# Transformation properties of Andrews-Beck $NT$ functions and generalized
Appell-Lerch series

Rong Chen  Department of Mathematics, Shanghai Normal University, Shanghai,
People’s Republic of China<EMAIL_ADDRESS>and  Xiao-Jie Zhu  School of
Mathematical Sciences, Key Laboratory of MEA(Ministry of Education) & Shanghai
Key Laboratory of PMMP, East China Normal University
<EMAIL_ADDRESS>
###### Abstract.

In 2021, Andrews mentioned that George Beck introduced a partition statistic
$NT(r,m,n)$ which is related to Dyson’s rank statistic. Motivated by Andrews’s
work, scholars have established a number of congruences and identities
involving $NT(r,m,n)$. In this paper, we strengthen and extend a recent work
of Mao on the transformation properties of the $NT$ function and provide an
analogy of Hickerson and Mortenson’s work on the rank function. As an
application, we demonstrate how one can deduce from our results many
identities involving $NT(r,m,n)$ and another crank-analog statistic
$M_{\omega}(r,m,n)$. As a related result, some new properties of generalized
Appell-Lerch series are given.

###### Key words and phrases:

Andrews-Beck $NT$ function, Dyson’s rank function, partitions, non-holomorphic
modular forms, Appell-Lerch series

###### 2020 Mathematics Subject Classification:

Primary 11P84; Secondary 05A17, 11F11, 11F30, 11F37, 11P83

##  1\. Introduction

###  1.1. Background and the main results

A partition of a positive integer $n$ is defined as a sequence of positive
integers in non-increasing order that sums to $n$. The number of all
partitions of $n$ is denoted by $p(n)$. The following distinguished
congruences were discovered by Ramanujan in [29]:

(1.1) $\displaystyle p(5n+4)\equiv 0$ $\displaystyle\pmod{5},$ (1.2)
$\displaystyle p(7n+5)\equiv 0$ $\displaystyle\pmod{7},$ (1.3) $\displaystyle
p(11n+6)\equiv 0$ $\displaystyle\pmod{11},$

where $n$ is any nonnegative integer.

In an effort to provide purely combinatorial interpretations of Ramanujan’s
famous congruences (1.1)-(1.3), two important partition statistics of ordinary
partitions, rank and crank, were introduced by Dyson, Andrews, and Garvan. In
1944, Dyson [14] defined the rank of a partition to be the largest part of the
partition minus the number of parts. Dyson also conjectured that for $k=5,7$,

$N(r,k,kn-s_{k})=\frac{p(kn-s_{k})}{k},$

where $N(r,k,n)$ counts the number of partitions of $n$ with rank congruent to
$r$ modulo $k$ and $s_{k}$ denotes $s_{k}=(k^{2}-1)/24$ throughout the paper.
In 1954, Dyson’s conjectures were proved by Atkin and Swinnerton-Dyer [4].
Therefore, the ranks of partitions provide purely combinatorial descriptions
of Ramanujan’s congruences (1.1) and (1.2). Unfortunately, Dyson’s rank failed
to account for Ramanujan’s third congruence (1.3) combinatorially. Dyson
conjectured the existence of an unknown partition statistic, which he
whimsically called “the crank”, to explain Ramanujan’s third congruence modulo
11. The crank was found by Andrews and Garvan [3] who defined it as the
largest part, if the partition has no ones, and otherwise as the difference
between the number of parts larger than the number of ones and the number of
ones. In 1987, Garvan [17] proved that for $k=5,7,11$,

$M(r,k,kn-s_{k})=\frac{p(kn-s_{k})}{k},$

where $M(r,k,n)$ counts the number of partitions of $n$ with crank congruent
to $r$ modulo $k$. Garvan’s results imply that the crank of partitions
provides purely combinatorial descriptions of Ramanujan’s three congruences
(1.1)-(1.3).

The transformation properties of the rank function was studied by Bringmann,
Ono and Rhoades [9] and later by many authors such as Ahlgren, Garvan,
Hickerson, Mortenson and Treneer [1], [18], [19]. For example, let

$D(a,M):=\sum_{n=0}^{\infty}\left(N(a,M,n)-\frac{p(n)}{M}\right)q^{n}.$

Hickerson and Mortenson [19, Theorem 4.1] state that

(1.4) $D(a,M)=d(a,M)+T_{a,M},$

where $d(a,M)$ are some given Appell-Lerch series (under $M$-dissection when
$\gcd(M,6)=1$) and $T_{a,M}$ is a theta function.

In 2021, Andrews [2] mentioned that George Beck had introduced the partition
statistics $NT(r,m,n)$ and $M_{\omega}(r,m,n)$, which count the total number
of parts in the partitions of $n$ with rank congruent to $r$ modulo $m$, and
the total number of ones in the partitions of $n$ with crank congruent to $r$
modulo $m$, respectively. Namely,

$\displaystyle NT(r,m,n)=\sum_{\begin{subarray}{c}\pi\vdash n,\\\
\text{rank}(\pi)\equiv r\pmod{m}\end{subarray}}\sharp(\pi)$

and

$\displaystyle M_{\omega}(r,m,n)=\sum_{\begin{subarray}{c}\pi\vdash n,\\\
\text{crank}(\pi)\equiv r\pmod{m}\end{subarray}}\omega(\pi),$

where $\sharp(\pi)$ denotes the number of parts of $\pi$, and $\omega(\pi)$
the number of ones. The following Andrews-Beck type congruence, conjectured by
Beck, was proved by Andrews [2]:

$\sum_{m=1}^{4}mNT(m,5,5n+1)\equiv\sum_{m=1}^{4}mNT(m,5,5n+4)\equiv
0\pmod{5}.$

Motivated by Andrews’ works, variants of Andrews-Beck type congruence were
studied by many authors. For example, Chern [11] found and proved a list of
this type congruences modulo 5,7,11, and 13. Moreover, identities and
equalities involving these new partition statistics have been established by
many authors through $q$-series approaches. For example, Mao [23] obtained
identities involving $NT(s,p,pn+d)$ such as

(1.5) $\displaystyle\sum_{n=0}^{\infty}$
$\displaystyle\big{(}NT(1,7,7n+5)-NT(6,7,7n+5)$
$\displaystyle+3NT(2,7,7n+5)-3NT(5,7,7n+5\big{)}q^{n}=-7\frac{(q^{7};q^{7})_{\infty}(q^{3},q^{4};q^{7})_{\infty}}{(q,q^{6};q^{7})_{\infty}(q^{2},q^{5};q^{7})_{\infty}^{2}}.$

which is analogue of the well-known identity

$\sum_{n=1}^{\infty}p(5n+4)q^{n}=5\frac{(q^{5};q^{5})^{5}}{(q;q)^{6}},$

and also indicate the congruence

$\displaystyle NT(1,7,7n+5)-NT(6,7,7n+5)+3NT(2,7,7n+5)-3NT(5,7,7n+5)\equiv
0\pmod{7}.$

Here and throughout the paper

$(a;q)_{n}:=\prod_{k=0}^{n-1}(1-aq^{k}),$
$(a;q)_{\infty}:=\prod_{k=0}^{\infty}(1-aq^{k}),$
$(a_{1},a_{2},\cdots,a_{n};q)_{\infty}:=\prod_{k=1}^{n}(a_{k};q)_{\infty},$

and for convenience

$(q)_{\infty}:=(q;q)_{\infty}.$

Identities involving $NT$ and $M_{\omega}$ are also established by many
authors. For example, Jin, Liu and Xia [20] found the relations such as

(1.6) $NT(2,5,5n+1)-NT(3,5,5n+1)=M_{\omega}(2,5,5n+1)-M_{\omega}(3,5,5n+1).$

In a recent work with Chen and Yin, the first author [10] found the identity
for $p=11$,

(1.7)
$\sum_{m=1}^{5}m\left[M_{\omega}(m,11,11n+6)-M_{\omega}(11-m,11,11n+6)\right]=0.$

As stated in [2], the fact $N(s,k,n)=N(k-s,k,n)$ is generally false if $N$ is
replaced by $NT$. Recently, Mao [24] established the modular approach for
analyzing the difference $NT(s,k,n)-NT(k-s,k,n)$ and proved additional
identities of the form shown in (1.5)-(1.7) by applying the theory of mock
modular forms. Mao [24, Lemma 2.1] represented the difference
$NT(s,k,n)-NT(k-s,k,n)$ as the rank function and the (generalized) Appell-
Lerch sum.

(1.8)
$\displaystyle\sum_{n=0}^{\infty}\left(NT(s,k,n)-NT(k-s,k,n)\right)q^{n}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{k-1}\frac{\zeta_{k}^{j(s-1)}(1-\zeta_{k}^{j})}{k}\left\\{\frac{1}{2}+\frac{\zeta_{k}^{j/2}}{2\pi
i(q)_{\infty}}\cdot\frac{\partial}{\partial
u}\bigg{|}_{u=0}A_{3}\left(u-\frac{j}{k};\tau\right)\right\\}$
$\displaystyle-\sum_{j=1}^{k-1}\frac{\zeta_{k}^{js}(1+\zeta_{k}^{j})\mathcal{R}\left(\zeta_{k}^{j};q\right)}{2k(1-\zeta_{k}^{j})}.$

By analogy, the difference $M_{\omega}(m,k,n)-M_{\omega}(k-m,k,n)$ can be
expressed as [24, Lemma 2.2]

(1.9)
$\displaystyle\sum_{n=0}^{\infty}\left(M_{\omega}(s,k,n)-M_{\omega}(k-s,k,n)\right)q^{n}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{k-1}\frac{\zeta_{k}^{j(s-1/2)}(1-\zeta_{k}^{j})}{2k\pi
i(q)_{\infty}}\cdot\frac{\partial}{\partial
u}\bigg{|}_{u=0}A_{1}\left(u-\frac{j}{k};\tau\right)$
$\displaystyle-\sum_{j=1}^{k-1}\frac{\zeta_{k}^{js}(1+\zeta_{k}^{j})\mathcal{C}\left(\zeta_{k}^{j};q\right)}{2k(1-\zeta_{k}^{j})}.$

The function $\mathcal{R}(z;q)$ (resp. $\mathcal{C}(z;q)$) is the rank (resp.
crank) function,

$\mathcal{R}(z;q):=\sum_{n=0}^{\infty}\sum_{m=-\infty}^{\infty}N(m,n)z^{m}q^{n},$
$\mathcal{C}(z;q):=\sum_{n=0}^{\infty}\sum_{m=-\infty}^{\infty}M(m,n)z^{m}q^{n},$

and $A_{\ell}(u,v;\tau)$ is the (generalized) Appell-Lerch series

(1.10) $A_{\ell}(u,v;\tau):=a^{\ell/2}\sum_{n\in\mathbb{Z}}\frac{(-1)^{\ell
n}q^{\ell n(n+1)/2}b^{n}}{1-aq^{n}},$

where $a:=e^{2\pi iu}$, $b:=e^{2\pi iv}$ and $q:=e^{2\pi i\tau}$. We also
write $A_{\ell}(u;\tau)$:=$A_{\ell}(u,0;\tau)$.

In this paper, we start from (1.8) to find an analog of (1.4). Let $p\geq 5$
be prime and

$D_{NT}(s,p):=\sum_{n=0}^{\infty}\left(NT(s,p,n)-NT(p-s,p,n)\right)q^{n}-\sum_{r=1}^{p-1}\frac{p-2r}{2p}D(r-s,p),$

where we point out that

$\sum_{r=1}^{p-1}\frac{p-2r}{2p}D(r-s,p)=\sum_{r=1}^{p-1}\frac{p-2r}{2p}\sum_{n=0}^{\infty}N(r-s,p,n)q^{n}.$

Our main theorem (see Theorem 3.6 below) shows that

$D_{NT}(s,p)=d_{NT}(s,p)+t_{s,p},$

where $d_{NT}(s,p)$ represents some given Appell-Lerch series and their
derivatives, and $t_{s,p}$ is “modular” (weight $3/2$). For example, we have

$D_{NT}(1,5)=q^{8}L_{5}(2;q^{5})-q^{5}L_{5}(4;q^{5})+t_{1,5},$

where for prime $p$ and integer $0<v<p$, we define

(1.11) $\displaystyle L_{p}(v;q):=$ $\displaystyle L_{p}(v)$ $\displaystyle:=$
$\displaystyle
l_{p}(v)+\frac{(-1)^{v}}{(q^{p};q^{p})_{\infty}}\left(p\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}q^{3pn(n+1)/2}}{(1-q^{pn+v})^{2}}+\left(\frac{p}{2}-3v\right)\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}q^{3pn(n+1)/2}}{1-q^{pn+v}}\right),$

with $l_{p}(v)=(-1)^{v}\frac{p-6v}{2}q^{-v}$ if $0<v<\frac{p}{6}$,
$l_{p}(v)=(-1)^{v}\frac{5p-6v}{2}q^{v-p}$ if $\frac{5p}{6}<v<p$ and
$l_{p}(v)=0$ else. For $v=0$ we define

(1.12) $\displaystyle
L_{p}(0)=\frac{p}{(q^{p};q^{p})_{\infty}}\left(\sum_{n\neq
0}\frac{(-1)^{n}q^{3pn(n+1)/2}}{(1-q^{pn})^{2}}+3\sum_{n=1}^{\infty}\frac{nq^{pn}}{1-q^{pn}}-\frac{1}{12}\right).$

We remark that the function $L_{p}(0)$ is close to the Andrews $spt$ function

$\sum_{n=0}^{\infty}spt(n)q^{n}=\frac{1}{(q;q)_{\infty}}\left(\sum_{n\neq
0}\frac{(-1)^{n}q^{3n(n+1)/2}}{(1-q^{n})^{2}}+\sum_{n=1}^{\infty}\frac{nq^{pn}}{1-q^{pn}}\right),$

which is related to a weight 3/2 harmonic Maass form studied by Bringmann [5],
also Bringmann, Folsom and Ono [6].

Furthermore, we find that all $p$-dissections of the part $t_{s,p}$ are
modular functions on $\Gamma_{1}(p)$ when multiplied by certain eta-quotients
and give the expression of $t_{s,p}$ (under $p$-dissection) by the modular
function approach for $p=5,7$ (see Appendices A and B).

Then we show that Theorems 3.6 and 3.7 imply identities such as (1.5)–(1.7).

###  1.2. Appell-Lerch series of level $\ell$

In proving the main theorems (Theorems 3.6 and 3.7 below) and finding out the
exact forms of $N_{p}(s,k)$ and $M_{p}(s,k)$ in these theorems, we rely
heavily on the properties of the modular completions
$\widehat{A}_{\ell}(u;\tau)$ of generalized Appell-Lerch series
$A_{\ell}(u;\tau)$. The completion was obtained by Zwegers [34] which extends
a previous important result obtained in the same author’s Ph.D. Thesis.
Zwegers also gave the transformation equations of these functions as real
analytic Jacobi forms. In particular, $\widehat{A}_{\ell}(u;\tau)$ is a real
analytic Jacobi form of weight $1$ and index $-\ell/2$; c.f. [34, Remark 4].
The positive integer $\ell$ is called the level.

The functions $A_{\ell}(u;\tau)$ appear in several branches of mathematics,
e.g., in representation theory of affine Lie superalgebras, conformal field
theory, the theory of K3 surfaces, the study of vector bundles on elliptic
curves; c.f. [21], [31], [25], [28]. More related to our topic, generalized
Appell-Lerch series play an important role in the study of Ramanujan’s mock
theta functions; c.f. [8] and [32].

Mao [24] used $A_{1}(u;\tau)$ and $A_{3}(u;\tau)$ to derive some of his
theorems (see (1.8) and (1.9) above). In detail, Mao obtained the
transformation equations of $\frac{\partial}{\partial
u}|_{u=0}\widehat{A}_{\ell}\left(u-\frac{j}{k};\tau\right)$ and then proved
the holomorphicity of the “holomorphic parts” of these functions at cusps for
the level $\ell=1$ and $3$. These properties are key to the proof of his
results.

In this paper, we obtain some new results on $\widehat{A}_{\ell}(u;\tau)$
which in the cases $\ell=1$ and $3$ will be used in the proof of Theorems 3.6
and 3.7 and in determining how many terms one should check when using a
computer algebra system to prove identities on $N_{p}(s,k)$ and $M_{p}(s,k)$.
These results are more meticulous and comprehensive than Mao’s and may be of
independent interest due to the role of $\widehat{A}_{\ell}(u;\tau)$ in
mathematics. We list the results as follows.

  * •

We obtain the $q$-series expansions of both the holomorphic part and
nonholomorphic part of $\frac{\partial}{\partial
u}|_{u=0}\widehat{A}_{\ell}(u-x;\tau)$ at any cusp, where $x$ is a positive
nonintegral rational number. See Theorem 2.1 below.

  * •

We give an explicit formula for the order of the holomorphic part of
$\frac{\partial}{\partial u}|_{u=0}\widehat{A}_{\ell}(u-x;\tau)$ at any cusp.
See Corollary 2.5 below.

  * •

We establish the modularity of certain linear combination of
$\widehat{A}_{\ell}$ and its derivative (with respect to $u$) and give an
explicit formula for the orders of this function at cusps. See Proposition 2.6
and Corollary 2.7 below.

  * •

We show that the image of $\frac{\partial}{\partial
u}|_{u=0}\widehat{A}_{\ell}(u-x;\tau)$ under an operator $U_{p,k}$, multiplied
by some generalized Dedekind eta functions, is a real analytic modular form on
the congruence subgroup $\Gamma_{1}(p)$ with trivial multiplier system. See
Theorem 2.14 below. This exhibits a technique of how one can multiply a
(possibly nonholomorphic) modular form by some generalized Dedekind eta
functions to reduce the multiplier system to a trivial one.

In the proofs of the above results, we freely use the theory of Jacobi forms
for which the reader may consult [15]. Finally, we emphasize that the real
analytic modular forms mentioned above, in general, are not harmonic Maass
forms as the shapes of their nonholomorphic parts show.

###  1.3. Organization of the paper and notations

This paper is organized as follows. In Section 2, we derive some properties of
the generalized Appell-Lerch series $A_{\ell}(u;\tau)$ as explained in Section
1.2. In Section 3, we find out the functions to cancel the nonholomophic part
of each $p$-dissection of the $NT$ function and prove the main theorem. In
Section 4, we calculate the lower bounds of the orders at all cusps for
$N_{p}(s,k)$ and $M_{p}(s,k)$ in Theorems 3.6 and 3.7 and then use the valence
formula to express $N_{p}(s,k)$ and $M_{p}(s,k)$ as linear combinations of
certain generalized Dedekind eta quotients. The resulted expressions are
collected in Appendices A and B.

We list some common notations here. The symbol $\mathscr{H}$ denotes the upper
half plane $\\{z\in\mathbb{C}\colon\operatorname{Im}z>0\\}$ and $\tau$ is
tacitly assumed to take values in $\mathscr{H}$. The symbol $q$ always means
$e^{2\pi i\tau}$. If $a,b$ are two numbers, the notation $\delta_{a,b}$ refers
to the Kronecker $\delta$, that is, $\delta_{a,b}=1$ if and only if $a=b$. Let
$x$ be a rational number; we define $\delta_{x}=1$ if $x\in\mathbb{Z}$ and
$\delta_{x}=0$ otherwise; define $\textnormal{sgn}(x)=1$ or $-1$ according to
$x>0$ or $x<0$ respectively and set $\textnormal{sgn}(0)=0$. For integers $a$
and $b$, $(a,b)$ means the greatest common divisor. The notations
$\zeta_{k}=e^{\frac{2\pi i}{k}}$ and $s_{k}=\frac{k^{2}-1}{24}$ have been
mentioned in above. Let $p\geq 5$ be a prime; we set $\chi_{12}(p)=1$ if
$p\equiv 1,11\pmod{12}$ and $\chi_{12}(p)=-1$ if $p\equiv 5,7\pmod{12}$. For a
complex-valued real-differentiable function $f(u)$ of a complex variable $u$,
define $\frac{\partial}{\partial u}f=\frac{1}{2}\left(\frac{\partial
f}{\partial\operatorname{Re}u}-i\frac{\partial
f}{\partial\operatorname{Im}u}\right)$. If $f$ is holomorphic, this coincides
with the complex derivative.

