$\mathcal{U}_{a}(q)=\sum_{n\geq 0}MO(a;n)q^{n}\equiv P_{3}(q)\cdot\sum_{n\geq
0}A(a,\ell;\ell n)q^{\ell n}\pmod{\ell}.$

The power series on the right is in $(\mathbb{Z}/\ell\mathbb{Z})[q^{\ell}],$
and so the $MO(a;n)$ inherit the $c_{3}(n)$ congruences.

##  5\. Proof of Theorem 1.7

Here we prove Theorem 1.7 and Corollary 1.8.

###  5.1. Nuts and Bolts

We first recall the definition of Hecke operators. Let $m$ be a positive
integer and $f(z)=\sum_{n=0}^{\infty}a(n)q^{n}\in M_{k}$. Then the action of
Hecke operator $T_{m}$ on $f(z)$ is defined by

(5.1) $\displaystyle
f(z)\,|\,T_{m}:=\sum_{n=0}^{\infty}\left(\sum_{d\mid\gcd(n,m)}d^{k-1}a\left(\frac{nm}{d^{2}}\right)\right)q^{n}.$

In particular, if $m=p$ is a prime, we have

(5.2) $\displaystyle f(z)\,|\,T_{p}:=f(z)\,|\,U_{p}+p^{k-1}f(z)\,|\,V_{p},$

where $f(z)\,|\,U_{p}:=\sum_{n=0}^{\infty}a(pn)q^{n}$ and
$f(z)\,|\,V_{p}:=\sum_{n=0}^{\infty}a(n)q^{pn}$.

Let’s recall a result of Serre [14] (also see [9, Lemma 2.63 and Theorem
2.65]) on the action of Hecke operator on cusp forms. For a number field $K$,
let $\mathcal{O}_{K}$ denote its ring of integers.

######  Lemma 5.1.

For $1\leq i\leq t$, let $f_{i}(z)=\sum_{n=1}^{\infty}a_{i}(n)q^{n}\in M_{k}$
be a modular form with coefficients in the ring of integers of a number field
$O_{K}.$ Then the following are true.

(i) If $\mathfrak{m}\subset\mathcal{O}_{K}$ is an ideal of norm $M$, then a
positive proportion of the primes $p\equiv-1\pmod{M}$ satisfy

$f_{1}(z)\mid T_{p}\equiv f_{2}(z)\mid T_{p}\equiv\cdots f_{t}(z)\mid
T_{p}\equiv 0\pmod{\mathfrak{m}}.$

(ii) There is a constant $a>0$ such that for every $1\leq i\leq t$ we have

$\\#\left\\{n\leq X:a_{i}(n)\not\equiv
0\pmod{\mathfrak{m}}\right\\}=O\left(X/(\log{}X)^{a}\right).$

We next recall some facts about $p$-adic modular forms developed by Serre
[15]. Let $p$ be a prime. Consider the field of $p$-adic numbers
$\mathbb{Q}_{p}$, with its non-archimedean valuation $\nu_{p}$. We say
$x\in\mathbb{Q}_{p}$ is $p$-integral if $\nu_{p}(x)\geq 0$. Let $f=\sum
a(n)q^{n}\in\mathbb{Q}_{p}[[q]]$ be a formal power series, we define
$\nu_{p}(f):=\inf_{n}\nu_{p}(a_{n})$. If $\nu_{p}(f)\geq m$, we write as well
$f\equiv 0\pmod{p^{m}}$. Assume $\\{f_{i}\\}$ to be a sequence of elements in
$\mathbb{Q}_{p}[[q]]$. We say that $f_{i}\to f$ if the coefficients of $f_{i}$
tend uniformly to those of $f$, i.e., $\nu_{p}(f-f_{i})\to\infty$. A $p$-adic
modular form $f$ is a formal series with coefficients in $\mathbb{Q}_{p}$
which is the limit of classical modular forms $f_{i}$ of weights $k_{i}$.

In order to prove Theorem 1.7, we need the following preliminary result.