The slash operators $f|_{k}\gamma$ acting on modular forms or analogues $f$
are defined preceding Theorem 2.1 and after Corollary 2.7. The slash operators
$\phi|_{k}\gamma$ acting on Jacobi forms or analogues $\phi$ are used in some
proofs and their definition can be found in [15, Theorem 1.4]. The order of a
modular form or its analogue at a cusp,
$\operatorname{ord}\nolimits_{i\infty}f|_{r}\gamma$ or
$\operatorname{ord}_{a/c}f$, is defined preceding the statement of and after
the proof of Corollary 2.5. The divisor $\mathrm{div}_{a/c}f$ or
$\mathrm{div}_{\tau}f$ of a meromorphic modular form $f$ at $a/c$ or $\tau$,
each of which is a term in the left-hand side of the valence formula, is
defined in the beginning part of Section 4.2.

##  2\. Modularity

###  2.1. Appell-Lerch series

Following Zwegers [34], let

(2.1)
$\widehat{A}_{\ell}(u;\tau):=A_{\ell}(u;\tau)+\frac{i}{2}\sum_{m=0}^{\ell-1}e^{2\pi
imu}\theta\left(m\tau+\frac{\ell-1}{2};\ell\tau\right)R\left(\ell
u-m\tau-\frac{\ell-1}{2};\ell\tau\right),$

where $\tau\in\mathscr{H}$,
$u\in\mathbb{C}\setminus(\mathbb{Z}\tau\oplus\mathbb{Z})$,

$\displaystyle\theta(z;\tau):=$
$\displaystyle\sum_{\nu\in\frac{1}{2}+\mathbb{Z}}e^{\pi i\nu^{2}\tau+2\pi
i\nu(z+\frac{1}{2})}$ (2.2) $\displaystyle=$ $\displaystyle-
iq^{\frac{1}{8}}e^{-\pi iz}\prod_{n=1}^{\infty}(1-q^{n})(1-e^{2\pi
iz}q^{n-1})(1-e^{-2\pi iz}q^{n}),$

and

$\displaystyle R(u;\tau):=\sum_{\nu\in\frac{1}{2}+\mathbb{Z}}$
$\displaystyle\left\\{\textnormal{sgn}(v)-E\left((\nu+\frac{\mathrm{Im}(u)}{\mathrm{Im}(\tau)})\sqrt{2\mathrm{Im}(\tau)}\right)\right\\}$
$\displaystyle\times(-1)^{\nu-\frac{1}{2}}q^{\frac{-\nu^{2}}{2}}e^{-2\pi i\nu
u},$

with

$E(z):=2\int_{0}^{z}e^{-\pi
u^{2}}\mathrm{d}u=\textnormal{sgn}(z)(1-\beta(z^{2}))\quad(z\in\mathbb{R}),$
$\beta(x):=\int_{x}^{\infty}u^{-\frac{1}{2}}e^{-\pi
u}\mathrm{d}u\quad(x\in\mathbb{R}_{\geq 0}).$

One can easily verify that

$\widehat{A}_{1}(u;\tau)=A_{1}(u;\tau),$

and

(2.3) $\widehat{A}_{3}(u;\tau)=A_{3}(u;\tau)+\frac{i}{2}\sum_{m=1}^{2}e^{2\pi
imu}\theta(m\tau;3\tau)R(3u-m\tau;3\tau).$

Zwegers [34, Theorem 4] and Mao [24, Theorem 2.4] found the transformation
formulas for these $\widehat{A}_{\ell}$ and their derivatives which we recall
here. For $m,n\in\mathbb{Z}$,

$\widehat{A}_{\ell}(u+n\tau+m;\tau)=(-1)^{\ell(n+m)}e^{2\pi i\ell nu}q^{\ell
n^{2}/2}\widehat{A}_{\ell}(u;\tau).$

For $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{Z})$,

(2.4)
$\widehat{A}_{\ell}\left(\frac{u}{c\tau+d};\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)e^{\frac{-\pi
ic\ell u^{2}}{c\tau+d}}\widehat{A}_{\ell}(u;\tau).$

For $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{Z})$ and
$j\not\equiv 0\pmod{k}$,

$\displaystyle\frac{\partial}{\partial
u}\bigg{|}_{u=0}\widehat{A}_{\ell}\left(u-\frac{j}{k};\frac{a\tau+b}{c\tau+d}\right)=$
$\displaystyle\frac{2j\ell c\pi i(c\tau+d)^{2}}{k}e^{\frac{-j^{2}\ell c\pi
i(c\tau+d)}{k^{2}}}\widehat{A}_{\ell}\left(-\frac{j(c\tau+d)}{k};\tau\right)$
$\displaystyle+(c\tau+d)^{2}e^{\frac{-j^{2}\ell c\pi
i(c\tau+d)}{k^{2}}}\frac{\partial}{\partial
u}\bigg{|}_{u=0}\widehat{A}_{\ell}\left(u-\frac{j(c\tau+d)}{k};\tau\right).$

For $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\Gamma_{1}(k)$, we have111In
[24, Eq. (2.18)], the factor $e^{\frac{j^{2}\ell c\pi i(2-d)}{k^{2}}}$
occurred instead of $e^{\frac{j^{2}\ell cd\pi i}{k^{2}}}$. The two factors are
equal since $\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)\in\Gamma_{1}(k)$. However, we prefer the one
presented here since the transformation formulas can be generalized to
$\partial\widehat{A}_{\ell}(-j/k)|_{2}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)=(-1)^{\ell cj/k}e^{\frac{j^{2}\ell cd\pi
i}{k^{2}}}\partial\widehat{A}_{\ell}(-dj/k),\,\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)\in\Gamma_{0}(k)$ where we can not use the
other factor. For the notation see Section 2.2.

(2.5) $\frac{\partial}{\partial
u}\bigg{|}_{u=0}\widehat{A}_{\ell}\left(u-\frac{j}{k};\frac{a\tau+b}{c\tau+d}\right)=(-1)^{\frac{\ell
j(c+d-1)}{k}}(c\tau+d)^{2}e^{\frac{\pi i\ell
cdj^{2}}{k^{2}}}\frac{\partial}{\partial
u}\bigg{|}_{u=0}\widehat{A}_{\ell}\left(u-\frac{j}{k};\tau\right).$

###  2.2. Behavior of $\widehat{A}_{\ell}$ and $\partial\widehat{A}_{\ell}$ at
cusps

Let $x$ be a positive rational number that is not an integer (say $x=j/k$),
and $\gamma=\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)\in\mathrm{SL}_{2}(\mathbb{Z})$ with $c\geq 0$.
Let $\partial\widehat{A}_{\ell}$ denote the function $\frac{\partial}{\partial
u}\widehat{A}_{\ell}(u;\tau)$ where $\frac{\partial}{\partial
u}=\frac{1}{2}\left(\frac{\partial}{\partial\operatorname{Re}u}-i\frac{\partial}{\partial\operatorname{Im}u}\right)$.
Therefore
$\partial\widehat{A}_{\ell}(-x)=\partial\widehat{A}_{\ell}(-x;\tau):=\left.\frac{\partial}{\partial
u}\right|_{u=0}\widehat{A}_{\ell}(u-x;\tau)$. The weight $2$ slash operator is
defined by
$\partial\widehat{A}_{\ell}(-x)|_{2}\gamma(\tau):=(c\tau+d)^{-2}\partial\widehat{A}_{\ell}\left(-x;\frac{a\tau+b}{c\tau+d}\right)$.

######  Theorem 2.1.

The function $\frac{1}{2\pi i}e^{-\pi i\ell
cdx^{2}}\cdot\partial\widehat{A}_{\ell}(-x)|_{2}\gamma$ can be written as
$H_{1}+H_{2}$. In this decomposition, $H_{1}(\tau)$, the holomorphic part, is
given by

$C+\left(\sum_{n>-cx,\,m\geq M}-\sum_{n<-cx,\,m<M}\right)(-1)^{\ell n}e^{2\pi
i(m-M)dx}\cdot(m-M)q^{\frac{\ell}{2}(n+cx)^{2}+(n+cx)(m-M)},$

where the variables $n$ and $m$ take values in $\mathbb{Z}$, $M=\ell
cx+\frac{\ell}{2}$ and

$C=-\delta_{cx}\frac{1-\delta_{(\ell+1)/2}+\delta_{(\ell+1)/2}\cdot(-1)^{cx}\cos(\pi
dx)}{4\sin^{2}(\pi dx)}.$

On the other hand, the nonholomorphic part $H_{2}(\tau)$ is given by

$-\frac{1}{2\pi}\sum_{n,m\in\mathbb{Z}}(-1)^{\ell n}e^{2\pi
idx(\ell(n-cx)-(m+\frac{\ell}{2}))}f\left(\ell(n-cx)-(m+\frac{\ell}{2});\operatorname{Im}\tau\right)q^{-\frac{\ell}{2}(n-cx)^{2}+(n-cx)(m+\frac{\ell}{2})}$

where

$f(t;\operatorname{Im}\tau):=\pi\lvert
t\rvert\beta\left(t^{2}\cdot\frac{2\operatorname{Im}\tau}{\ell}\right)-e^{-\pi
t^{2}\cdot\frac{2\operatorname{Im}\tau}{\ell}}\cdot\sqrt{\frac{\ell}{2\operatorname{Im}\tau}}.$

###### Proof.

We shall use the theory of Jacobi forms in this proof, c.f. [15]. According to
Zwegers [34, Theorem 4], $\widehat{A}_{\ell}(u;\tau)$ transforms like a Jacobi
form of weight $1$ and index $-\ell/2$. One can verify by [15, Theorem 1.4]
that, for $\gamma_{1}=\left[\left(\begin{smallmatrix}{a_{1}}&{b_{1}}\\\
{c_{1}}&{d_{1}}\end{smallmatrix}\right),(\lambda,\mu),\xi\right]$,

(2.6)
$(\partial\widehat{A}_{\ell})|_{2,-\ell/2}\gamma_{1}(u;\tau)=\partial(\widehat{A}_{\ell}|_{1,-\ell/2}\gamma_{1})(u;\tau)-2\pi
i\left(\frac{c_{1}}{c_{1}\tau+d_{1}}\ell(u+\tau\lambda+\mu)-\ell\lambda\right)\widehat{A}_{\ell}|_{1,-\ell/2}\gamma_{1}(u;\tau),$

where $a_{1}d_{1}-b_{1}c_{1}=1$ and $\lambda,\mu\in\mathbb{R}$,
$\lvert\xi\rvert=1$. It follows that for the given
$\gamma=\left(\begin{smallmatrix}{a}&{b}\\\ {c}&{d}\end{smallmatrix}\right)$
(even when $c<0$), we have

$\displaystyle\partial\widehat{A}_{\ell}|_{2,-\ell/2}\left[(0,-x),1\right]|_{2,-\ell/2}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)(u;\tau)$ $\displaystyle=$
$\displaystyle\partial\widehat{A}_{\ell}|_{2,-\ell/2}\left[\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right),(-xc,-xd),1\right](u;\tau)$ (2.7)
$\displaystyle=$
$\displaystyle\partial(\widehat{A}_{\ell}|_{1,-\ell/2}[(-cx,-dx),1])(u;\tau)-2\pi
i\frac{c}{c\tau+d}\ell u\widehat{A}_{\ell}|_{1,-\ell/2}[(-cx,-dx),1](u;\tau),$

in the last equality of which we have used the fact
$\widehat{A}_{\ell}|_{1,-\ell/2}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)=\widehat{A}_{\ell}$ (c.f. [34, Theorem 4]).
Unfolding the slash operators, setting $u=0$ and using the chain rule for
$\frac{\partial}{\partial u}$ in the above equality we obtain

$\displaystyle\partial\widehat{A}_{\ell}(-x)|_{2}\gamma=q^{-\frac{\ell
c^{2}x^{2}}{2}}e^{-\pi i\ell cdx^{2}}\cdot\left(2\pi i\ell
cx\widehat{A}_{\ell}(-x(c\tau+d);\tau)+\partial\widehat{A}_{\ell}(-x(c\tau+d);\tau)\right)$
(2.8) $\displaystyle=$ $\displaystyle q^{-\frac{\ell c^{2}x^{2}}{2}}e^{-\pi
i\ell cdx^{2}}\cdot\left(2\pi i\ell
cx(A_{\ell}(-x(c\tau+d);\tau)+\widetilde{A}_{\ell}(-x(c\tau+d);\tau))\right.$
$\displaystyle+\left.\partial
A_{\ell}(-x(c\tau+d);\tau)+\partial\widetilde{A}_{\ell}(-x(c\tau+d);\tau)\right),$

where $\widetilde{A}_{\ell}=\widehat{A}_{\ell}-A_{\ell}$ is the nonholomorphic
correction term (c.f. (2.1)). It remains to compute the four terms

$A_{\ell}(-x(c\tau+d);\tau),\quad\partial
A_{\ell}(-x(c\tau+d);\tau),\quad\widetilde{A}_{\ell}(-x(c\tau+d);\tau),\quad\partial\widetilde{A}_{\ell}(-x(c\tau+d);\tau).$

By the definition,

$A_{\ell}(-x(c\tau+d);\tau)=e^{-\pi i\ell dx}q^{-\frac{\ell
cx}{2}}\sum_{n\in\mathbb{Z}}\frac{(-1)^{\ell n}q^{\ell n(n+1)/2}}{1-e^{-2\pi
idx}q^{n-cx}}.$

Splitting the sum $\sum_{n\in\mathbb{Z}}$ into three sums $\sum_{n>cx}$,
$\sum_{n<cx}$ and $\sum_{n=cx}$ and expanding the geometric series
$(1-z)^{-1}=1+z+z^{2}+\dots$ we find that

(2.9) $A_{\ell}(-x(c\tau+d);\tau)=e^{-\pi i\ell dx}q^{-\frac{\ell
cx}{2}}\cdot\left(\sum_{n>cx,\,m\geq 0}(-1)^{\ell n}q^{\ell n(n+1)/2}e^{-2\pi
imdx}q^{m(n-cx)}\right.\\\ +\sum_{n<cx,\,m\geq 0}(-1)^{\ell n+1}q^{\ell
n(n+1)/2}e^{-2\pi idx}q^{cx-n}e^{2\pi imdx}q^{m(cx-n)}\\\
\left.+\delta_{cx}(-1)^{\ell cx}q^{\ell cx(cx+1)/2}(1-e^{-2\pi
idx})^{-1}\right).$

The two series on the right-hand side converge normally222All the series in
this proof converge normally (on compact sets of $\mathscr{H}$) and we will
never mention this hereafter. so the equality actually holds for
$\tau\in\mathscr{H}$.

For the term $\partial A_{\ell}(-x(c\tau+d);\tau)$, we have

(2.10) $\partial A_{\ell}(-x(c\tau+d);\tau)=\pi i\ell
A_{\ell}(-x(c\tau+d);\tau)+2\pi ie^{-2\pi
ix(c\tau+d)(\ell/2+1)}\sum_{n\in\mathbb{Z}}\frac{(-1)^{\ell n}q^{(\ell
n^{2}+\ell n+2n)/2}}{(1-e^{-2\pi idx}q^{n-cx})^{2}}.$

The first term on the right-side hand has been computed in (2.9) and second
term, the sum $\sum_{n\in\mathbb{Z}}$ is equal to

(2.11) $\sum_{n>cx,\,m\geq 0}(-1)^{\ell n}q^{n(\ell n+\ell+2)/2}e^{-2\pi
imdx}(m+1)q^{m(n-cx)}\\\ +\sum_{n<cx,\,m\geq 0}(-1)^{\ell n}q^{n(\ell
n+\ell+2)/2}e^{4\pi idx}q^{2(cx-n)}e^{2\pi imdx}(m+1)q^{m(cx-n)}\\\
+\delta_{cx}(-1)^{\ell cx}q^{cx(\ell cx+\ell+2)/2}(1-e^{-2\pi idx})^{-2}.$

Combining (2.9), (2.10) and (2.11) we obtain

(2.12) $q^{-\frac{\ell c^{2}x^{2}}{2}}e^{-2\pi i\ell cdx^{2}}\left(\ell
cx(A_{\ell}(-x(c\tau+d);\tau)+\frac{1}{2\pi i}\partial
A_{\ell}(-x(c\tau+d);\tau)\right)=-S_{1}+S_{2}+C_{1},$

where

$\displaystyle S_{1}$ $\displaystyle=\sum_{n<-cx,\,m\leq 0}(-1)^{\ell
n}e^{2\pi i(m-M)dx}\cdot(m-M)q^{\frac{\ell}{2}(n+cx)^{2}+(n+cx)(m-M)},$
$\displaystyle S_{2}$ $\displaystyle=\sum_{n>-cx,\,m\geq 1}(-1)^{\ell
n}e^{2\pi i(m-M)dx}\cdot(m-M)q^{\frac{\ell}{2}(n+cx)^{2}+(n+cx)(m-M)},$
$\displaystyle C_{1}$ $\displaystyle=\delta_{cx}(-1)^{\ell cx}e^{-2\pi i\ell
cdx^{2}}\left(\ell\left(cx+\frac{1}{2}\right)\frac{e^{-\pi i\ell
dx}}{1-e^{-2\pi idx}}+\frac{e^{-\pi i(\ell+2)dx}}{(1-e^{-2\pi
idx})^{2}}\right).$

Now we consider the terms $\widetilde{A}_{\ell}(-x(c\tau+d);\tau)$ and
$\partial\widetilde{A}_{\ell}(-x(c\tau+d);\tau)$. As a prerequisite, note that
for $\lambda,\mu\in\mathbb{Q}$ we have

(2.13)
$R(\lambda\tau+\mu;\tau)=\sum_{\nu\in\frac{1}{2}+\mathbb{Z}}(\textnormal{sgn}(\nu)-\textnormal{sgn}(\nu+\lambda))(-1)^{\nu-\frac{1}{2}}e^{-2\pi
i\nu\mu}q^{-\frac{1}{2}\nu^{2}-\lambda\nu}\\\
+\sum_{\nu\in\frac{1}{2}+\mathbb{Z}}\textnormal{sgn}(\nu+\lambda)\beta((\nu+\lambda)^{2}\cdot
2\operatorname{Im}\tau)(-1)^{\nu-\frac{1}{2}}e^{-2\pi
i\nu\mu}q^{-\frac{1}{2}\nu^{2}-\lambda\nu}$

where the first sum on the right-hand side is the holomorphic part and the
second sum the nonholomorphic part. On the other hand, since

$\displaystyle\frac{\partial}{\partial
u}E\left((\nu+\frac{\mathrm{Im}(u)}{\mathrm{Im}(\tau)})\sqrt{2\mathrm{Im}(\tau)}\right)$
$\displaystyle=-\frac{i}{2}\frac{\partial}{\partial\operatorname{Im}u}E\left((\nu+\frac{\mathrm{Im}(u)}{\mathrm{Im}(\tau)})\sqrt{2\mathrm{Im}(\tau)}\right)$
$\displaystyle=-ie^{-\pi(\nu+\frac{\operatorname{Im}z}{\operatorname{Im}\tau})^{2}\cdot
2\operatorname{Im}\tau}\cdot\sqrt{\frac{2}{\operatorname{Im}\tau}},$

we have

(2.14) $\partial
R(\lambda\tau+\mu;\tau)=\sum_{\nu\in\frac{1}{2}+\mathbb{Z}}(\textnormal{sgn}(\nu)-\textnormal{sgn}(\nu+\lambda))(-1)^{\nu-\frac{1}{2}}(-2\pi
i\nu)e^{-2\pi i\nu\mu}q^{-\frac{1}{2}\nu^{2}-\lambda\nu}\\\
+\sum_{\nu\in\frac{1}{2}+\mathbb{Z}}\textnormal{sgn}(\nu+\lambda)\beta((\nu+\lambda)^{2}\cdot
2\operatorname{Im}\tau)(-1)^{\nu-\frac{1}{2}}(-2\pi i\nu)e^{-2\pi
i\nu\mu}q^{-\frac{1}{2}\nu^{2}-\lambda\nu}\\\
+i\sum_{\nu\in\frac{1}{2}+\mathbb{Z}}e^{-\pi(\nu+\lambda)^{2}\cdot
2\operatorname{Im}\tau}\cdot\sqrt{\frac{2}{\operatorname{Im}\tau}}(-1)^{\nu-\frac{1}{2}}e^{-2\pi
i\nu\mu}q^{-\frac{1}{2}\nu^{2}-\lambda\nu}$

where the first sum on the right-hand side is the holomorphic part and the
remaining two sums the nonholomorphic part. By definition

$\ell cx(\widetilde{A}_{\ell}(-x(c\tau+d);\tau)+\frac{1}{2\pi
i}\partial\widetilde{A}_{\ell}(-x(c\tau+d);\tau)\\\ =\frac{1}{2\pi
i}\sum_{k=0}^{\ell-1}q^{-kcx}e^{-2\pi
ikdx}\theta\left(k\tau+\frac{\ell-1}{2};\ell\tau\right)\cdot\left(-\pi(\ell
cx+k)R\left(-(cx+\frac{k}{\ell})\ell\tau-\ell
dx-\frac{\ell-1}{2};\ell\tau\right)\right.\\\ +\left.\frac{i\ell}{2}\partial
R\left(-(cx+\frac{k}{\ell})\ell\tau-\ell
dx-\frac{\ell-1}{2};\ell\tau\right)\right).$

It follows from this, (2.13), (2.14) and the definition (2.2) of $\theta$ that
the holomorphic part of $q^{-\frac{\ell c^{2}x^{2}}{2}}e^{-2\pi i\ell
cdx^{2}}\left(\ell cx(\widetilde{A}_{\ell}(-x(c\tau+d);\tau)+\frac{1}{2\pi
i}\partial\widetilde{A}_{\ell}(-x(c\tau+d);\tau)\right)$ is equal to

(2.15) $\frac{1}{2\pi i}q^{-\frac{\ell c^{2}x^{2}}{2}}e^{-2\pi i\ell
cdx^{2}}\sum_{0\leq
k<\ell}\sum_{n\in\frac{1}{2}+\mathbb{Z}}\sum_{\nu\in\frac{1}{2}+\mathbb{Z}}e^{-2\pi
ikdx}q^{-kcx}e^{\pi i\ell n}q^{\frac{\ell}{2}n^{2}+kn}\\\
\cdot\left(\textnormal{sgn}(\nu)-\textnormal{sgn}(\nu-
cx-\frac{k}{\ell})\right)(-1)^{\nu-\frac{1}{2}}\pi(\ell\nu-\ell cx-k)e^{2\pi
i\nu(\ell dx+\frac{\ell-1}{2})}q^{-\frac{\ell}{2}\nu^{2}+(\ell cx+k)\nu}.$

Note that $\textnormal{sgn}(\nu)-\textnormal{sgn}(\nu-cx-\frac{k}{\ell})=0$
unless $0<\nu\leq cx+\frac{k}{\ell}$ since $cx+\frac{k}{\ell}\geq 0$.
Therefore, applying the changes of variables $n=\frac{1}{2}+n_{1}$ and
$\nu=\frac{1}{2}+\nu_{1}$ we find that (2.15) is equal to

(2.16) $(-1)^{\ell+1}e^{-2\pi i\ell cdx^{2}}\sum_{0\leq
k<\ell}\sum_{n_{1}\in\mathbb{Z}}\sum_{\begin{subarray}{c}{0\leq\nu_{1}\in\mathbb{Z}}\\\
{\nu_{1}<cx+k/\ell-1/2}\end{subarray}}e^{\pi i\ell
n_{1}}q^{\frac{\ell}{2}(n_{1}+\frac{1}{2}-cx)+(\ell
cx+k)(n_{1}+\frac{1}{2}-cx)}\\\
\cdot(-1)^{\nu_{1}}(\ell\nu_{1}+\frac{\ell}{2}-\ell cx-k)e^{2\pi
i\left(\nu_{1}(\ell
dx+\frac{\ell-1}{2})+(\frac{\ell}{2}-k)dx\right)}q^{-\frac{\ell}{2}(\nu+\frac{1}{2})^{2}+(\ell
cx+k)(\nu+\frac{1}{2})}.$

Set $n=n_{1}-\nu_{1}-1$ and $m=\ell\nu+\ell-k-1$, which gives a bijection333It
may happen that the domain and codomain of this bijection are the empty set.
from

$\\{(k,n_{1},\nu_{1})\in\mathbb{Z}^{3}\colon 0\leq
k<\ell,\,0\leq\nu_{1}<cx+k/\ell-1/2\\}$

onto

$\\{(n,m)\in\mathbb{Z}^{2}\colon 0\leq m<M-1\\}.$

Hence (2.16) becomes

$-S_{3}:=-\sum_{\begin{subarray}{c}{n\in\mathbb{Z}}\\\ {1\leq
m<M}\end{subarray}}(-1)^{\ell n}e^{2\pi
i(m-M)dx}\cdot(m-M)q^{\frac{\ell}{2}(n+cx)^{2}+(n+cx)(m-M)}.$

It follows from this, (2.12) and (2.8) that the holomorphic part of
$\frac{1}{2\pi i}e^{-\pi i\ell
cdx^{2}}\cdot\partial\widehat{A}_{\ell}(-x)|_{2}\gamma$ is equal to
$C_{1}-S_{1}+S_{2}-S_{3}$. Now if $cx\not\in\mathbb{Z}$, then $C=C_{1}=0$ and

$-S_{1}+S_{2}-S_{3}=\sum_{n>-cx,\,m\geq M}-\sum_{n<-cx,\,m<M}.$

Thus, the holomorphic part of $\frac{1}{2\pi i}e^{-\pi i\ell
cdx^{2}}\cdot\partial\widehat{A}_{\ell}(-x)|_{2}\gamma$ equals $H_{1}(\tau)$
as required. Otherwise if $cx\in\mathbb{Z}$, then the holomorphic part is
equal to

$C_{1}-S_{1}+S_{2}-S_{3}=C_{1}-\sum_{n=-cx,\,1\leq m<M}+\sum_{n>-cx,\,m\geq
M}-\sum_{n<-cx,\,m<M}.$

Since $C_{1}-\sum_{n=-cx,\,1\leq m<M}=C$, the holomorphic part is equal to
$H_{1}(\tau)$ as well in this case.

It remains to compute the nonholomorphic part of $\frac{1}{2\pi i}e^{-\pi
i\ell cdx^{2}}\cdot\partial\widehat{A}_{\ell}(-x)|_{2}\gamma$ which is equal
to the nonholomorphic part of $q^{-\frac{\ell c^{2}x^{2}}{2}}e^{-2\pi i\ell
cdx^{2}}\left(\ell cx(\widetilde{A}_{\ell}(-x(c\tau+d);\tau)+\frac{1}{2\pi
i}\partial\widetilde{A}_{\ell}(-x(c\tau+d);\tau)\right)$. This is equal to, by
(2.13) and (2.14),

$\frac{1}{2\pi i}q^{-\frac{\ell c^{2}x^{2}}{2}}e^{-2\pi i\ell
cdx^{2}}\sum_{0\leq k<\ell}q^{-kcx}e^{-2\pi
ikdx}\theta\left(k\tau+\frac{\ell-1}{2};\ell\tau\right)\\\
\cdot\sum_{\nu\in\frac{1}{2}+\mathbb{Z}}(-1)^{\nu-\frac{1}{2}}e^{2\pi
i\nu(\ell dx+\frac{\ell-1}{2})}f\left(\ell\nu-\ell
cx-k;\operatorname{Im}\tau\right)q^{-\frac{\ell}{2}\nu^{2}+(\ell cx+k)\nu}.$

Substituting the definition of $\theta$ in the above expression and applying
the change of variable $\nu=\nu^{\prime}+\frac{1}{2}$ we find that the
nonholomorphic part equals

$-\frac{1}{2\pi}e^{\pi i\ell
cx}\sum_{\begin{subarray}{c}{\nu^{\prime},m^{\prime}\in\mathbb{Z}}\\\ {0\leq
k<\ell}\end{subarray}}e^{2\pi
i\left(dx(\ell\nu^{\prime}+\ell-k-1-\frac{\ell}{2}-\ell cx+1)+\frac{1}{2}(\ell
m^{\prime}+k+\frac{\ell}{2}+\ell\nu^{\prime}+\ell-k-1-\frac{\ell}{2}-\ell
cx+1)\right)}\\\ \cdot f\left(\ell\nu^{\prime}+\frac{\ell}{2}-\ell
cx-k;\operatorname{Im}\tau\right)q^{\frac{1}{2\ell}\left((\ell
m^{\prime}+k+\frac{\ell}{2})^{2}-(\ell\nu^{\prime}+\ell-k-1-\frac{\ell}{2}-\ell
cx+1)^{2}\right)}$

Setting $\nu_{1}=\ell\nu^{\prime}+\ell-k-1$ and $m=\ell m^{\prime}+k$ we
simplify the above expression further to

$-\frac{1}{2\pi}e^{\pi i\ell
cx}\cdot\sum_{\begin{subarray}{c}\nu_{1},m\in\mathbb{Z}\\\
\nu_{1}+m\equiv-1\bmod{\ell}\end{subarray}}e^{2\pi
i\left((dx+\frac{1}{2})(\nu_{1}-M)+\frac{1}{2}(m+\frac{\ell}{2})\right)}f(\nu_{1}-M;\operatorname{Im}\tau)q^{\frac{1}{2\ell}\left((m+\frac{\ell}{2})^{2}-(\nu_{1}-M)^{2}\right)}.$

Finally, the above expression is actually equal to $H_{2}(\tau)$ which can be
seen by the change of variables $\nu_{1}=\ell n-m-1$. This concludes the whole
proof. ∎

######  Remark 2.2.

The terms “holomorphic part” and “nonholomorphic part” need to be further
clarified since a decomposition of a real analytic function into a sum of a
holomprhic one and a nonholomorphic one is not unique. Let $g$ be a real
analytic function of period $N$, say
$g=\partial\widehat{A}_{\ell}(-x)|_{2}\gamma$. Then $g(\tau)=\sum_{n\in
N^{-1}\mathbb{Z}}c_{n}(\operatorname{Im}\tau)q^{n}$ which converges
unconditionally for any fixed $\operatorname{Im}\tau$ with respect to the
inner product on $\mathbb{R}/N\mathbb{Z}$ by elementary Fourier analysis. The
uniquely defined coefficients $c_{n}(y)=N^{-1}\int_{0}^{N}g(x+iy)e^{-2\pi
in(x+iy)}\,\mathrm{d}x$ are real analytic since $g$ is. Applying integration
by parts to this integral we obtain $c_{n}(y)e^{-2\pi iny}=O(n^{-2})$ for $y$
in any compact set, from which it follows that the series of $g(\tau)$
converges absolutely and uniformly on any compact subset of $\mathscr{H}$. If
$\lim_{\operatorname{Im}\tau\to+\infty}c_{n}(\operatorname{Im}\tau)=c_{n}(+\infty)$
exists in $\mathbb{C}$ for each $n$ (this is the case for
$g=\partial\widehat{A}_{\ell}(-x)|_{2}\gamma$), then formally $g$ can be
decomposed as $H_{1}+H_{2}$ where

$\displaystyle H_{1}(\tau)$ $\displaystyle=\sum_{n\in
N^{-1}\mathbb{Z}}c_{n}(+\infty)q^{n},$ $\displaystyle H_{2}(\tau)$
$\displaystyle=\sum_{n\in
N^{-1}\mathbb{Z}}\left(c_{n}(\operatorname{Im}\tau)-c_{n}(+\infty)\right)q^{n}.$

However, the series $H_{1}(\tau)$ (or $H_{2}(\tau)$) not necessarily
converges. To ensure this decomposition makes sense, we should _assume_ that
$H_{2}(\tau)$ converges absolutely and uniformly on the sets
$\\{\tau\in\mathscr{H}\colon\operatorname{Im}\tau\geq Y_{0}\\}$ with $Y_{0}$
being any positive real number from which the absolute and compact uniform
convergence of $H_{1}$ follows. (This is the case for
$g=\partial\widehat{A}_{\ell}(-x)|_{2}\gamma$ and as well for $g$ being any
Harmonic Maass form.) Under these assumptions we call $H_{1}$, $H_{2}$ the
holomorphic part and the nonholomorphic part of $g$ respectively. Note that
the nonholomorphic part $H_{2}$ of $g$ is characterized by the conditions
$g-H_{2}$ is holomorphic on $\mathscr{H}$ and the Fourier expansion
$H_{2}(\tau)=\sum_{n\in
N^{-1}\mathbb{Z}}c_{n}^{\prime}(\operatorname{Im}\tau)q^{n}$ satisfies
$\lim_{\operatorname{Im}\tau\to+\infty}c_{n}^{\prime}(\operatorname{Im}\tau)=0$
for each $n$. Also note that our definition does not apply to all cases, e.g.,
to Harmonic Maass forms of manageable growth of weight $\leq 1$.

The following property is immediate according to Remark 2.2.

######  Corollary 2.3.

Let $f_{1}$ and $f_{2}$ have well defined holomorphic parts $H_{11},H_{21}$
and nonholomorphic parts $H_{12},H_{22}$ respectively. Then the holomorphic
part of $f_{1}+f_{2}$ is $H_{11}+H_{21}$ and the nonholomorphic part of
$f_{1}+f_{2}$ is $H_{12}+H_{22}$.

A function series $\sum_{n}f_{n}(z)$ is said to converge normally for $z\in
A\subseteq\mathbb{C}$ if there is a sequence $c_{n}\geq 0$ with the property
$\lvert f_{n}(z)\rvert\leq c_{n},\,z\in A$ and $\sum_{n}c_{n}<+\infty$. Note
that normal convergence implies absolute and uniform convergence and that the
product of two normally convergent function series still converges normally.
Also note that $H_{1}(\tau)$ and $H_{2}(\tau)$ in Theorem 2.1 both converge
normally on $\\{\tau\in\mathscr{H}\colon\operatorname{Im}\tau\geq Y_{0}\\}$
where $Y_{0}>0$.

######  Proposition 2.4.

Let $f_{1},\,f_{2}$ be real analytic (or of class $C^{2}$ in general)
functions on $\mathscr{H}$ of period $N$. ($N$ is a positive integer.) Suppose

  1. (1)

$f_{1}$ has well defined holomorphic part $H_{1}$ and nonholomorphic part
$H_{2}$,

  2. (2)

The Fourier expansion of $H_{2}(\tau)$ converges normally on
$\\{\tau\in\mathscr{H}\colon\operatorname{Im}\tau\geq Y_{0}\\}$ for any
positive real number $Y_{0}$,

  3. (3)

$f_{2}$ is holomorphic and there is $n_{0}\in N^{-1}\mathbb{Z}$ such that
$f_{2}=\sum_{n_{0}\leq n\in N^{-1}\mathbb{Z}}a_{n}q^{n}$.

Then the holomorphic part of $f_{1}f_{2}$ is $H_{1}f_{2}$ and the
nonholomorphic part is $H_{2}f_{2}$.

###### Proof.

Set $y=\operatorname{Im}\tau$. We write

$H_{1}(\tau)=\sum_{n\in N^{-1}\mathbb{Z}}c_{n}q^{n},\qquad
H_{2}(\tau)=\sum_{n\in N^{-1}\mathbb{Z}}d_{n}(y)q^{n}$

according to Remark 2.2. Since $H_{1}$ and $f_{2}$ converge on $\lvert
q\rvert<1$, so is $H_{1}f_{2}$ by the theory of Laurent series. By assumption
$H_{2}$ and by the theory of Laurent series $f_{2}$, converge normally on
$\\{\tau\in\mathscr{H}\colon\operatorname{Im}\tau\geq Y_{0}\\}$. Therefore the
double series

$H_{2}f_{2}=\sum_{n\in
N^{-1}\mathbb{Z}}\left(\sum_{n_{1}+n_{2}=n}d_{n_{1}}(y)a_{n_{2}}\right)q^{n}$

converges normally. Set $b_{n}(y)=\sum_{n_{1}+n_{2}=n}d_{n_{1}}(y)a_{n_{2}}$.
It remains to prove $\lim_{y\to+\infty}b_{n}(y)=0$. For each $n\in
N^{-1}\mathbb{Z}$, define

$Y_{n}:=\sup\left\\{\frac{n_{1}-n}{n_{1}}\colon n_{1}\in
N^{-1}\mathbb{Z},\,n_{1}<\min\\{0,n\\}\right\\}.$

Clearly, $0<Y_{n}<+\infty$. We split $b_{n}(y)$ into three parts:

$b_{n}(y)=\sum_{\begin{subarray}{c}n_{1}+n_{2}=n\\\ 0\leq n_{1}\leq
n-n_{0}\end{subarray}}+\sum_{\begin{subarray}{c}n_{1}+n_{2}=n\\\ n\leq
n_{1}<0\end{subarray}}+\sum_{\begin{subarray}{c}n_{1}+n_{2}=n\\\
n_{1}<\min\\{0,n\\}\end{subarray}}$

and use $b_{n}^{(1)}(y)$, $b_{n}^{(2)}(y)$ and $b_{n}^{(3)}(y)$ to denote the
three sums on the right-hand side. We want to show that we can interchange the
limit $y\to+\infty$ and the summation $\sum_{n_{1}+n_{2}=n}$. The sums
$b_{n}^{(1)}(y)$ and $b_{n}^{(2)}(y)$ are finite so let us consider
$b_{n}^{(3)}(y)$. If $y\geq Y_{n}$, then for any $n_{1}\in N^{-1}\mathbb{Z}$
we have $e^{2\pi n_{1}y}\leq e^{-2\pi n_{2}}$ where $n_{2}=n-n_{1}$. It
follows that for $y\geq Y_{n}$

$\displaystyle\lvert b_{n}^{(3)}(y)\rvert$
$\displaystyle\leq\sum_{n_{1}<\min\\{0,n\\}}\lvert
d_{n_{1}}(y)a_{n_{2}}\rvert=\sum_{n_{1}<\min\\{0,n\\}}\lvert
d_{n_{1}}(y)\rvert e^{-2\pi n_{1}y}\cdot\lvert a_{n_{2}}\rvert e^{2\pi
n_{1}y}$ $\displaystyle\leq\sum_{n_{1}<\min\\{0,n\\}}\lvert d_{n_{1}}(y)\rvert
e^{-2\pi n_{1}y}\cdot\lvert a_{n_{2}}\rvert e^{-2\pi n_{2}}$
$\displaystyle\leq\sum_{n_{1}\in N^{-1}\mathbb{Z}}\lvert d_{n_{1}}(y)\rvert
e^{-2\pi n_{1}y}\cdot\sum_{n_{2}\in N^{-1}\mathbb{Z}}\lvert a_{n_{2}}\rvert
e^{-2\pi n_{2}}<+\infty$

since $H_{2}$, respectively $f_{2}$, converge normally on
$\\{\tau\in\mathscr{H}\colon\operatorname{Im}\tau\geq Y_{n}\\}$, respectively
$\\{\tau\in\mathscr{H}\colon\operatorname{Im}\tau=1\\}$. Thus the series
$b_{n}(y)$ converges uniformly on $y\geq Y_{n}$ and hence
$\lim_{y\to+\infty}b_{n}(y)=0$ by interchanging limits and uniform limits for
each $n$. This concludes the whole proof. ∎

The order
$\operatorname{ord}_{i\infty}\partial\widehat{A}_{\ell}(-x)|_{2}\gamma$, which
is defined to be the exponent $n$ (possibly nonintegral) of the leading term
$c_{n}q^{n}$ in the Fourier expansion of the holomorphic part of
$\partial\widehat{A}_{\ell}(-x)|_{2}\gamma$, can be computed easily using
Theorem 2.1:

######  Corollary 2.5.

Recall that $M=\ell cx+\frac{\ell}{2}$. If

$(\ell+1)\cdot(\lceil-cx\rceil+cx)+\lceil M\rceil-M>\frac{\ell}{2}+1,$

then we have

$\operatorname{ord}\nolimits_{i\infty}\partial\widehat{A}_{\ell}(-x)|_{2}\gamma=\frac{\ell}{2}(\lceil-
cx\rceil+cx-1)^{2}+(\lceil-cx\rceil+cx-1)\cdot(\lceil M\rceil-M-1).$

On the other hand, if the reversed inequality holds, then we have

$\operatorname{ord}\nolimits_{i\infty}\partial\widehat{A}_{l}(-x)|_{2}\gamma=\frac{\ell}{2}(\lceil-
cx\rceil+cx)^{2}+(\lceil-cx\rceil+cx)\cdot(\lceil M\rceil-M).$

###### Proof.

Recall that $H_{1}(\tau)$ in Theorem 2.1 is the holomorphic part of
$\frac{1}{2\pi i}e^{-\pi i\ell
cdx^{2}}\cdot\partial\widehat{A}_{\ell}(-x)|_{2}\gamma$. Thus we need to find
the leading term of $H_{1}(\tau)$. If $cx\in\mathbb{Z}$ then
$dx\not\in\mathbb{Z}$ and hence $C\neq 0$. Since the exponent of $q$ of any
term in the sum $\sum_{n>-cx,\,m\geq M}-\sum_{n<-cx,\,m<M}$ is positive, we
have
$\operatorname{ord}\nolimits_{i\infty}\partial\widehat{A}_{\ell}(-x)|_{2}\gamma=0$
and hence the required assertion holds in this case. Now assume that
$cx\not\in\mathbb{Z}$. Then $C=0$. The exponent of $q$ of the leading term of
the sum $\sum_{n>-cx,\,m\geq M}$ is equal to

$e_{1}=\frac{\ell}{2}(\lceil-cx\rceil+cx)^{2}+(\lceil-cx\rceil+cx)\cdot(\lceil
M\rceil-M)$

and that of $\sum_{n<-cx,\,m<M}$ is equal to

$e_{2}=\frac{\ell}{2}(\lceil-cx\rceil+cx-1)^{2}+(\lceil-
cx\rceil+cx-1)\cdot(\lceil M\rceil-M-1).$

Therefore
$\operatorname{ord}\nolimits_{i\infty}\partial\widehat{A}_{l}(-x)|_{2}\gamma=\min\\{e_{1},e_{2}\\}$
since $e_{1}\neq e_{2}$. It remains to compare $e_{1}$ and $e_{2}$. Since

$e_{1}-e_{2}=(\ell+1)(\lceil-cx\rceil+cx)+\lceil M\rceil-M-(\frac{\ell}{2}+1)$

the desired assertion holds. ∎

Note that the case $(\ell+1)\cdot(\lceil-cx\rceil+cx)+\lceil
M\rceil-M=\frac{\ell}{2}+1$ could not happen.