######  Lemma 5.2.

The following are true:

(i) If $m$ is a positive integer, then we have that

$E_{2}(z)\equiv\frac{1}{(2^{m}-1)}\sum_{i=1}^{m}2^{i-1}E_{2+3\cdot
2^{m+1}}(z)\,|\,V_{2^{i-1}}\pmod{2^{m}}.$

Moreover, $E_{2}(z)\pmod{2^{m}}$ is the reduction of a weight $2+3\cdot
2^{m+1}$ modular form on $\mathrm{SL}_{2}(\mathbb{Z})$.
(ii) If $m$ is a positive integer, then we have that

$E_{2}(z)\equiv\frac{2}{(3^{m}-1)}\sum_{i=1}^{m}3^{i-1}E_{2+4\cdot
3^{m}}(z)\,|\,V_{3^{i-1}}\pmod{3^{m}}.$

Moreover, $E_{2}(z)\pmod{3^{m}}$ is the reduction of a weight $2+4\cdot 3^{m}$
modular form on $\mathrm{SL}_{2}(\mathbb{Z})$.
(iii) If $p\geq 5$ is prime and $m$ is a positive integer, then we have that

$E_{2}(z)\equiv\frac{(p-1)}{(p^{m}-1)}\sum_{i=1}^{m}p^{i-1}E_{2+(p-1)p^{m-1}}(z)\,|\,V_{p^{m-1}}\pmod{p^{m}}.$

In particular, $E_{2}(z)\pmod{p^{m}}$ is the reduction of a weight
$2+p^{m-1}(p-1)$ modular form on $\mathrm{SL}_{2}(\mathbb{Z})$.

###### Proof.

Let $g(z)=\sum_{n=0}^{\infty}b(n)q^{n}$ be a weight $k$ modular form on
$\mathrm{SL}_{2}(\mathbb{Z})$ with $p$-integral coefficients. Then
$g(z)\,|\,T_{p}=\sum_{n=0}^{\infty}b(pn)q^{n}+p^{k-1}b(n)q^{pn}\in M_{k}$.
Since $E_{p^{m-1}(p-1)}(z)\equiv 1\pmod{p^{m}}$, we have that
$\\{g(z)E_{p^{m-1}(p-1)}(z)\\}$ converges to $g(z)$ $p$-adically. Hence,
$g(z)$ is a $p$-adic modular form. Also, we have the convergence

$g(z)E_{p^{m-1}(p-1)}(z)\,|\,T_{p}\longrightarrow
g(z)\,|\,U_{p}=\sum_{n=0}^{\infty}b(pn)q^{n}.$

Hence, $U_{p}$ is an operator on $M_{k}$ and so is $V_{p}$, defined as
$g(z)\,|\,V_{p}=p^{1-k}(g(z)\,|\,T_{p}-g(z)\,|\,U_{p})$. Now, our proof of the
lemma follows from [15, Example on Page 210]. ∎

###  5.2. Proof of Theorem 1.7

By Theorem 1.3 and 1.4, we have

$\displaystyle\mathcal{U}_{a}(q)$
$\displaystyle=\sum_{t=0}^{a}w_{t}(a)\sum_{\begin{subarray}{c}\alpha,\beta,\gamma\geq
0\\\
\alpha+2\beta+3\gamma=t\end{subarray}}c(\alpha,\beta,\gamma)E_{2}(q)^{\alpha}E_{4}(q)^{\beta}E_{6}(q)^{\gamma}=F_{0}(q)+F_{2}(q)+\cdots+F_{2a}(q),$
(5.3) $\displaystyle\mathcal{U}_{a}^{\star}(q)$
$\displaystyle=\sum_{t=0}^{a}w_{t}^{\star}(a)\cdot\mathbb{E}_{2t}^{\star}(q)=F_{0}^{\star}(q)+F_{2}^{\star}(q)+\cdots+F_{2a}^{\star}(q),$

where $F_{2i}(q)$ and $F_{2i}^{\star}(q)$, for $0\leq i\leq a$, are
quasimodular forms of weight $2i$ on $\mathrm{SL}_{2}(\mathbb{Z})$. Using
Lemma 5.2 and the Chinese Remainder Theorem, we find that $F_{2i}(q)$ and
$F_{2i}^{\star}(q)$, for $0\leq i\leq a$, are modular forms modulo any integer
$m$ on $\mathrm{SL}_{2}(\mathbb{Z})$. Employing Lemma 5.1 $(i)$ on (5.2), we
complete the proof of first and second parts of Theorem 1.7 and finally
applying Lemma 5.1 $(ii)$ to (5.2), claim $(iii)$ follows.