For any coprime integers $a,\,c$, we define
$\operatorname{ord}_{a/c}f=\operatorname{ord}_{i\infty}f|_{2}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)$, where $b,\,d$ are any integers satisfying
$ad-bc=1$. This is well defined provided that $f$ satisfies the modular
transformation equations on a subgroup of $\mathrm{SL}_{2}(\mathbb{Z})$ of
finite index. Thus Corollary 2.5 amounts to giving
$\operatorname{ord}_{a/c}\partial\widehat{A}_{l}(-x)$ which, as one can easily
see, is independent of $a$.

In the remainder we need another function related to
$\partial\widehat{A}_{\ell}$, namely, the function

(2.17)
$g_{\ell,x}(\tau):=q^{-\frac{\ell}{2}x^{2}}\left(\partial\widehat{A}_{\ell}(x\tau;\tau)-2\pi
i\ell x\widehat{A}_{\ell}(x\tau;\tau)\right).$

As before, $x$ is a nonintegral positive rational number and $k$ is a positive
integer such that $kx\in\mathbb{Z}$.

######  Proposition 2.6.

We have
$g_{\ell,x}=\partial\widehat{A}_{l}(-x)|_{2}\left(\begin{smallmatrix}{0}&{-1}\\\
{1}&{0}\end{smallmatrix}\right)$. As a consequence, for any
$\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)\in\Gamma_{1}(k)$ we have

$g_{\ell,x}\left(k\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{2}e^{-\pi i\ell
abkx^{2}}\cdot(-1)^{\ell((a-1)x+kbx)}g_{\ell,x}(k\tau).$

###### Proof.

As in the proof of Theorem 2.1, we shall freely use the theory of Jacobi forms
here. We can express $g_{\ell,x}$ in terms of $\widehat{A}_{\ell}$ as

(2.18) $g_{\ell,x}(\tau)=(\partial\widehat{A}_{\ell}-2\pi i\ell
x\widehat{A}_{\ell})|_{2,-\ell/2}[(x,0),1](0;\tau).$

Setting $u=0$ and $\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)=\left(\begin{smallmatrix}{0}&{1}\\\
{-1}&{0}\end{smallmatrix}\right)$ in (2.7) we find that

(2.19)
$\partial\widehat{A}_{\ell}(-x;\tau)|_{2}\left(\begin{smallmatrix}{0}&{-1}\\\
{1}&{0}\end{smallmatrix}\right)=\partial\widehat{A}_{\ell}(-x;\tau)|_{2}\left(\begin{smallmatrix}{0}&{1}\\\
{-1}&{0}\end{smallmatrix}\right)=\partial(\widehat{A}_{\ell}|_{1,-\ell/2}[(x,0),1])(0;\tau).$

By the chain rule for $\frac{\partial}{\partial u}$,

(2.20) $\partial(\widehat{A}_{\ell}|_{1,-\ell/2}[(x,0),1])=-2\pi i\ell
x\widehat{A}_{\ell}|_{2,-\ell/2}[(x,0),1]+(\partial\widehat{A}_{\ell})|_{2,-\ell/2}[(x,0),1]).$

Combining (2.18), (2.19) and (2.20) we obtain
$g_{\ell,x}=\partial\widehat{A}_{l}(-x)|_{2}\left(\begin{smallmatrix}{0}&{-1}\\\
{1}&{0}\end{smallmatrix}\right)$ as required. To prove the last assertion, let
$\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)\in\Gamma_{1}(k)$ be arbitrary. Then

$\displaystyle(c\tau+d)^{-2}g_{\ell,x}\left(k\frac{a\tau+b}{c\tau+d}\right)$
$\displaystyle=(c\tau+d)^{-2}\left(k\frac{a\tau+b}{c\tau+d}\right)^{-2}\partial\widehat{A}_{\ell}\left(-x;-\frac{c\tau+d}{k(a\tau+b)}\right)$
$\displaystyle=\partial\widehat{A}_{\ell}(-x)|_{2}\left(\begin{smallmatrix}{-c/k}&{-d}\\\
{a}&{bk}\end{smallmatrix}\right)(k\tau)$
$\displaystyle=\partial\widehat{A}_{\ell}(-x)|_{2}\left(\begin{smallmatrix}{d}&{-c/k}\\\
{-bk}&{a}\end{smallmatrix}\right)|_{2}\left(\begin{smallmatrix}{0}&{-1}\\\
{1}&{0}\end{smallmatrix}\right)(k\tau)$ $\displaystyle=(-1)^{\ell
x(a-bk-1)}e^{-\pi i\ell abkx^{2}}g_{\ell,x}(k\tau),$

where we have used (2.5) in the last equality. ∎

The holomorphic part and nonholomorphic part of $g_{\ell,x}(k\tau)$ at any
cusp can be derived from the above proposition and Theorem 2.1 of which we
omit the details. What we need below are formulas for the orders of
$g_{\ell,x}(k\tau)$ at cusps:

######  Corollary 2.7.

Let $a,c$ be nonnegative coprime integers. Set $c^{\prime}=\frac{ka}{(ka,c)}$
and $M^{\prime}=\ell c^{\prime}x+\frac{\ell}{2}$. If

$(\ell+1)\cdot(\lceil-c^{\prime}x\rceil+c^{\prime}x)+\lceil
M^{\prime}\rceil-M^{\prime}>\frac{\ell}{2}+1,$

then

$\operatorname{ord}\nolimits_{\frac{a}{c}}(g_{\ell,x}(k\tau))=\frac{(ka,c)^{2}}{k}\left(\frac{\ell}{2}(\lceil-c^{\prime}x\rceil+c^{\prime}x-1)^{2}+(\lceil-c^{\prime}x\rceil+c^{\prime}x-1)\cdot(\lceil
M^{\prime}\rceil-M^{\prime}-1)\right).$

On the other hand, if the reversed inequality holds, then

$\operatorname{ord}\nolimits_{\frac{a}{c}}(g_{\ell,x}(k\tau))=\frac{(ka,c)^{2}}{k}\left(\frac{\ell}{2}(\lceil-c^{\prime}x\rceil+c^{\prime}x)^{2}+(\lceil-c^{\prime}x\rceil+c^{\prime}x)\cdot(\lceil
M^{\prime}\rceil-M^{\prime})\right).$

Before giving the proof, we extend the definition of slash operators a bit.
Let $r\in\mathbb{Z}$. For a real matrix $\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)$ with positive determinant and a function $f$
on $\mathscr{H}$, set $f|_{r}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)(\tau)=(ad-
bc)^{r/2}\cdot(c\tau+d)^{-r}f\left(\frac{a\tau+b}{c\tau+d}\right)$. One can
easily check that
$f|_{r}(\gamma_{1}\gamma_{2})=(f|_{r}\gamma_{1})|_{r}\gamma_{2}$ for matrices
$\gamma_{1}$ and $\gamma_{2}$ of positive determinants and that
$g_{\ell,x}(k\tau)=k^{-1}g_{\ell,x}|_{2}\left(\begin{smallmatrix}{k}&{0}\\\
{0}&{1}\end{smallmatrix}\right)(\tau)$.

###### Proof of Corollary 2.7.

Let $a^{\prime},b^{\prime}$ be integers satisfying
$(ka,c)=kaa^{\prime}+cb^{\prime}$ and $b,d$ be integers satisfying $ad-bc=1$.
Then

(2.21) $\begin{pmatrix}{0}&{-1}\\\
{1}&{0}\end{pmatrix}\begin{pmatrix}{k}&{0}\\\
{0}&{1}\end{pmatrix}\begin{pmatrix}{a}&{b}\\\
{c}&{d}\end{pmatrix}=\begin{pmatrix}{-\frac{c}{(ka,c)}}&{-a^{\prime}}\\\
{\frac{ka}{(ka,c)}}&{-b^{\prime}}\end{pmatrix}\begin{pmatrix}{(ka,c)}&{ka^{\prime}b+b^{\prime}d}\\\
{0}&{\frac{k}{(ka,c)}}\end{pmatrix}$

with the first factor on the right-hand side in $\mathrm{SL}_{2}(\mathbb{Z})$.
Thus, by Proposition 2.6 we have

$\displaystyle\operatorname{ord}\nolimits_{\frac{a}{c}}(g_{\ell,x}(k\tau))$
$\displaystyle=\operatorname{ord}\nolimits_{i\infty}(g_{\ell,x}|_{2}\left(\begin{smallmatrix}{k}&{0}\\\
{0}&{1}\end{smallmatrix}\right)|_{2}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right))$
$\displaystyle=\operatorname{ord}\nolimits_{i\infty}(\partial\widehat{A}_{\ell}(-x)|_{2}\left(\begin{smallmatrix}{0}&{-1}\\\
{1}&{0}\end{smallmatrix}\right)\left(\begin{smallmatrix}{k}&{0}\\\
{0}&{1}\end{smallmatrix}\right)\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right))$
$\displaystyle=\operatorname{ord}\nolimits_{i\infty}\left(\partial\widehat{A}_{\ell}(-x)|_{2}\left(\begin{smallmatrix}{-\frac{c}{(ka,c)}}&{-a^{\prime}}\\\
{\frac{ka}{(ka,c)}}&{-b^{\prime}}\end{smallmatrix}\right)\left(\begin{smallmatrix}{(ka,c)}&{ka^{\prime}b+b^{\prime}d}\\\
{0}&{\frac{k}{(ka,c)}}\end{smallmatrix}\right)\right)$
$\displaystyle=\frac{(ka,c)^{2}}{k}\cdot\operatorname{ord}\nolimits_{-\frac{c}{ka}}\partial\widehat{A}_{\ell}(-x).$

The assertions follow from this and Corollary 2.5. ∎

###  2.3. The $U_{p,k}$ operator and its variant

Let $p$ be a positive number and $k$ be an integer. The operator $U_{p,k}$ is
given by

$U_{p,k}(f(\tau)):=\frac{1}{p}\sum_{m=0}^{p-1}\zeta_{p}^{-mk}f\left(\frac{\tau+m}{p}\right),$

where $f(\tau)$ is a function defined on $\mathscr{H}$. If $f(\tau)$ has the
Fourier expansion

$f(\tau)=\sum_{n\geq n_{0}}a(n)q^{n},$

then one can easily verify

(2.22) $\displaystyle U_{p,k}(g(q^{p})f(\tau))=q^{\frac{k}{p}}g(q)\sum_{n\geq
n_{0}-\left[\frac{k}{p}\right]}a(pn+k)q^{n}$

for any $q$-series

$g(q)=\sum_{m\geq m_{0}}b(m)q^{m}.$

The following lemma relates the transformation equations of $U_{p,k}f$ to that
of $f$. Let $\Gamma(p)$ denote the principal congruence subgroup of level $p$
consisting of integral modular matrices $\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)$ with $a\equiv d\equiv 1\pmod{p}$ and $b\equiv
c\equiv 0\pmod{p}$.

######  Lemma 2.8.

Let $r$ be an even integer. Suppose $f$ satisfies $f|_{r}\gamma=\chi(\gamma)f$
for any $\gamma\in\Gamma_{1}(p)$ where $\chi$ is a linear character of
$\Gamma_{1}(p)$ such that $\chi\left(\begin{smallmatrix}{a}&{b/p}\\\
{cp}&{d}\end{smallmatrix}\right)=1$ for any $\begin{pmatrix}{a}&{b}\\\
{c}&{d}\end{pmatrix}\in\Gamma(p)$. Then

$(U_{p,k}f)|_{r}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)=\zeta_{p}^{bk}\cdot
U_{p,k}f,\quad\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)\in\Gamma_{1}(p).$

###### Proof.

As in the proof of [24, Lemma 2.8], for $\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)\in\Gamma_{1}(p)$,

(2.23) $\begin{pmatrix}{1}&{m}\\\
{0}&{p}\end{pmatrix}\begin{pmatrix}{a}&{b}\\\
{c}&{d}\end{pmatrix}=\begin{pmatrix}{a+cm}&{\frac{(d-a)m+b(1-a)-cm(m+b)}{p}}\\\
{pc}&{-c(m+b)+d}\end{pmatrix}\begin{pmatrix}{1}&{m+b}\\\
{0}&{p}\end{pmatrix},$

with the second factor on the right-hand side, denoted by $\gamma_{m}$ here,
in $\Gamma_{1}(p)$. It follows directly from the assumption that
$\chi(\gamma_{m})=1$ since $\left(\begin{smallmatrix}{1}&{0}\\\
{0}&{p}\end{smallmatrix}\right)^{-1}\gamma_{m}\left(\begin{smallmatrix}{1}&{0}\\\
{0}&{p}\end{smallmatrix}\right)\in\Gamma(p)$. Therefore,

$\displaystyle(U_{p,k}f)|_{r}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)$ $\displaystyle=p^{\frac{r}{2}-1}\sum_{0\leq
m<p}\zeta_{p}^{-mk}f|_{r}\left(\begin{smallmatrix}{1}&{m}\\\
{0}&{p}\end{smallmatrix}\right)\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)$ $\displaystyle=p^{\frac{r}{2}-1}\sum_{0\leq
m<p}\zeta_{p}^{-mk}f|_{r}\gamma_{m}|_{r}\left(\begin{smallmatrix}{1}&{m+b}\\\
{0}&{p}\end{smallmatrix}\right)$
$\displaystyle=\zeta_{p}^{bk}p^{\frac{r}{2}-1}\sum_{0\leq
m<p}\zeta_{p}^{-(m+b)k}\chi(\gamma_{m})f|_{r}\left(\begin{smallmatrix}{1}&{m+b}\\\
{0}&{p}\end{smallmatrix}\right)$ $\displaystyle=\zeta_{p}^{bk}\cdot U_{p,k}f.$

The last equality relies on the fact that
$\zeta_{p}^{-(m+b)k}f|_{r}\left(\begin{smallmatrix}{1}&{m+b}\\\
{0}&{p}\end{smallmatrix}\right)=\zeta_{p}^{-(m+b-p)k}f|_{r}\left(\begin{smallmatrix}{1}&{m+b-p}\\\
{0}&{p}\end{smallmatrix}\right)$ because
$\chi\left(\begin{smallmatrix}{1}&{1}\\\ {0}&{1}\end{smallmatrix}\right)=1$ by
assumption and $\zeta_{p}^{pk}=1$. ∎

######  Remark 2.9.

According to (2.5), when $p$ is odd and $j\in\mathbb{Z}$ is not divisible by
$p$, $f=\partial\widehat{A}_{\ell}(-j/p)$ satisfies the condition of the above
lemma. Moreover, Lemma 2.8 of Mao [24] is a special case of the above lemma.
Our proof is essentially identical to Mao’s.

If $\chi$ is different from the one in the above lemma, it is still possible
that $U_{p,k}f$ is modular on $\Gamma_{1}(p)$ provided that we modify
$U_{p,k}$ slightly according to $\chi$. We make no attempt to present the most
general definition but give the following one which is precisely what we need
in the proof of the main theorem:

$U^{\prime}_{p,k}(f(\tau)):=\frac{1}{p}\sum_{m=0}^{p-1}\zeta_{24p}^{-m(p-1)}\zeta_{p}^{-mk}f\left(\frac{\tau+m}{p}\right).$

One can check immediately that

(2.24) $U^{\prime}_{p,k}(q^{\frac{p-1}{24}}(q^{p};q^{p})_{\infty}\cdot
f)=q^{\frac{p-1}{24p}}(q;q)_{\infty}\cdot U_{p,k}f,$

so $U^{\prime}_{p,k}$ is just a reformulation of $U_{p,k}$. However, when $f$
itself is not modular while $q^{-1/24}f$ is, then it is necessary to shift the
focus from $U_{p,k}$ to $U^{\prime}_{p,k}$ for proving modularity. We now give
the analogue of Lemma 2.8, which plays a key role in the proof of the main
theorems, for $U^{\prime}_{p,k}$ acting on certain concrete functions. Recall
that the Dedekind eta function is defined by
$\eta(\tau)=q^{1/24}(q;q)_{\infty}$.

######  Lemma 2.10.

Let $p$ be an odd positive integer not divisible by $3$ (say, $p$ is a prime
$>3$) and $0\leq k<p$, $0<j<p$. Set

$f=U_{p,k}^{\prime}\left(q^{\frac{p-1}{24}}\frac{(q^{p};q^{p})_{\infty}}{(q;q)_{\infty}}\partial\widehat{A}_{\ell}(-j/p)\right)=U_{p,k}^{\prime}\left(\frac{\eta(p\tau)}{\eta(\tau)}\partial\widehat{A}_{\ell}(-j/p;\tau)\right).$

If there exists an integer $v$ and a positive integer $\ell_{1}$ such that
$12\ell_{1}v^{2}\equiv-24k+1\pmod{p}$ and $v\not\equiv 0\pmod{p}$, then $f$
transforms like $\eta(\tau)\eta(p\tau)^{-1}g_{\ell_{1},v/p}(p\tau)$ (c.f.
(2.17)), that is,

$(c\tau+d)^{-2}\eta\left(p\frac{a\tau+b}{c\tau+d}\right)\eta\left(\frac{a\tau+b}{c\tau+d}\right)^{-1}f\left(\frac{a\tau+b}{c\tau+d}\right)\\\
=e^{-\frac{\pi
i\ell_{1}abv^{2}}{p}}\cdot(-1)^{\ell_{1}v((a-1)/p+b)}\eta(p\tau)\eta(\tau)^{-1}f(\tau),\quad\begin{pmatrix}{a}&{b}\\\
{c}&{d}\end{pmatrix}\in\Gamma_{1}(p).$

In particular, if $-24k+1\equiv 0\pmod{p}$, then $f$ transforms like
$\eta(\tau)\eta(p\tau)^{-1}$.

###### Proof.

It is possible to produce a proof similar to that of Lemma 2.8 but such a
proof is rather tedious. The most concise proof makes use of the concept of
generalized double coset operators introduced in [33, Section 3]. Set
$f_{0}(\tau)=\frac{\eta(p\tau)}{\eta(\tau)}\partial\widehat{A}_{\ell}(-j/p;\tau)$.
Let $\chi_{1^{-1}p^{1}}$, $\chi_{1^{1}p^{-1}}$, $\phi_{\ell,j/p}$,
$\psi_{\ell_{1},v/p}$ be the characters444If a nonzero function $f$ satisfies
$f|_{r}\gamma=\chi(\gamma)f$ for any $\gamma\in G$ where
$\chi(\gamma)\in\mathbb{C}$, $G$ is a finite index subgroup of
$\mathrm{SL}_{2}(\mathbb{Z})$ and $r$ is an integer, then the character of $f$
is defined to be the map $\gamma\mapsto\chi(\gamma)$ on $G$ which turns out to
be a complex linear group character. of $\eta(\tau)^{-1}\eta(p\tau)$,
$\eta(\tau)\eta(p\tau)^{-1}$, $\partial\widehat{A}_{\ell}(-j/p;\tau)$ and
$g_{\ell_{1},v/p}(p\tau)$ respectively. The formulas for $\phi_{\ell,j/p}$ and
$\psi_{\ell_{1},v/p}$ have been given in (2.5) and Proposition 2.6
respectively. For the formulas for $\chi_{1^{-1}p^{1}}$, $\chi_{1^{1}p^{-1}}$
see [33, Eq. (15) and (16)] or (2.28) which is due to H. Petersson. One can
verify that $\Gamma_{1}(p)\left(\begin{smallmatrix}{1}&{0}\\\
{0}&{p}\end{smallmatrix}\right)\Gamma_{1}(p)=\cup_{0\leq
m<p}\Gamma_{1}(p)\left(\begin{smallmatrix}{1}&{0}\\\
{0}&{p}\end{smallmatrix}\right)\left(\begin{smallmatrix}{1}&{m}\\\
{0}&{1}\end{smallmatrix}\right)$ which is a disjoint union using, for
instance, (2.23). Thus, if we can prove that

(2.25) $\chi_{1^{-1}p^{1}}\begin{pmatrix}{a}&{b/p}\\\
{cp}&{d}\end{pmatrix}\phi_{\ell,j/p}\begin{pmatrix}{a}&{b/p}\\\
{cp}&{d}\end{pmatrix}=\chi_{1^{1}p^{-1}}\begin{pmatrix}{a}&{b}\\\
{c}&{d}\end{pmatrix}\psi_{\ell_{1},v/p}\begin{pmatrix}{a}&{b}\\\
{c}&{d}\end{pmatrix}$

for $\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)\in\left(\begin{smallmatrix}{1}&{0}\\\
{0}&{p}\end{smallmatrix}\right)^{-1}\Gamma_{1}(p)\left(\begin{smallmatrix}{1}&{0}\\\
{0}&{p}\end{smallmatrix}\right)\cap\Gamma_{1}(p)=\Gamma(p)$, then

$T_{p}f_{0}:=\frac{1}{p}\sum_{0\leq
m<p}\chi_{1^{1}p^{-1}}\cdot\psi_{\ell_{1},v/p}\begin{pmatrix}{1}&{m}\\\
{0}&{1}\end{pmatrix}^{-1}f_{0}\left(\frac{\tau+m}{p}\right)$

transforms like $\eta(\tau)\eta(p\tau)^{-1}g_{\ell_{1},v/p}(p\tau)$ according
to [33, Proposition 3.2(3) and (4)]. Since
$12\ell_{1}v^{2}\equiv-24k+1\pmod{p}$ and $p^{2}-1\equiv 0\pmod{24}$ we have
$\chi_{1^{1}p^{-1}}\cdot\psi_{\ell_{1},v/p}\left(\begin{smallmatrix}{1}&{m}\\\
{0}&{1}\end{smallmatrix}\right)^{-1}=\zeta_{24p}^{-m(p-1)}\zeta_{p}^{-mk}$
which means $T_{p}=U^{\prime}_{p,k}$. It remains to prove (2.25). Let
$\chi_{\eta}$ be the multiplier system of $\eta(\tau)$ (c.f. (2.28), the
Petersson’s formula). Then

$\displaystyle\chi_{1^{-1}p^{1}}\left(\begin{smallmatrix}{a}&{b/p}\\\
{cp}&{d}\end{smallmatrix}\right)$
$\displaystyle=\chi_{\eta(p\tau)}\left(\begin{smallmatrix}{a}&{b/p}\\\
{cp}&{d}\end{smallmatrix}\right)\chi_{\eta}^{-1}\left(\begin{smallmatrix}{a}&{b/p}\\\
{cp}&{d}\end{smallmatrix}\right)$
$\displaystyle=\chi_{\eta}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)\chi_{\eta}^{-1}\left(\begin{smallmatrix}{a}&{bp}\\\
{c/p}&{d}\end{smallmatrix}\right)$
$\displaystyle=\chi_{\eta}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)\chi_{\eta(p\tau)}^{-1}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)$
$\displaystyle=\chi_{1^{1}p^{-1}}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right),$

where $\chi_{\eta}^{-1}\left(\begin{smallmatrix}{a}&{b/p}\\\
{cp}&{d}\end{smallmatrix}\right)=\chi_{\eta}^{-1}\left(\begin{smallmatrix}{a}&{bp}\\\
{c/p}&{d}\end{smallmatrix}\right)$ follows from the fact $24\mid p^{2}-1$ and
(2.28). Thus the desired equality (2.25) is equivalent to
$\phi_{\ell,j/p}\left(\begin{smallmatrix}{a}&{b/p}\\\
{cp}&{d}\end{smallmatrix}\right)=\psi_{\ell_{1},v/p}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)$ which actually holds since

$\displaystyle\phi_{\ell,j/p}\left(\begin{smallmatrix}{a}&{b/p}\\\
{cp}&{d}\end{smallmatrix}\right)$ $\displaystyle=(-1)^{\frac{\ell
j(pc+d-1)}{p}}e^{\frac{\pi i\ell j^{2}cd}{p}}=(-1)^{\ell
j\left(c+\frac{d-1}{p}+\frac{cd}{p}\right)}=1$

and

$\displaystyle\psi_{\ell_{1},v/p}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)$
$\displaystyle=(-1)^{\ell_{1}v\left(\frac{a-1}{p}+b\right)}e^{\frac{-\pi
i\ell_{1}v^{2}ab}{p}}=(-1)^{\ell_{1}v\left(b+\frac{a-1}{p}-\frac{ab}{p}\right)}=1.$

∎

We end this subsection with a lemma concerning the orders of
$U^{\prime}_{p,k}f$ at cusps.

######  Lemma 2.11.

Let the notation be as in Lemma 2.10 and let $a,c$ be coprime integers. Set
$f_{0}=q^{\frac{p-1}{24}}\frac{(q^{p};q^{p})_{\infty}}{(q;q)_{\infty}}\partial\widehat{A}_{\ell}(-j/p)$.
We have

$\operatorname{ord}\nolimits_{\frac{a}{c}}U^{\prime}_{p,k}f_{0}\geq\min_{0\leq
m<p}\frac{\gcd(a+cm,cp)^{2}}{p}\cdot\operatorname{ord}\nolimits_{\frac{a+cm}{cp}}f_{0}.$

###### Proof.

Let $m\in\mathbb{Z}$ and $\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)\in\mathrm{SL}_{2}(\mathbb{Z})$. Set
$g=\gcd(a+cm,cp)$. Then

$\begin{pmatrix}{1}&{m}\\\ {0}&{p}\end{pmatrix}\begin{pmatrix}{a}&{b}\\\
{c}&{d}\end{pmatrix}=\begin{pmatrix}{(a+cm)g^{-1}}&{-y}\\\
{cpg^{-1}}&{x}\end{pmatrix}\begin{pmatrix}{g}&{(b+dm)x+dpy}\\\
{0}&{pg^{-1}}\end{pmatrix},$

where $x,y$ are integers with the property $(a+cm)x+cpy=g$. The first factor
on the right-hand side is in $\mathrm{SL}_{2}(\mathbb{Z})$ and the second is
rational. Therefore

$\displaystyle\operatorname{ord}\nolimits_{\frac{a}{c}}U^{\prime}_{p,k}f_{0}$
$\displaystyle=\operatorname{ord}\nolimits_{i\infty}(U^{\prime}_{p,k}f_{0})|_{2}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)$ $\displaystyle\geq\min_{0\leq
m<p}\operatorname{ord}\nolimits_{i\infty}f_{0}|_{2}\left(\begin{smallmatrix}{1}&{m}\\\
{0}&{p}\end{smallmatrix}\right)|_{2}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)$ $\displaystyle=\min_{0\leq
m<p}\operatorname{ord}\nolimits_{i\infty}f_{0}|_{2}\left(\begin{smallmatrix}{(a+cm)g^{-1}}&{-y}\\\
{cpg^{-1}}&{x}\end{smallmatrix}\right)|_{2}\left(\begin{smallmatrix}{g}&{(b+dm)x+dpy}\\\
{0}&{pg^{-1}}\end{smallmatrix}\right)$ $\displaystyle=\min_{0\leq
m<p}\frac{\gcd(a+cm,cp)^{2}}{p}\cdot\operatorname{ord}\nolimits_{\frac{a+cm}{cp}}f_{0}.$

∎

######  Remark 2.12.

To calculate
$\operatorname{ord}\nolimits_{\frac{a+cm}{cp}}f_{0}=\operatorname{ord}_{i\infty}f_{0}|_{2}\gamma$
where $\gamma\in\mathrm{SL}_{2}(\mathbb{Z})$ such that
$\gamma(i\infty)=\frac{a+cm}{cp}$, let $H_{1}$ be the holomorphic part of
$\partial\widehat{A}_{\ell}(-j/p)|_{2}\gamma$. By Theorem 2.1, the series
expression of $H_{1}$ and that of the nonholomorphic part converge normally on
$\\{\tau\in\mathscr{H}\colon\operatorname{Im}\tau\geq Y_{0}\\}$ for any
$Y_{0}>0$. Therefore, the holomorphic part of $f_{0}|_{2}\gamma$ is equal to
$\eta(p\gamma\tau)\eta(\gamma\tau)^{-1}\cdot H_{1}(\tau)$ by Proposition 2.4
and consequently, one can calculate
$\operatorname{ord}\nolimits_{\frac{a+cm}{cp}}f_{0}$ using Corollary 2.5.
Finally, note that the fact
$\operatorname{ord}\nolimits_{\frac{a}{c}}U^{\prime}_{p,k}f_{0}$ is well
defined follows from Corollary 2.3 and Proposition 2.4.

###  2.4. Generalized Dedekind eta functions

To cancel the multiplier system of $N_{p}(s,k)$ in Theorem 3.6, we need the
generalized Dedekind eta functions. Let $p$ be a positive integer and
$\delta\in\mathbb{Z}$. Set

(2.26)
$\eta_{p,\delta}(\tau)=q^{\frac{p}{2}P_{2}\left(\frac{\delta}{p}\right)}\cdot\prod_{\begin{subarray}{c}n\equiv-\delta\bmod{p}\\\
n\geq
p-\delta\end{subarray}}(1-q^{n})\prod_{\begin{subarray}{c}n\equiv\delta\bmod{p}\\\
n\geq\delta\end{subarray}}(1-q^{n})=q^{\frac{p}{2}P_{2}\left(\frac{\delta}{p}\right)}\cdot(q^{\delta},q^{p-\delta};q^{p})_{\infty},$

where $P_{2}(x)=x^{2}-x+\frac{1}{6}$ is the second Bernoulli polynomial. Since
$\eta_{p,\delta}=-\eta_{p,\delta+p}$ we assume $0\leq\delta<p$ without loss of
generality. When $p\mid\delta$ we have $\eta_{p,\delta}=0$ which differs from
some authors’ definition555We adopt the definition (2.26) since they are
specializations of the weight $1/2$, index $1/2$ Jacobi form $\theta(z;\tau)$
uniformly for all $p$ and $\delta$.. It is known that $\eta_{p,\delta}$ is a
modular function on $\Gamma_{1}(p)$ and is holomorphic on $\mathscr{H}$,
possibly with a multiplier system (character) which is denoted by
$\chi_{\eta,p,\delta}$ hereafter (c.f. [30]). S. Robins [30] also gives the
order of $\eta_{p,\delta}$ (for $p\nmid\delta$) at the cusp $\frac{a}{c}$,
namely,

(2.27)
$\operatorname{ord}\nolimits_{\frac{a}{c}}\eta_{p,\delta}=\frac{(p,c)^{2}}{2p}\overline{P}_{2}\left(\frac{a\delta}{(p,c)}\right),$

where $\overline{P}_{2}(x)=P_{2}(x-[x])$ is the second Bernoulli function. On
the other hand, although there are known formulas for $\chi_{\eta,p,\delta}$
expressed in terms of Meyer sums (c.f. [26]), we need an alternative exact
formula which is expressed in terms of Petersson’s formula for the multiplier
system $\chi_{\eta}$ of Dedekind eta function:

(2.28) $\chi_{\eta}\begin{pmatrix}{a}&{b}\\\
{c}&{d}\end{pmatrix}=\begin{cases}\genfrac{(}{)}{}{}{d}{\lvert
c\rvert}\exp{\frac{\pi i}{12}\left((a+d-3)c-bd(c^{2}-1)\right)}&\text{if
}2\nmid c,\\\ \genfrac{(}{)}{}{}{c}{d}\exp{\frac{\pi
i}{12}\left((a-2d)c-bd(c^{2}-1)+3d-3\right)}&\text{if }2\mid c,\end{cases}$

where $\genfrac{(}{)}{}{}{\cdot}{\cdot}$ is the Kronecker-Jacobi symbol. For a
proof of (2.28), see [22].

######  Lemma 2.13.

Let $\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)\in\Gamma_{1}(p)$. Then

$\chi_{\eta,p,\delta}\begin{pmatrix}{a}&{b}\\\
{c}&{d}\end{pmatrix}:=\left.\eta_{p,\delta}\left(\frac{a\tau+b}{c\tau+d}\right)\middle/\eta_{p,\delta}(\tau)\right.=\chi_{\eta}^{2}\begin{pmatrix}{a}&{pb}\\\
{c/p}&{d}\end{pmatrix}e^{\frac{\pi
iab\delta^{2}}{p}}(-1)^{\frac{(a-1)\delta}{p}+b\delta}$

###### Proof.

Set
$f_{\delta/p}(\tau)=q^{\frac{\delta^{2}}{2p^{2}}}\theta(\frac{\delta}{p}\tau;\tau)$.
We have $f_{\delta/p}(p\tau)=-i\eta(p\tau)\eta_{p,\delta}(\tau)$ by (2.2). The
required assertion will follow from the well known modular transformation
equations of $\theta$:

$\displaystyle\theta\left(\frac{z}{c\tau+d};\frac{a\tau+b}{c\tau+d}\right)$
$\displaystyle=(c\tau+d)^{1/2}e^{\pi
i\frac{c}{c\tau+d}z^{2}}\cdot\chi_{\eta}^{3}\begin{pmatrix}{a}&{b}\\\
{c}&{d}\end{pmatrix}\theta(z;\tau),\quad\begin{pmatrix}{a}&{b}\\\
{c}&{d}\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{Z}),$
$\displaystyle\theta(z+\lambda\tau+\mu;\tau)$
$\displaystyle=q^{-\frac{1}{2}\lambda^{2}}e^{-2\pi i\lambda
z}(-1)^{\lambda+\mu}\theta(z;\tau),\quad\lambda,\mu\in\mathbb{Z},$

which can be proved, for instance, using [7, Lemma 2.2] and the facts
$\chi_{\eta}\left(\begin{smallmatrix}{1}&{1}\\\
{0}&{1}\end{smallmatrix}\right)=\zeta_{24}$ and
$\chi_{\eta}\left(\begin{smallmatrix}{0}&{-1}\\\
{1}&{0}\end{smallmatrix}\right)=\zeta_{8}^{-1}$. Note that $z^{1/2}$ must be
understood as the principal branch, that is, $z^{1/2}=\exp\frac{1}{2}\log z$
with $-\pi<\operatorname{Im}\log z\leq\pi$. It follows that, for
$\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)\in\Gamma_{1}(p)$,

$\displaystyle f_{\delta/p}$
$\displaystyle\left(\frac{a\tau+bp}{cp^{-1}\tau+d}\right)=(cp^{-1}\tau+d)^{1/2}e^{\frac{\pi
i\delta^{2}(a\tau+bp)a}{p^{2}}}\chi_{\eta}^{3}\left(\begin{smallmatrix}{a}&{bp}\\\
{cp^{-1}}&{d}\end{smallmatrix}\right)\theta\left(\frac{\delta}{p}\tau+\frac{a-1}{p}\delta\tau+b\delta;\tau\right)$
$\displaystyle=(cp^{-1}\tau+d)^{1/2}e^{\frac{\pi
i\delta^{2}(a\tau+bp)a}{p^{2}}}\chi_{\eta}^{3}\left(\begin{smallmatrix}{a}&{bp}\\\
{cp^{-1}}&{d}\end{smallmatrix}\right)q^{-\frac{1}{2}\left(\frac{a-1}{p}\delta\right)^{2}}e^{-2\pi
i\frac{(a-1)\delta^{2}\tau}{p^{2}}}(-1)^{\frac{a-1}{p}\delta+b\delta}\theta\left(\frac{\delta}{p}\tau;\tau\right)$
$\displaystyle=(cp^{-1}\tau+d)^{1/2}\chi_{\eta}^{3}\left(\begin{smallmatrix}{a}&{bp}\\\
{cp^{-1}}&{d}\end{smallmatrix}\right)e^{\frac{\pi
iab\delta^{2}}{p}}(-1)^{\frac{a-1}{p}\delta+b\delta}f_{\delta/p}(\tau).$

Therefore,

$\displaystyle\eta_{p,\delta}|_{0}\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)$
$\displaystyle=i\cdot(f_{\delta/p}\eta^{-1})|_{0}\left(\begin{smallmatrix}{p}&{0}\\\
{0}&{1}\end{smallmatrix}\right)\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)$
$\displaystyle=i\cdot(f_{\delta/p}\eta^{-1})|_{0}\left(\begin{smallmatrix}{a}&{bp}\\\
{c/p}&{d}\end{smallmatrix}\right)\left(\begin{smallmatrix}{p}&{0}\\\
{0}&{1}\end{smallmatrix}\right)$
$\displaystyle=\chi_{\eta}^{2}\left(\begin{smallmatrix}{a}&{bp}\\\
{cp^{-1}}&{d}\end{smallmatrix}\right)e^{\frac{\pi
iab\delta^{2}}{p}}(-1)^{\frac{a-1}{p}\delta+b\delta}\cdot
i(f_{\delta/p}\eta^{-1})|_{0}\left(\begin{smallmatrix}{p}&{0}\\\
{0}&{1}\end{smallmatrix}\right)$
$\displaystyle=\chi_{\eta}^{2}\left(\begin{smallmatrix}{a}&{bp}\\\
{cp^{-1}}&{d}\end{smallmatrix}\right)e^{\frac{\pi
iab\delta^{2}}{p}}(-1)^{\frac{a-1}{p}\delta+b\delta}\eta_{p,\delta},$

which concludes the proof. ∎

Combining the techniques developed above in this section we obtain:

######  Theorem 2.14.

Let $p$ be an odd positive integer not divisible by $3$ and $0\leq k<p$,
$0<j<p$. Let $r_{1},\dots,r_{(p-1)/2}$ be integers. If

(2.29)
$12\cdot\sum_{\delta=1}^{(p-1)/2}r_{\delta}\cdot\delta^{2}\equiv-24k+1\pmod{p},$

then the real analytic function

$f:=U_{p,k}\left(\frac{\partial\widehat{A}_{\ell}(-j/p)}{(q;q)_{\infty}}\right)\cdot
q^{n_{0}}(q^{p};q^{p})_{\infty}^{1-2\sum
r_{\delta}}\cdot\prod_{\delta=1}^{(p-1)/2}(q^{\delta},q^{p-\delta};q^{p})_{\infty}^{r_{\delta}},$

where

$n_{0}=\frac{p^{2}-1}{24p}+\frac{1}{2p}\sum_{\delta=1}^{(p-1)/2}r_{\delta}\cdot\left(\delta^{2}-p\delta\right),$

satisfies that

(2.30)
$f\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{2-\sum\nolimits_{\delta}r_{\delta}}\cdot
f(\tau),\quad\begin{pmatrix}{a}&{b}\\\ {c}&{d}\end{pmatrix}\in\Gamma_{1}(p),$

and vice versa.