###  5.3. Proof of Corollary 1.8

By Theorem 1.3, we find that

$\displaystyle\mathcal{U}_{2}(q)$ $\displaystyle=\frac{1}{2^{3}}\sum_{n\geq
0}[(-2n+1)\sigma_{1}(n)+\sigma_{3}(n)]q^{n},$
$\displaystyle\mathcal{U}_{3}(q)$ $\displaystyle=\frac{1}{2^{7}\cdot 3\cdot
5}\sum_{n\geq
0}[(40n^{2}-100n+37)\sigma_{1}(n)+(-30n+50)\sigma_{3}(n)+3\sigma_{5}(n)]q^{n},$
$\displaystyle\mathcal{U}_{4}(q)$ $\displaystyle=\frac{1}{2^{10}3^{3}\cdot
5\cdot 7}\sum_{n\geq 0}\big{[}(-840n^{3}+5880n^{2}-9870n+3229)\sigma_{1}$
$\displaystyle\quad\quad\quad+(756n^{2}-4410n+4935)\sigma_{3}+(-126n+441)\sigma_{5}+5\sigma_{7}\big{]}q^{n},$
$\displaystyle\mathcal{U}_{5}(q)$
$\displaystyle=\frac{1}{2^{15}3^{3}5^{2}\cdot 7}\sum_{n\geq
0}[(3360n^{4}-50400n^{3}+223440n^{2}-314200n+96111)\sigma_{1}(n)$
$\displaystyle\quad\quad\quad+(-3360n^{3}+45360n^{2}-167580n+157100)\sigma_{3}(n)$
$\displaystyle\quad\quad\quad+(720n^{2}-7560n+16758)\sigma_{5}(n)+(-50n+300)\sigma_{7}(n)+\sigma_{9}(n)]q^{n}.$

Let $s$ and $t$ be non-negative integers. If $k$ is a positive odd, then for
primes $p\equiv-1\pmod{s}$ we have

(5.4)
$\displaystyle(pn)^{t}\sigma_{k}(pn)=(pn)^{t}\sigma_{k}(n)\sigma_{k}(p)=(pn)^{t}\sigma_{k}(n)(1+p^{k})\equiv
0\pmod{s}$

for all $n$ coprime to $p$. Corollary 1.8 follows by applying (5.4)
appropriately in each case.

## References

  * [1] T. Amdeberhan, G. E. Andrews, R. Tauraso, _Extensions of MacMahon’s sums of divisors_ , preprint available at https://arxiv.org/pdf/2309.03191.pdf (2023).
  * [2] G. E. Andrews, S. C. F. Rose, _MacMahon’s sum-of-divisors functions, Chebyshev polynomials, and quasimodular forms_ , J. Reine Angew. Math. 676 (2013), 97–103.
  * [3] B. Brindle, _A unified approch to qMZVs_ , preprint available at https://arxiv.org/pdf/2111.00051.pdf (2021).
  * [4] B. Gordon, _Ramanujan congruences for $p_{-k}\pmod{11^{r}}$_, Glasgow J. Math. 24 (1983), 107-123.
  * [5] G.-H. Han, _The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications_ , Annales de l’Institut Fourier, Volume 60 (2010) no. 1, 1-29.
  * [6] M. Kaneko, D. Zagier, _A generalized Jacobi theta function and quasimodular forms_ , The moduli space of curves (Texas Island, 1994), Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, 165–172.
  * [7] P. A. MacMahon, _Divisors of Numbers and their Continuations in the Theory of Partitions_ , Proc. London Math. Soc. (2) 19 (1920), no.1, 75-113 [also in Percy Alexander MacMahon Collected Papers, Vol.2, pp. 303–341 (ed. G.E. Andrews), MIT Press, Cambridge, 1986].
  * [8] N. A. Nekrasov, A. Okounkov, _Seiberg-Witten theory and random partitions, in The unity of mathematics_ , Progress in Mathematics, Birkhäuser Boston, 2006, vol. 244, 525-596.
  * [9] K. Ono, _The web of modularity: arithmetic of the coefficients of modular forms and $q$-series_, CBMS Regional Conference Series in Mathematics, 102, Amer. Math. Soc., Providence, RI, 2004.
  * [10] G. Pólya, _Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen_ , Acta Mathematica, 68 (0), (1937), 145-254.
  * [11] S. Ramanujan, _On certain arithmetical functions_ , Trans. Camb. Phil. Soc., 22 (1916), 159–184.
  * [12] S. C. F. Rose, _Quasimodularity of generalized sum-of-divisors functions_ , Research in Number Theory 1 (2015), Paper No. 18, 11 pp.
  * [13] J.-P. Serre, _A course in arithmetic_ , Springer-Verlag, New York, 1973.
  * [14] J.-P. Serre, _Divisibilité de certaines fonctions arithmétiques_ , L’Enseignement Math. 22 (1976), 227–260.
  * [15] J.-P. Serre, _Formes modulaires et fonctions zêta $p$-adiques_, Springer Lect. Notes 350 (1973), 191–268.
  * [16] J. Sturm, _On the congruence of modular forms_ , Springer Lect. Notes in Math. 1240 (1984), 275–280.
  * [17] H. P. F. Swinnerton-Dyer, _On $\ell$-adic representations and congruences for coefficients of modular forms_, Springer Lect. Notes. 350 (1973), 1–55.