###### Proof.

By (2.24) and (2.26) we can express $f$ in terms of generalized Dedekind eta
functions as $f=f_{1}f_{2}f_{3}$ where

$\displaystyle f_{1}(\tau)$
$\displaystyle=\frac{\eta(p\tau)}{\eta(\tau)}U_{p,k}^{\prime}\left(\frac{\eta(p\tau)}{\eta(\tau)}\partial\widehat{A}_{\ell}(-j/p;\tau)\right),$
$\displaystyle f_{2}(\tau)$
$\displaystyle=\eta(p\tau)^{-2\sum_{\delta}r_{\delta}},$ $\displaystyle
f_{3}(\tau)$
$\displaystyle=\prod_{\delta=1}^{(p-1)/2}\eta_{p,\delta}(\tau)^{r_{\delta}}.$

Let $\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)\in\Gamma_{1}(p)$ be arbitrary. By Lemma 2.10
we have

$(c\tau+d)^{-2}\cdot f_{1}\left(\frac{a\tau+b}{c\tau+d}\right)=e^{-\frac{\pi
i\ell_{1}abv^{2}}{p}}\cdot(-1)^{\ell_{1}v((a-1)/p+b)}f_{1}(\tau),$

where $l_{1}$ and $v$ are any integers with $\ell_{1}>0$,
$12\ell_{1}v^{2}\equiv-24k+1\pmod{p}$ and $v\not\equiv 0\pmod{p}$. By the
definition of $\chi_{\eta}$ we have

$(c\tau+d)^{\sum_{\delta}r_{\delta}}\cdot
f_{2}\left(\frac{a\tau+b}{c\tau+d}\right)=\chi_{\eta}^{-2\sum_{\delta}r_{\delta}}\begin{pmatrix}{a}&{pb}\\\
{c/p}&{d}\end{pmatrix}f_{2}(\tau).$

According to Lemma 2.13, we have

$f_{3}\left(\frac{a\tau+b}{c\tau+d}\right)=\chi_{\eta}^{2\sum_{\delta}r_{\delta}}\begin{pmatrix}{a}&{pb}\\\
{c/p}&{d}\end{pmatrix}\prod_{\delta=1}^{(p-1)/2}\left(e^{\frac{\pi
iabr_{\delta}\delta^{2}}{p}}(-1)^{((a-1)/p+b)r_{\delta}\delta}\right)f_{3}(\tau).$

Taking the product of these three identities we find that (2.30) is equivalent
to

$e^{-\frac{\pi
i\ell_{1}abv^{2}}{p}}\cdot(-1)^{\ell_{1}v((a-1)/p+b)}\prod_{\delta=1}^{(p-1)/2}\left(e^{\frac{\pi
iabr_{\delta}\delta^{2}}{p}}(-1)^{((a-1)/p+b)r_{\delta}\delta}\right)=1,$

that is,

(2.31)
$\frac{ab}{p}\left(\sum_{\delta=1}^{(p-1)/2}r_{\delta}\delta^{2}-\ell_{1}v^{2}\right)+\left(\frac{a-1}{p}+b\right)\left(\sum_{\delta=1}^{(p-1)/2}r_{\delta}\delta+\ell_{1}v\right)\in
2\mathbb{Z}$

for any $\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)\in\Gamma_{1}(p)$. Now assume (2.29) holds;
then $\sum_{\delta}r_{\delta}\delta^{2}\equiv\ell_{1}v^{2}\pmod{p}$. If
$\sum_{\delta}r_{\delta}\delta\equiv\ell_{1}v\pmod{2}$, then
$\sum_{\delta}r_{\delta}\delta^{2}\equiv\ell_{1}v^{2}\pmod{2p}$ from which
(2.31) follows. Otherwise, if
$\sum_{\delta}r_{\delta}\delta\equiv\ell_{1}v+1\pmod{2}$, then
$\sum_{\delta}r_{\delta}\delta^{2}\equiv\ell_{1}v^{2}+p\pmod{2p}$ in which
case (2.31) is equivalent to

(2.32) $ab+\frac{a-1}{p}+b\in
2\mathbb{Z},\quad\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)\in\Gamma_{1}(p).$

If $\frac{a-1}{p}$ is odd, then $a$ is even and hence $b$ is odd in which case
(2.32) holds. If $\frac{a-1}{p}$ is even, then $a$ is odd in which case (2.32)
holds as well whenever $b$ is odd or even. We have proved that (2.29) implies
(2.30). To prove the converse, just set $\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)=\left(\begin{smallmatrix}{1}&{1}\\\
{0}&{1}\end{smallmatrix}\right)$ in (2.31). ∎

##  3\. Generating functions

###  3.1. Mock modular part of $NT$

We consider the case $p\geq 5$ being prime and rewrite (1.8) as follows.

######  Lemma 3.1.

For prime $p\geq 5$ and $0<s<\frac{p}{2}$. We have

(3.1)
$\displaystyle\sum_{n=0}^{\infty}\left(NT(s,p,n)-NT(p-s,p,n)\right)q^{n}=\mathcal{F}_{p,s}(\tau)+\sum_{r=1}^{p-1}\frac{p-2r}{2p}D(r-s,p),$

where

$\mathcal{F}_{p,s}(\tau)=\frac{\delta_{s,1}}{2}+\sum_{j=1}^{p-1}\frac{\zeta_{p}^{j(s-1/2)}(1-\zeta_{p}^{j})}{2p\pi
i(q)_{\infty}}\cdot\frac{\partial}{\partial
u}\bigg{|}_{u=0}A_{3}\left(u-\frac{j}{p};\tau\right).$

###### Proof.

One can check that

(3.2) $\sum_{j=1}^{p-1}\zeta_{p}^{j(s-1)}(1-\zeta_{p}^{j})=p\delta_{s,1},$

since $0<s<p/2$. Let

$g_{k}=\sum_{j=1}^{p-1}\frac{\zeta_{p}^{jk}}{1-\zeta_{p}^{j}}.$

If $k\equiv 0\pmod{p}$ then

$g_{k}=\sum_{j=1}^{p-1}\frac{1}{1-\zeta_{p}^{j}}=\sum_{j=1}^{(p-1)/2}\left(\frac{1}{1-\zeta_{p}^{j}}+\frac{1}{1-\zeta_{p}^{-j}}\right)=\frac{p-1}{2}.$

For $0<k<p$ we have

$g_{k}-g_{k+1}=\sum_{j=1}^{p-1}\left(\frac{\zeta_{p}^{jk}}{1-\zeta_{p}^{j}}-\frac{\zeta_{p}^{jk+j}}{1-\zeta_{p}^{j}}\right)=\sum_{j=1}^{p-1}\zeta_{p}^{jk}=-1.$

Hence $g_{k}=k-(p+1)/2$ if $0<k\leq p$ and $g_{0}=g_{p}=(p-1)/2$. By

$\mathcal{R}\left(\zeta_{p}^{j};q\right)=\sum_{n=0}^{\infty}\sum_{m=0}^{p-1}\zeta_{p}^{mj}N(m,p,n)q^{n},$

and noting that $N(r,p,n)=N(r+p,p,n)$, we have

(3.3)
$\displaystyle-\sum_{j=1}^{p-1}\frac{\zeta_{p}^{js}(1+\zeta_{p}^{j})\mathcal{R}\left(\zeta_{p}^{j};q\right)}{(1-\zeta_{p}^{j})}=$
$\displaystyle-\sum_{j=1}^{p-1}\frac{\zeta_{p}^{js}(1+\zeta_{p}^{j})}{1-\zeta_{p}^{j}}\sum_{n=0}^{\infty}\sum_{r=0}^{p-1}\zeta_{p}^{rj}N(r,p,n)q^{n}$
$\displaystyle=$
$\displaystyle-\sum_{n=0}^{\infty}\sum_{r=0}^{p-1}(g_{r}+g_{r+1})N(r-s,p,n)q^{n}$
$\displaystyle=$
$\displaystyle\sum_{r=1}^{p-1}(p-2r)\sum_{n=0}^{\infty}N(r-s,p,n)q^{n}$
$\displaystyle=$ $\displaystyle\sum_{r=1}^{p-1}(p-2r)D(r-s,p).$

Substituting (3.2) and (3.3) to (1.8) we have (3.1). ∎

The part $\mathcal{F}_{p,s}(\tau)$ is a weight $3/2$ mock modular form. The
completed form is given by

(3.4)
$\widehat{\mathcal{F}}_{p,s}(\tau):=\sum_{j=1}^{p-1}\frac{\zeta_{p}^{j(s-1/2)}(1-\zeta_{p}^{j})}{2p\pi
i(q)_{\infty}}\cdot\frac{\partial}{\partial
u}\bigg{|}_{u=0}\widehat{A}_{3}\left(u-\frac{j}{p};\tau\right).$

Following [24, Lemma 2.9], define

$\displaystyle T_{m,k}(q):=\sum_{n=-\infty}^{\infty}$
$\displaystyle\left[\textnormal{sgn}\left(\frac{1}{2}+kn+m\right)-E\left(\left(kn+m+\frac{1}{6}\right)\sqrt{\frac{6y}{k}}\right)\right]$
$\displaystyle\times(-1)^{n}(6kn+6m+1)q^{\frac{-3kn^{2}+(6m+1)n}{2}},$

and

$\displaystyle S_{m,k}(q):=\sum_{n=-\infty}^{\infty}$
$\displaystyle\left[\textnormal{sgn}\left(\frac{1}{2}-kn-m\right)+E\left(\left(kn+m+\frac{1}{6}\right)\sqrt{\frac{6y}{k}}\right)\right]$
$\displaystyle\times(-1)^{n}(6kn+6m+1)q^{\frac{-3kn^{2}+(6m+1)n}{2}},$

with $y=\mathrm{Im(\tau)}$. We mention that $T_{m,k}$ and $S_{m,k}$ can not be
expanded into a normal q-series so that we can not use (2.22) directly. But
similarly, by a straightforward calculation, we have

(3.5) $\displaystyle U_{p,k}(q^{n}T_{m,p}(q^{p}))=$ $\displaystyle
q^{n/p}T_{m,p}(q),$ (3.6) $\displaystyle U_{p,k}(q^{n}S_{m,p}(q^{p}))=$
$\displaystyle q^{n/p}S_{m,p}(q),$

if $n\equiv k\pmod{p}$ and

(3.7) $U_{p,k}(q^{n}T_{m,p}(q^{p}))=U_{p,k}(q^{n}S_{m,p}(q^{p}))=0,$

if $n\not\equiv k\pmod{p}$.

######  Lemma 3.2.

For prime $p\geq 5$, the completed form of
$U_{p,k}\left(\mathcal{F}_{p,s}(\tau)\right)$ is given by the following.

(1)If $6k+s^{2}-s\equiv 0\pmod{p}$ then

$U_{p,k}\left(\widehat{\mathcal{F}}_{p,s}(\tau)\right)=U_{p,k}\left(\mathcal{F}_{p,s}(\tau)\right)+\frac{1}{4}R_{p,s-1}-\frac{\delta_{s,1}}{2},$

(2)If $6k+s^{2}+s\equiv 0\pmod{p}$ then

$U_{p,k}\left(\widehat{\mathcal{F}}_{p,s}(\tau)\right)=U_{p,k}\left(\mathcal{F}_{p,s}(\tau)\right)-\frac{1}{4}R_{p,s},$

(3)If $6k+s^{2}\pm s\not\equiv 0\pmod{p}$ then

$U_{p,k}\left(\widehat{\mathcal{F}}_{p,s}(\tau)\right)=U_{p,k}\left(\mathcal{F}_{p,s}(\tau)\right),$

where

$R_{p,n}:=(-1)^{m_{1}}q^{-\frac{3m_{1}^{2}+m_{1}}{2p}}S_{m_{1},p}(q)-(-1)^{m_{2}}q^{-\frac{3m_{2}^{2}+m_{2}}{2p}}T_{m_{2},p}(q),$

with $n\equiv 3m_{1}\pmod{p}$ and $n\equiv-3m_{2}-1\pmod{p}$.

######  Remark 3.3.

One can easily verify that $R_{p,n}$ does not depend on the choice of $m_{1}$
and $m_{2}$.

###### Proof.

Following the proof of [24, Lemma 2.9], we have

$\displaystyle\widehat{A}_{3}\left(u-\frac{j}{k};\tau\right)-A_{3}\left(u-\frac{j}{k};\tau\right)$
$\displaystyle=$ $\displaystyle\frac{\pi
i(q)_{\infty}}{2}\sum_{m=0}^{p-1}(-1)^{m}q^{-\frac{3m^{2}+m}{2}}\left(\zeta_{p}^{-(6m+1)j/2}S_{m,p}(q^{p})-\zeta_{p}^{(6m+1)j/2}T_{m,p}(q^{p})\right).$

Hence

$\displaystyle\widehat{\mathcal{F}}_{p,s}(\tau)-\mathcal{F}_{p,s}(\tau)+\frac{\delta_{s,1}}{2}$
$\displaystyle=$
$\displaystyle\frac{1}{4p}\sum_{j=1}^{p-1}\zeta_{p}^{j(s-1/2)}(1-\zeta_{p}^{j})\sum_{m=0}^{p-1}(-1)^{m}q^{-\frac{3m^{2}+m}{2}}\left(\zeta_{p}^{-(6m+1)j/2}S_{m,p}(q^{p})-\zeta_{p}^{(6m+1)j/2}T_{m,p}(q^{p})\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4p}\sum_{m=0}^{p-1}(-1)^{m}q^{-\frac{3m^{2}+m}{2}}\sum_{j=1}^{p-1}\left(\zeta_{p}^{(s-3m-1)j}(1-\zeta_{p}^{j})S_{m,p}(q^{p})-\zeta_{p}^{(s+3m)j}(1-\zeta_{p}^{j})T_{m,p}(q^{p})\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4}\big{[}(-1)^{m_{1}}q^{-\frac{3m_{1}^{2}+m_{1}}{2}}S_{m_{1},p}(q^{p})-(-1)^{m_{2}}q^{-\frac{3m_{2}^{2}+m_{2}}{2}}T_{m_{2},p}(q^{p})$
$\displaystyle-(-1)^{n_{1}}q^{-\frac{3n_{1}^{2}+n_{1}}{2}}S_{n_{1},p}(q^{p})+(-1)^{n_{2}}q^{-\frac{3n_{2}^{2}+n_{2}}{2}}T_{n_{2},p}(q^{p})\big{]},$

where $s-3m_{1}-1\equiv 0\pmod{p}$, $s+3m_{2}\equiv 0\pmod{p}$,
$s-3n_{1}-1\equiv-1\pmod{p}$ and $s+3n_{1}\equiv-1\pmod{p}$. The last equation
holds by the following.

$\sum_{j=1}^{p-1}\zeta_{p}^{jk}(1-\zeta_{p}^{j})$

is equal to $p$ if $k\equiv 0\pmod{p}$, $-p$ if $k\equiv-1\pmod{p}$ and 0
else. For $i=1,2$ we have

$-\frac{3m_{i}^{2}+m_{i}}{2}\equiv\frac{s^{2}-s}{6}\pmod{p},$

and

$-\frac{3n_{i}^{2}+n_{i}}{2}\equiv\frac{s^{2}+s}{6}\pmod{p}.$

Hence the Lemma holds via the fact (3.5)-(3.7). ∎

To find certain q-series to cancel the nonholomorphic part of
$\widehat{\mathcal{F}}_{p,s}(\tau)$, we let

$\mathcal{L}_{p}(v):=\frac{(-1)^{v}q^{\frac{-3v^{2}-2s_{p}}{2p}}}{(q^{p};q^{p})_{\infty}}\left(\frac{\partial}{\partial
u}\bigg{|}_{u=0}\frac{p}{2\pi
i}A_{3}(u+v\tau;p\tau)-3vA_{3}(v\tau;p\tau)\right),$

with integer $0<v<p$, and its completed form

$\widehat{\mathcal{L}}_{p}(v):=\frac{(-1)^{v}q^{\frac{-3v^{2}-2s_{p}}{2p}}}{(q^{p};q^{p})_{\infty}}\left(\frac{\partial}{\partial
u}\bigg{|}_{u=0}\frac{p}{2\pi
i}\widehat{A}_{3}(u+v\tau;p\tau)-3v\widehat{A}_{3}(v\tau;p\tau)\right).$

For the case $v=0$, since $A_{3}(0;\tau)$ is not well-defined, we define

$\mathcal{L}_{p}(0):=\frac{pq^{\frac{-s_{p}}{p}}}{(q^{p};q^{p})_{\infty}}\left(\frac{1}{2\pi
i}\frac{\partial}{\partial u}\bigg{|}_{u=0}\left(A_{3}(u;p\tau)-\frac{e^{3\pi
iu}}{1-e^{2\pi iu}}\right)-\frac{1}{8}E_{2}(p\tau)-\frac{11}{24}\right),$

and

$\widehat{\mathcal{L}}_{p}(0):=\frac{pq^{\frac{-s_{p}}{p}}}{(q^{p};q^{p})_{\infty}}\left(\frac{1}{2\pi
i}\frac{\partial}{\partial
u}\bigg{|}_{u=0}\left(\widehat{A}_{3}(u;p\tau)-\frac{e^{3\pi iu}}{1-e^{2\pi
iu}}\right)-\frac{1}{8}E_{2}(p\tau)-\frac{11}{24}\right).$

The function $E_{2}(\tau)$ is the weight 2 Eisenstein series

$E_{2}(\tau):=1-24\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{n}}.$

By (2.3) we have

$\displaystyle\frac{1}{2\pi i(q^{p};q^{p})_{\infty}}\frac{\partial}{\partial
u}\bigg{|}_{u=0}\left(\widehat{A}_{3}(u+v\tau;p\tau)-A_{3}(u+v\tau;p\tau)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4\pi(q^{p};q^{p})_{\infty}}\frac{\partial}{\partial
u}\bigg{|}_{u=0}\big{(}e^{2\pi
i(u+v\tau)}\theta(p\tau;3p\tau)R(3u+3v\tau-p\tau;3p\tau)$
$\displaystyle+e^{4\pi
i(u+v\tau)}\theta(2p\tau;3p\tau)R(3u+3v\tau-2p\tau;3p\tau)\big{)}$
$\displaystyle=$
$\displaystyle-\frac{1}{4}q^{-\frac{1}{2}v}\sum_{n\in\mathbb{Z}}\left(\textnormal{sgn}\left(n+\frac{1}{2}\right)-E\left(\left(pn+v+\frac{p}{6}\right)\sqrt{\frac{6y}{p}}\right)\right)(-1)^{n}(6n+1)q^{-\frac{3pn^{2}+(6v+p)n}{2}}$
$\displaystyle-\frac{1}{4}q^{\frac{1}{2}v}\sum_{n\in\mathbb{Z}}\left(\textnormal{sgn}\left(n+\frac{1}{2}\right)-E\left(\left(pn+v-\frac{p}{6}\right)\sqrt{\frac{6y}{p}}\right)\right)(-1)^{n}(6n-1)q^{-\frac{3pn^{2}+(6v-p)n}{2}}.$

Similarly,

$\displaystyle\frac{1}{(q^{p};q^{p})_{\infty}}\left(\widehat{A}_{3}(v\tau;p\tau)-A_{3}(u+v\tau;p\tau)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}q^{-\frac{1}{2}v}\sum_{n\in\mathbb{Z}}\left(\textnormal{sgn}\left(n+\frac{1}{2}\right)-E\left(\left(pn+v+\frac{p}{6}\right)\sqrt{\frac{6y}{p}}\right)\right)(-1)^{n}q^{-\frac{3pn^{2}+(6v+p)n}{2}}$
$\displaystyle+\frac{1}{2}q^{\frac{1}{2}v}\sum_{n\in\mathbb{Z}}\left(\textnormal{sgn}\left(n+\frac{1}{2}\right)-E\left(\left(pn+v-\frac{p}{6}\right)\sqrt{\frac{6y}{p}}\right)\right)(-1)^{n}q^{-\frac{3pn^{2}+(6v-p)n}{2}}.$

Then we have

$\displaystyle\widehat{\mathcal{L}}_{p}(v)-\mathcal{L}_{p}(v)$
$\displaystyle=$
$\displaystyle-\frac{(-1)^{v}}{4}q^{\frac{-3v^{2}-pv-2s_{p}}{2p}}\sum_{n\in\mathbb{Z}}\left(\textnormal{sgn}\left(n+\frac{1}{2}\right)-E\left(\left(pn+v+\frac{p}{6}\right)\sqrt{\frac{6y}{p}}\right)\right)$
$\displaystyle\times(-1)^{n}(6np+6v+p)q^{-\frac{3pn^{2}+(6v+p)n}{2}}$
$\displaystyle-\frac{(-1)^{v}}{4}q^{\frac{-3v^{2}+pv-2s_{p}}{2p}}\sum_{n\in\mathbb{Z}}\left(\textnormal{sgn}\left(n+\frac{1}{2}\right)-E\left(\left(pn+v-\frac{p}{6}\right)\sqrt{\frac{6y}{p}}\right)\right)$
$\displaystyle\times(-1)^{n}(6np+6v-p)q^{-\frac{3pn^{2}+(6v-p)n}{2}}.$

If $p\equiv 1\pmod{6}$, then we let $m_{1}=\frac{p-1}{6}-v$ and
$m_{2}=\frac{p-1}{6}+v$. We have

(3.8)
$\widehat{\mathcal{L}}_{p}(v)-\mathcal{L}_{p}(v)=(-1)^{\frac{p-1}{6}}\frac{1}{4}R_{p,n}+\epsilon^{*}_{p}(v),$

where $2n+6v+1\equiv 0\pmod{p}$, $R_{p,n}$ was defined in Lemma 3.2 and
$\epsilon^{*}_{p}(v)=(-1)^{v}\frac{p-6v}{2}q^{\frac{v(p-3v)-2s_{p}}{2p}}$ if
$0\leq v<\frac{p-1}{6}$,
$\epsilon^{*}_{p}(v)=(-1)^{v}\frac{5p-6v}{2}q^{\frac{(p-v)(3v-2p)-2s_{p}}{2p}}$
if $\frac{5p+1}{6}\leq v<1$ and $\epsilon^{*}_{p}(v)=0$ if $\frac{p+1}{6}\leq
v<\frac{5p-1}{6}$. If $p\equiv-1\pmod{6}$, then we let
$m_{1}=-\frac{p+1}{6}-v$ and $m_{2}=-\frac{p+1}{6}+v$. Similarly we have

(3.9)
$\widehat{\mathcal{L}}_{p}(v)-\mathcal{L}_{p}(v)=(-1)^{\frac{p+1}{6}}\frac{1}{4}R_{p,n}+\epsilon^{\\#}_{p}(v),$

where
$\epsilon^{\\#}_{p}(v)=(-1)^{v}\frac{p-6v}{2}q^{\frac{v(p-3v)-2s_{p}}{2p}}$ if
$0\leq v<\frac{p+1}{6}$,
$\epsilon^{\\#}_{p}(v)=(-1)^{v}\frac{5p-6v}{2}q^{\frac{(p-v)(3v-2p)-2s_{p}}{2p}}$
if $\frac{5p-1}{6}\leq v<1$ and $\epsilon^{\\#}_{p}(v)=0$ if
$\frac{p-1}{6}\leq v<\frac{5p+1}{6}$. By (3.8), (3.9) and Lemma 3.2, we
arrived at the following theorem.

######  Theorem 3.4.

Let $0\leq v_{n}<p$ such that $2n+6v_{n}+1\equiv 0\pmod{p}$.

(1)If $6k+s^{2}-s\equiv 0\pmod{p}$ then

$U_{p,k}(\widehat{\mathcal{F}}_{p,s}(\tau))-\chi_{12}(p)\widehat{\mathcal{L}}_{p}(v_{s-1})=U_{p,k}(\mathcal{F}_{p,s}(\tau))-\chi_{12}(p)(\mathcal{L}_{p}(v_{s-1})+\epsilon_{p}(v_{s-1})),$

(2)If $6k+s^{2}+s\equiv 0\pmod{p}$ then

$U_{p,k}(\widehat{\mathcal{F}}_{p,s}(\tau))+\chi_{12}(p)\widehat{\mathcal{L}}_{p}(v_{s})=U_{p,k}(\mathcal{F}_{p,s}(\tau))+\chi_{12}(p)(\mathcal{L}_{p}(v_{s})+\epsilon_{p}(v_{s})),$

(3)If $6k+s^{2}\pm s\not\equiv 0\pmod{p}$ then

$U_{p,k}(\widehat{\mathcal{F}}_{p,s}(\tau))=U_{p,k}(\mathcal{F}_{p,s}(\tau)),$

where $\epsilon_{p}(v)=(-1)^{v}\frac{p-6v}{2}q^{\frac{v(p-3v)-2s_{p}}{2p}}$ if
$0\leq v<\frac{p}{6}$,
$\epsilon_{p}(v)=(-1)^{v}\frac{5p-6v}{2}q^{\frac{(p-v)(3v-2p)-2s_{p}}{2p}}$ if
$\frac{5p}{6}<v<1$ and $\epsilon_{p}(v)=0$ if $\frac{p}{6}<v<\frac{5p}{6}$.

###  3.2. The main theorems

######  Lemma 3.5.

Let

$F(\tau)=\frac{1}{2\pi i}\frac{\partial}{\partial
u}\bigg{|}_{u=0}\left(\widehat{A}_{3}(u;\tau)-\frac{e^{3\pi iu}}{1-e^{2\pi
iu}}\right)-\frac{1}{8}E_{2}(\tau)-\frac{11}{24}.$

Then for $\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{Z})$, we have

(3.10) $F\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{2}F(\tau).$

###### Proof.

Let

$f(u,\tau)=\widehat{A}_{3}(u;\tau)-\frac{e^{3\pi iu}}{1-e^{2\pi iu}}.$

By (2.4) we have

(3.11) $\displaystyle
f\left(\frac{u}{c\tau+d},\frac{a\tau+b}{c\tau+d}\right)=$
$\displaystyle\widehat{A}_{3}\left(\frac{u}{c\tau+d};\frac{a\tau+b}{c\tau+d}\right)-\frac{e^{\frac{3\pi
iu}{c\tau+d}}}{1-e^{\frac{2\pi iu}{c\tau+d}}}$ $\displaystyle=$
$\displaystyle(c\tau+d)e^{\frac{-3\pi
icu^{2}}{c\tau+d}}\widehat{A}_{3}(u;\tau)-\frac{e^{\frac{3\pi
iu}{c\tau+d}}}{1-e^{\frac{2\pi iu}{c\tau+d}}}$ $\displaystyle=$
$\displaystyle(c\tau+d)e^{\frac{-3\pi
icu^{2}}{c\tau+d}}f(u,\tau)+(c\tau+d)e^{\frac{-3\pi
icu^{2}}{c\tau+d}}\frac{e^{3\pi iu}}{1-e^{2\pi iu}}-\frac{e^{\frac{3\pi
iu}{c\tau+d}}}{1-e^{\frac{2\pi iu}{c\tau+d}}}.$

Let

$g(u,\tau)=(c\tau+d)e^{\frac{-3\pi icu^{2}}{c\tau+d}}\frac{e^{3\pi
iu}}{1-e^{2\pi iu}}-\frac{e^{\frac{3\pi iu}{c\tau+d}}}{1-e^{\frac{2\pi
iu}{c\tau+d}}}.$

We calculate that

$\frac{1}{2\pi i}\frac{\partial}{\partial u}\bigg{|}_{u\rightarrow
0}g(u,\tau)=-\frac{3ic}{4\pi}+\frac{11}{24}\left((c\tau+d)^{-1}-(c\tau+d)\right).$

Applying $\frac{c\tau+d}{2\pi i}\frac{\partial}{\partial
u}\bigg{|}_{u\rightarrow 0}$ on both sides of (3.11), we have

(3.12) $\displaystyle\frac{1}{2\pi i}\frac{\partial}{\partial
u}\bigg{|}_{u=0}f\left(u,\frac{a\tau+b}{c\tau+d}\right)$ $\displaystyle=$
$\displaystyle(c\tau+d)^{2}\frac{1}{2\pi i}\frac{\partial}{\partial
u}\bigg{|}_{u=0}f(u,\tau)-\frac{3ic}{4\pi}(c\tau+d)+\frac{11}{24}\left(1-(c\tau+d)^{2}\right).$

Let

$h(\tau)=\frac{3}{8\pi y}+\frac{11}{24},$

with $\tau=x+yi$. Then

(3.13)
$h\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{2}h(\tau)-\frac{3ic}{4\pi}(c\tau+d)+\frac{11}{24}\left(1-(c\tau+d)^{2}\right).$

Let

$\widehat{E}_{2}(\tau)=-\frac{3}{\pi y}+E_{2}(\tau).$

It is well-known that

(3.14)
$\widehat{E}_{2}\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{2}\widehat{E}_{2}(\tau).$

Combining (3.12)-(3.14) we have (3.10). ∎

Now we give the proof of the main theorem. Let $D(a,M,k)$ be the
$M$-dissection of $D(a,M)$, namely

$D(a,M,k):=\sum_{n=0}^{\infty}\left(N(a,M,Mn+k)-\frac{p(Mn+k)}{M}\right)q^{n},$

and recall the definition of $L_{p}(v)$ (1.11) and (1.12)

######  Theorem 3.6.

Let $p\geq 5$ be a prime and $0<s<\frac{p}{2}$, $0\leq k<p$. For each integer
$m$ let $v_{m}$ be the integer such that $0\leq v_{m}<p$ and
$2m+6v_{m}+1\equiv 0\pmod{p}$. Let
$c(k,v)=\frac{3v(p-v)}{2p}-\frac{k+s_{p}}{p}$.

(1)If $6k+s^{2}-s\equiv 0\pmod{p}$ then

$\displaystyle\sum_{n\geq n_{0}}\left(NT(s,p,pn+k)-NT(p-s,p,pn+k)\right)q^{n}$
$\displaystyle=$
$\displaystyle\sum_{r=1}^{p-1}\frac{p-2r}{2p}D(r-s,p,k)+\chi_{12}(p)q^{c(k,v_{s-1})}L_{p}(v_{s-1})+N_{p}(s,k).$

(2)If $6k+s^{2}+s\equiv 0\pmod{p}$ then

$\displaystyle\sum_{n\geq n_{0}}\left(NT(s,p,pn+k)-NT(p-s,p,pn+k)\right)q^{n}$
$\displaystyle=$
$\displaystyle\sum_{r=1}^{p-1}\frac{p-2r}{2p}D(r-s,p,k)-\chi_{12}(p)q^{c(k,v_{s})}L_{p}(v_{s})+N_{p}(s,k).$

(3)If $6k+s^{2}\pm s\not\equiv 0\pmod{p}$ then

$\displaystyle\sum_{n\geq n_{0}}\left(NT(s,p,pn+k)-NT(p-s,p,pn+k)\right)q^{n}$
$\displaystyle=$
$\displaystyle\sum_{r=1}^{p-1}\frac{p-2r}{2p}D(r-s,p,k)+N_{p}(s,k).$

The part $N_{p}(s,k)$ has the property that

(3.15)
$\frac{(q,q^{p-1};q^{p})_{\infty}^{2}}{q(q^{p};q^{p})_{\infty}^{3}}G_{p}^{k+s_{p}+1}N_{p}(s,k)$

is a modular function on $\Gamma_{1}(p)$ (with trivial multiplier system),
where

$G_{p}=\frac{(q^{(p-1)/2},q^{(p+1)/2};q^{p})_{\infty}}{(q^{(p-3)/2},q^{(p+3)/2};q^{p})_{\infty}}.$

Before giving the proof, we recall the definition of modular functions: we say
that $f$ is a modular function on $\Gamma_{1}(p)$ (with trivial multiplier
system) if $f$ is meromorphic on $\mathscr{H}$ and
$f\left(\frac{a\tau+b}{c\tau+d}\right)=f(\tau)$ for any
$\left(\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\right)\in\Gamma_{1}(p)$ and
$\operatorname{ord}_{a/c}f>-\infty$ for any coprime integers $a$ and $c$.

###### Proof.

First we consider the case $6k+s^{2}\pm s\not\equiv 0\pmod{p}$. Applying the
operator $q^{-\frac{k}{p}}U_{p,k}$ to both sides of (3.1) we find that
$N_{p}(s,k)=q^{-k/p}U_{p,k}\mathcal{F}_{p,s}$. Therefore, by Theorem 3.4(3),
$N_{p}(s,k)=q^{-k/p}U_{p,k}\widehat{\mathcal{F}}_{p,s}$ which is holomorphic
on $\mathscr{H}$. Taking into account of (3.4), we have

$\displaystyle N_{p}(s,k)=\frac{q^{-k/p}}{2p\pi
i}\sum_{j=1}^{p-1}\zeta_{p}^{j(s-1/2)}(1-\zeta_{p}^{j})\cdot
U_{p,k}\left(\frac{\partial\widehat{A}_{3}\left(-j/p\right)}{(q)_{\infty}}\right).$

Thus, to prove the modularity of (3.15) it suffices to prove the modularity of

(3.16)
$\frac{(q,q^{p-1};q^{p})_{\infty}^{2}}{q(q^{p};q^{p})_{\infty}^{3}}\cdot\frac{(q^{(p-1)/2},q^{(p+1)/2};q^{p})_{\infty}^{k+s_{p}+1}}{(q^{(p-3)/2},q^{(p+3)/2};q^{p})_{\infty}^{k+s_{p}+1}}q^{-\frac{k}{p}}U_{p,k}\left(\frac{\partial\widehat{A}_{3}\left(-j/p\right)}{(q;q)_{\infty}}\right)$

for each $0<j<p$. When $p\geq 7$, the above function is the function $f$ in
Theorem 2.14 with $\ell=3$,

$r_{1}=2,\quad r_{(p-1)/2}=k+\frac{p^{2}-1}{24}+1,\quad
r_{(p-3)/2}=-\left(k+\frac{p^{2}-1}{24}+1\right)$

and $r_{j}=0$ for other $j$. Since in this case

$\displaystyle 12\cdot\sum_{\delta=1}^{(p-1)/2}r_{\delta}\cdot\delta^{2}$
$\displaystyle=24+12\cdot\left(k+\frac{p^{2}-1}{24}+1\right)\cdot\left(\left(\frac{p-1}{2}\right)^{2}-\left(\frac{p-3}{2}\right)^{2}\right)$
$\displaystyle\equiv-24k+1\pmod{p},$

(3.16) and hence (3.15) satisfy the modular transformation equations on
$\Gamma_{1}(p)$ with trivial multiplier system by Theorem 2.14. It follows
from this, the fact $N_{p}(s,k)$ is holomorphic on $\mathscr{H}$, Lemma 2.11
and (2.27) that (3.15) is a modular function on $\Gamma_{1}(p)$. The case
$p=5$ is proved similarly with the only change

$r_{1}=r_{(p-3)/2}=2-\left(k+\frac{p^{2}-1}{24}+1\right).$

Next we consider the case $6k+s^{2}-s\equiv 0\pmod{p}$. Applying the operator
$q^{-\frac{k}{p}}U_{p,k}$ to both sides of (3.1) and using Theorem 3.4(1) we
find that

$\displaystyle N_{p}(s,k)$
$\displaystyle=q^{-k/p}U_{p,k}\mathcal{F}_{p,s}-\chi_{12}(p)q^{c(k,v_{s-1})}L_{p}(v_{s-1})$
$\displaystyle=q^{-k/p}\cdot\left(U_{p,k}\widehat{\mathcal{F}}_{p,s}-\chi_{12}(p)\widehat{\mathcal{L}}_{p}(v_{s-1})\right),$

which is holomorphic on $\mathscr{H}$. Note that for $v_{s-1}\not\equiv
0\pmod{p}$,

$\widehat{\mathcal{L}}_{p}(v_{s-1})=\frac{(-1)^{v_{s-1}}p}{2\pi i}\cdot
q^{\frac{1}{24p}}\eta(p\tau)^{-1}g_{3,v_{s-1}/p}(p\tau)$

where $g_{3,v_{s-1}/p}$ is defined in (2.17). It follows that

$N_{p}(s,k)=\frac{q^{-k/p}}{2p\pi
i}\sum_{j=1}^{p-1}\zeta_{p}^{j(s-1/2)}(1-\zeta_{p}^{j})\cdot
U_{p,k}\left(\frac{\partial\widehat{A}_{3}\left(-j/p\right)}{(q;q)_{\infty}}\right)\\\
-\chi_{12}(p)\frac{(-1)^{v_{s-1}}p}{2\pi i}\cdot
q^{\frac{1}{24p}-\frac{k}{p}}\eta(p\tau)^{-1}g_{3,v_{s-1}/p}(p\tau),$

Since $6v_{s-1}+2s-1\equiv 0\pmod{p}$ and $6k+s^{2}-s\equiv 0\pmod{p}$, we
have $36v_{s-1}^{2}\equiv-24k+1\pmod{p}$. Hence, according to Lemma 2.10,
$q^{-\frac{1}{24p}}U_{p,k}\left(\frac{\partial\widehat{A}_{3}\left(-j/p\right)}{(q;q)_{\infty}}\right)$
and $\eta(p\tau)^{-1}g_{3,v_{s-1}/p}(p\tau)$ satisfy the same modular
transformation equations on $\Gamma_{1}(p)$ (i.e., they have the same
multiplier system). Therefore, proving the modularity of (3.15), as in the
last case, is equivalent to proving the modularity of (3.16) which can be done
in the same manner. Note that when proving the holomorphicity at cusps, we
need Corollaries 2.3 and 2.7 in addition. This concludes the proof of the case
$v_{s-1}\not\equiv 0\pmod{p}$. Otherwise, let us consider the case $v_{s-1}=0$
in which situation $0=36v_{s-1}^{2}\equiv-24k+1\pmod{p}$. By the last
assertion of Lemma 2.10 and Lemma 3.5 proving the modularity of (3.15) is
still equivalent to proving the modularity of (3.16) which can be done as
above.

Finally, for the case $6k+s^{2}+s\equiv 0\pmod{p}$, the proof is almost the
same as the last case $6k+s^{2}-s\equiv 0\pmod{p}$. We omit the redundant
details. ∎

Let

$D_{C}(a,M):=\sum_{n=0}^{\infty}\left(M(a,M,n)-\frac{p(n)}{M}\right)q^{n},$

and

$D_{C}(a,M,k):=\sum_{n=0}^{\infty}\left(M(a,M,Mn+k)-\frac{p(Mn+k)}{M}\right)q^{n}.$

From (1.9) we deduce the following theorem which is the $M_{\omega}$-analogue
of Theorem 3.6.

######  Theorem 3.7.

Under the same condition and notation of Theorem 3.6,

$\displaystyle\sum_{n\geq
n_{0}}\left(M_{\omega}(s,p,pn+k)-M_{\omega}(p-s,p,pn+k)\right)q^{n}$
$\displaystyle=$
$\displaystyle\sum_{r=1}^{p-1}\frac{p-2r}{2p}D_{C}(r-s,p,k)+M_{p}(s,k).$

The part $M_{p}(s,k)$ has the property that

(3.17)
$\frac{(q,q^{p-1};q^{p})_{\infty}^{2}}{q(q^{p};q^{p})_{\infty}^{3}}G_{p}^{k+s_{p}+1}M_{p}(s,k)$

is a modular function on $\Gamma_{1}(p)$.

The proof proceeds as that of Theorem 3.6, with the roles $A_{3}$ and
$\widehat{A}_{3}$ replaced by $A_{1}$ and $\widehat{A}_{1}$ (according to
(1.9)). Since $A_{1}$ is holomorphic, that is, $A_{1}=\widehat{A}_{1}$, no
nonholomorphic correction term is needed. Thus, to prove the modularity of
(3.17) it suffices to prove the modularity of (3.16) with $\widehat{A}_{3}$
replaced by $\widehat{A}_{1}$ which follows from Theorem 2.14 (with $\ell=1$)
as well.

##  4\. Examples and representations of $N_{p}(s,k)$ and $M_{p}(s,k)$

###  4.1. Examples

In this section, we show how Theorems 3.6 and 3.7 work on proving identities
such as (1.5)-(1.7). The part $D(s,p,k)$ was obtained by Hickerson and
Mortenson [19]. Denote

$J_{k}:=(q^{k};q^{k})_{\infty},$
$J_{k,a}:=(q^{a},q^{k-a},q^{k};q^{k})_{\infty},$

and

$g(x,q):=x^{-1}\left(-1+\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{(x;q)_{n+1}(q/x;q)_{n}}\right)$

be a universal mock theta function. They state that for $M=5$ [19,
Eq(12)-Eq(14)]

(4.1) $\displaystyle D(0,5)=$
$\displaystyle-2q^{5}g(q^{5},q^{25})+\frac{4}{5}\cdot\frac{J_{5}J_{25}^{3}}{J_{25,5}^{3}}$
$\displaystyle+\frac{4q}{5}\cdot\frac{J_{25}^{2}}{J_{25,5}}-\frac{2q^{2}}{5}\cdot\frac{J_{25}^{2}}{J_{25,10}}+\frac{2q^{3}}{5}\cdot\frac{J_{5}J_{25}^{3}}{J_{25,10}^{3}},$
(4.2) $\displaystyle D(1,5)=$ $\displaystyle
D(4,5)=q^{5}g(q^{5},q^{25})-q^{8}g(q^{10},q^{25})-\frac{1}{5}\cdot\frac{J_{5}J_{25}^{3}}{J_{25,5}^{3}}$
$\displaystyle-\frac{q}{5}\cdot\frac{J_{25}^{2}}{J_{25,5}}+\frac{3q^{2}}{5}\cdot\frac{J_{25}^{2}}{J_{25,10}}-\frac{3q^{3}}{5}\cdot\frac{J_{5}J_{25}^{3}}{J_{25,10}^{3}},$
(4.3) $\displaystyle D(2,5)=$ $\displaystyle
D(3,5)=q^{8}g(q^{10},q^{25})-\frac{1}{5}\cdot\frac{J_{5}J_{25}^{3}}{J_{25,5}^{3}}$
$\displaystyle-\frac{q}{5}\cdot\frac{J_{25}^{2}}{J_{25,5}}-\frac{2q^{2}}{5}\cdot\frac{J_{25}^{2}}{J_{25,10}}+\frac{2q^{3}}{5}\cdot\frac{J_{5}J_{25}^{3}}{J_{25,10}^{3}},$

and for $M=7$ [19, Eq(34)-Eq(37)]

(4.4) $\displaystyle D(0,7)=$ $\displaystyle 2+2q^{7}g(q^{7},q^{49})$
$\displaystyle-\frac{8}{7}\cdot\frac{J_{49,21}^{2}}{J_{7}}+\frac{6q}{7}\cdot\frac{J_{49}^{2}}{J_{49,7}}-\frac{2q^{2}}{7}\cdot\frac{J_{49,14}^{2}}{J_{7}}$
$\displaystyle+\frac{4q^{3}}{7}\cdot\frac{J_{49}^{2}}{J_{49,14}}+\frac{2q^{4}}{7}\cdot\frac{J_{49}^{2}}{J_{49,21}}-\frac{4q^{6}}{7}\cdot\frac{J_{49,7}^{2}}{J_{7}},$
(4.5) $\displaystyle D(1,7)=$ $\displaystyle
D(6,7)=-1-q^{7}g(q^{7},q^{49})+q^{16}g(q^{21},q^{49})$
$\displaystyle+\frac{6}{7}\cdot\frac{J_{49,21}^{2}}{J_{7}}-\frac{q}{7}\cdot\frac{J_{49}^{2}}{J_{49,7}}+\frac{5q^{2}}{7}\cdot\frac{J_{49,14}^{2}}{J_{7}}$
$\displaystyle-\frac{3q^{3}}{7}\cdot\frac{J_{49}^{2}}{J_{49,14}}+\frac{2q^{4}}{7}\cdot\frac{J_{49}^{2}}{J_{49,21}}+\frac{3q^{6}}{7}\cdot\frac{J_{49,7}^{2}}{J_{7}},$
(4.6) $\displaystyle D(2,7)=$ $\displaystyle
D(5,7)=q^{13}g(q^{14},q^{49})-q^{16}g(q^{21},q^{49})$
$\displaystyle-\frac{1}{7}\cdot\frac{J_{49,21}^{2}}{J_{7}}-\frac{q}{7}\cdot\frac{J_{49}^{2}}{J_{49,7}}-\frac{2q^{2}}{7}\cdot\frac{J_{49,14}^{2}}{J_{7}}$
$\displaystyle+\frac{4q^{3}}{7}\cdot\frac{J_{49}^{2}}{J_{49,14}}-\frac{5q^{4}}{7}\cdot\frac{J_{49}^{2}}{J_{49,21}}+\frac{3q^{6}}{7}\cdot\frac{J_{49,7}^{2}}{J_{7}},$
(4.7) $\displaystyle D(3,7)=$ $\displaystyle D(4,7)=-q^{13}g(q^{14},q^{49})$
$\displaystyle-\frac{1}{7}\cdot\frac{J_{49,21}^{2}}{J_{7}}-\frac{q}{7}\cdot\frac{J_{49}^{2}}{J_{49,7}}-\frac{2q^{2}}{7}\cdot\frac{J_{49,14}^{2}}{J_{7}}$
$\displaystyle-\frac{3q^{3}}{7}\cdot\frac{J_{49}^{2}}{J_{49,14}}+\frac{2q^{4}}{7}\cdot\frac{J_{49}^{2}}{J_{49,21}}-\frac{4q^{6}}{7}\cdot\frac{J_{49,7}^{2}}{J_{7}}.$

Analogously, Mortenson [27, Theorem 10.1] calculated the generating functions
of $D_{C}(a,M,k)$ by

(4.8) $\displaystyle D_{C}(0,5)=$
$\displaystyle\frac{4}{5}\cdot\frac{J_{5}J_{25}^{3}}{J_{25,5}^{3}}-\frac{6q}{5}\cdot\frac{J_{25}^{2}}{J_{25,5}}$
$\displaystyle-\frac{2q^{2}}{5}\cdot\frac{J_{25}^{2}}{J_{25,10}}+\frac{2q^{3}}{5}\cdot\frac{J_{5}J_{25}^{3}}{J_{25,10}^{3}},$
(4.9) $\displaystyle D_{C}(1,5)=$ $\displaystyle
D_{C}(4,5)=-\frac{1}{5}\cdot\frac{J_{5}J_{25}^{3}}{J_{25,5}^{3}}+\frac{4q}{5}\cdot\frac{J_{25}^{2}}{J_{25,5}}$
$\displaystyle-\frac{2q^{2}}{5}\cdot\frac{J_{25}^{2}}{J_{25,10}}-\frac{3q^{3}}{5}\cdot\frac{J_{5}J_{25}^{3}}{J_{25,10}^{3}},$
(4.10) $\displaystyle D_{C}(2,5)=$ $\displaystyle
D_{C}(3,5)=-\frac{1}{5}\cdot\frac{J_{5}J_{25}^{3}}{J_{25,5}^{3}}-\frac{q}{5}\cdot\frac{J_{25}^{2}}{J_{25,5}}$
$\displaystyle\frac{3q^{2}}{5}\cdot\frac{J_{25}^{2}}{J_{25,10}}+\frac{2q^{3}}{5}\cdot\frac{J_{5}J_{25}^{3}}{J_{25,10}^{3}},$

and [27, Theorem 10.2]

(4.12) $\displaystyle D_{C}(0,7)=$
$\displaystyle\frac{6}{7}\cdot\frac{J_{49,21}^{2}}{J_{7}}-\frac{8q}{7}\cdot\frac{J_{49}^{2}}{J_{49,7}}-\frac{2q^{2}}{7}\cdot\frac{J_{49,14}^{2}}{J_{7}}$
$\displaystyle+\frac{4q^{3}}{7}\cdot\frac{J_{49}^{2}}{J_{49,14}}+\frac{2q^{4}}{7}\cdot\frac{J_{49}^{2}}{J_{49,21}}-\frac{4q^{6}}{7}\cdot\frac{J_{49,7}^{2}}{J_{7}},$
(4.13) $\displaystyle D_{C}(1,7)=$ $\displaystyle
D_{C}(6,7)=-\frac{1}{7}\cdot\frac{J_{49,21}^{2}}{J_{7}}+\frac{6q}{7}\cdot\frac{J_{49}^{2}}{J_{49,7}}-\frac{2q^{2}}{7}\cdot\frac{J_{49,14}^{2}}{J_{7}}$
$\displaystyle-\frac{3q^{3}}{7}\cdot\frac{J_{49}^{2}}{J_{49,14}}-\frac{5q^{4}}{7}\cdot\frac{J_{49}^{2}}{J_{49,21}}+3\frac{q^{6}}{7}\cdot\frac{J_{49,7}^{2}}{J_{7}},$
(4.14) $\displaystyle D_{C}(2,7)=$ $\displaystyle
D_{C}(5,7)=-\frac{1}{7}\cdot\frac{J_{49,21}^{2}}{J_{7}}-\frac{q}{7}\cdot\frac{J_{49}^{2}}{J_{49,7}}+\frac{5q^{2}}{7}\cdot\frac{J_{49,14}^{2}}{J_{7}}$
$\displaystyle-\frac{3q^{3}}{7}\cdot\frac{J_{49}^{2}}{J_{49,14}}+\frac{2q^{4}}{7}\cdot\frac{J_{49}^{2}}{J_{49,21}}-\frac{4q^{6}}{7}\cdot\frac{J_{49,7}^{2}}{J_{7}},$
(4.15) $\displaystyle D_{C}(3,7)=$ $\displaystyle
D_{C}(4,7)=-\frac{1}{7}\cdot\frac{J_{49,21}^{2}}{J_{7}}-\frac{q}{7}\cdot\frac{J_{49}^{2}}{J_{49,7}}-\frac{2q^{2}}{7}\cdot\frac{J_{49,14}^{2}}{J_{7}}$
$\displaystyle\frac{4q^{3}}{7}\cdot\frac{J_{49}^{2}}{J_{49,14}}+\frac{2q^{4}}{7}\cdot\frac{J_{49}^{2}}{J_{49,21}}+\frac{3q^{6}}{7}\cdot\frac{J_{49,7}^{2}}{J_{7}}.$

By the representations of $N_{p}(s,k)$ and $M_{p}(s,k)$ for $p=5$ and $p=7$ in
the appendices, one can obtain (1.6) from Theorem 3.6, Theorem 3.7,
(4.1)-(4.3) and (4.8)-(4.10) immediately. Similarly we have

$\displaystyle\sum_{n=0}^{\infty}\big{(}NT(1,7,7n+5)-NT(6,7,7n+5)+3NT(2,7,7n+5)-3NT(5,7,7n+5\big{)}q^{n}$
$\displaystyle=$
$\displaystyle-7\frac{J_{7}^{5}J_{7,2}^{10}}{J_{7,1}^{5}J_{7,3}^{7}}(1-t)^{4},$

where $t=q\frac{J_{7,1}^{3}}{J_{7,2}^{2}J_{7,3}}$. We use Frye and Garvan’s
MAPLE package [16] to verify

$1-t=\frac{J_{7,1}J_{7,3}^{2}}{J_{7,2}^{3}}$

computationally and then arrive at (1.5). We do not list the case
$M_{11}(s,k)$ but we mention that the relations, which have been proved in
[10, Appendix], can also be demonstrated by our method. It follows directly
that the equation

$\sum_{m=1}^{(p-1)/2}m\left[M_{\omega}(m,p,pn-s_{p})-M_{\omega}(p-m,p,pn-
s_{p})\right]=0,$

holds for $p=5,7$, and $11$.

###  4.2. Behavior of $N_{p}(s,k)$ and $M_{p}(s,k)$ at cusps

In this subsection, we derive the representations of $N_{p}(s,k)$ and
$M_{p}(s,k)$ presented in Appendices A and B. Similar representations of
$M_{11}(s,k)$ were previously obtained in [10], where the authors employ
complicated $q$-series techniques in their proof. Here we prove identities
involving $N_{p}(s,k)$ or $M_{p}(s,k)$ using the valence formula. We need to
know up to which power of $q$ we must check that the Fourier expansions of
(3.15) and of some modular function are the same.

Recall the cusps of $\Gamma_{1}(p)\backslash\mathscr{H}$ can be represented by
the set

$\displaystyle\mathscr{R}_{p}:=$
$\displaystyle\left\\{\frac{f(a_{0},c)}{c}\colon 1\leq c<\frac{p}{2},\quad
0\leq a_{0}<(c,p),\quad(a_{0},c,p)=1\right\\}$ $\displaystyle\cup$
$\displaystyle\left\\{\frac{f(a_{0},c)}{c}\colon c=\frac{p}{2}\text{ or
}c=p,\quad 0\leq a_{0}\leq\frac{(c,p)}{2},\quad(a_{0},c,p)=1\right\\}$

where $c,\,a_{0}$ take values in $\mathbb{Z}$ and $f(a_{0},c)$ is any integer
such that $(f(a_{0},c),c)=1$ and $f(a_{0},c)\equiv a_{0}\pmod{p}$; c.f. [12,
Corollary 6.3.19]. Let $a_{1}/c_{1}$ and $a_{2}/c_{2}$ be two points in the
projective line over $\mathbb{Q}$ where $a_{1},c_{1},a_{2},c_{2}$ are integers
with $(a_{1},c_{1})=(a_{2},c_{2})=1$. Then they represent the same cusp of
$\Gamma_{1}(p)\backslash\mathscr{H}$, that is, there exists
$\gamma\in\Gamma_{1}(p)$ such that
$\gamma\frac{a_{1}}{c_{1}}=\frac{a_{2}}{c_{2}}$, if and only if there is an
$\varepsilon\in\\{\pm 1\\}$ such that $c_{1}\equiv\varepsilon c_{2}\pmod{p}$
and $a_{1}\equiv\varepsilon a_{2}\pmod{(p,c_{1})}$. Moreover, if $p\neq 4$,
then the width of the cusp representative $\frac{a}{c}$ equals
$\frac{p}{(c,p)}$ where $a$, $c$ are coprime integers. In particular, if $p$
is an odd prime, then

$\mathscr{R}_{p}=\left\\{\frac{p}{c}\colon 1\leq
c\leq\frac{p-1}{2}\right\\}\cup\left\\{\frac{a}{p}\colon 1\leq
a\leq\frac{p-1}{2}\right\\}.$

The width of any cusp in the former set of the right-hand side above equals
$p$ and that in the latter set equals $1$. Note that the cusp $i\infty$ is
represented by $\frac{1}{p}$.

For a meromorphic function $f$ on $\mathscr{H}$ that satisfies the modular
transformation equations of $\Gamma_{1}(p)$ ($p\neq 4$) of an integral or
half-integral weight (possibly with a multiplier system) we set
$\mathrm{div}_{a/c}f=\frac{p}{(c,p)}\operatorname{ord}_{a/c}f$ (the values
$-\infty$ and $+\infty$, which represent that $f$ has an essential singularity
at $a/c$ and $f=0$ respectively, are allowed). For $\tau_{0}\in\mathscr{H}$,
if $f(\tau)=c\cdot(\tau-\tau_{0})^{n}(1+o(1))$ as $\tau\to\tau_{0}$, set
$\mathrm{div}_{\tau_{0}}f=\frac{n}{e_{\tau_{0}}}$ where

$e_{\tau_{0}}:=\\#\\{\gamma\in\Gamma_{1}(p)/\\{\pm
I\\}\colon\gamma\tau_{0}=\tau_{0}\\}.$

It is known that when $p>3$ we have $e_{\tau_{0}}=1$ for any
$\tau_{0}\in\mathscr{H}$ (c.f. [13, Ex 2.3.7, p.57]). Clearly,
$\mathrm{div}_{x}(fg)=\mathrm{div}_{x}(f)+\mathrm{div}_{x}(g)$ where
$x\in\mathscr{H}\cup\mathscr{R}_{p}$ and $g$ satisfies the same conditions for
$f$ but with possibly different weight and multiplier system, while
$\mathrm{div}_{x}(f+g)\geq\min\\{\mathrm{div}_{x}(f),\mathrm{div}_{x}(g)\\}$
where the weight and multiplier system of $f$ and $g$ are required to be the
same. (A sufficient condition for
$\mathrm{div}_{x}(f+g)=\min\\{\mathrm{div}_{x}(f),\mathrm{div}_{x}(g)\\}$ is
$\mathrm{div}_{x}(f)\neq\mathrm{div}_{x}(g)$.) A special case of the well
known valence formula states that, if $f$ is nonzero, of weight $0$ and
meromorphic at all cusps, then

(4.17)
$\sum_{a/c\in\mathscr{R}_{p}}\mathrm{div}_{a/c}f+\sum_{\tau\in\Gamma_{1}(p)\backslash\mathscr{H}}\mathrm{div}_{\tau}f=0.$

For a proof of the valence formula that allows half-integral weights and any
multiplier systems, see [33, Theorem 2.1 and Eq. (13)].

More generally, if $f$ is only real analytic on $\mathscr{H}$ instead of
meromorphic and has well defined holomorphic part $H_{1}$ and nonholomorphic
part $H_{2}$ (c.f. Remark 2.2), then for $\tau\in\mathscr{H}$, define
$\mathrm{div}_{\tau}f=\mathrm{div}_{\tau}H_{1}$. For the cusp representative
$a/c\in\mathscr{R}_{p}$, let $\gamma\in\mathrm{SL}_{2}(\mathbb{Z})$ satisfy
$\gamma(i\infty)=a/c$. If $f|_{r}\gamma$ ($r$ is the weight) has well defined
holomorphic part $H_{\gamma,1}$ and nonholomorphic part $H_{\gamma,2}$, then
define $\mathrm{div}_{a/c}f=\frac{p}{(c,p)}\mathrm{ord}_{i\infty}H_{\gamma,1}$
($p\neq 4$). Clearly, this is independent of the choice of $\gamma$.

######  Proposition 4.1.

Let the notation be as in Theorem 3.6 and let $f$ denote the function (3.15).
Then

(4.18) $\displaystyle\mathrm{div}_{\frac{p}{c}}f$ $\displaystyle>-s_{p},\quad$
for $\displaystyle 1\leq c\leq\frac{p-1}{2},$ (4.19)
$\displaystyle\mathrm{div}_{\frac{a}{p}}f$
$\displaystyle>-\frac{(k+s_{p}+3)p}{8},\quad$ for $\displaystyle 1\leq
a\leq\frac{p-1}{2},$ (4.20) $\displaystyle\mathrm{div}_{\tau}f$
$\displaystyle\geq 0,\quad$ for $\displaystyle\tau\in\mathscr{H}.$

###### Proof.

Set

$\displaystyle f_{j}(\tau)$
$\displaystyle=\frac{\eta(p\tau)}{\eta(\tau)}U^{\prime}_{p,k}\left(\frac{\eta(p\tau)}{\eta(\tau)}\partial\widehat{A}_{3}\left(-\frac{j}{p}\right)\right),\quad
j=1,2,\dots,p-1,$ $\displaystyle g_{v}(\tau)$
$\displaystyle=g_{3,v/p}(p\tau),\quad v=1,2,\dots,p-1,$ $\displaystyle
g_{0}(\tau)$ $\displaystyle=F(p\tau),$ $\displaystyle h(\tau)$
$\displaystyle=\eta_{p,1}(\tau)^{2}\eta_{p,(p-1)/2}(\tau)^{k+s_{p}+1}\eta_{p,(p-3)/2}(\tau)^{-(k+s_{p}+1)}\eta(p\tau)^{-4}.$

See (2.17) for the definition of $g_{3,v/p}(p\tau)$ and Lemma 3.5 for
$F(\tau)$. We have seen in the proof of Theorem 3.6 that there is a
$v\in\\{0,1,\dots,p-1\\}$ and a sequence
$c,c_{1},c_{2},\dots,c_{p-1}\in\mathbb{C}$ ($c_{j}\neq 0$ for $1\leq j\leq
p-1$) such that $\sum_{j=1}^{p-1}c_{j}f_{j}+cg_{v}$ is holomorphic on
$\mathscr{H}$ and

(4.21) $f=\left(\sum_{j=1}^{p-1}c_{j}f_{j}+cg_{v}\right)\cdot h$

where we have used (2.24) to express $U_{p,k}$ in terms of $U^{\prime}_{p,k}$.
Thus, we need to estimate the orders of $f_{j}$, $g_{v}$ and $h$ at cusps. It
is known that $\operatorname{ord}_{a/c}\eta(p\tau)=\frac{(p,c)^{2}}{24p}$.
This fact together with (2.27) gives

(4.22) $\operatorname{ord}_{p/c}h=0,\quad 1\leq c\leq\frac{p-1}{2}.$

On the other hand, for $1\leq a\leq\frac{p-1}{2}$,

$\operatorname{ord}_{\frac{a}{p}}h=p\left(\left\\{\frac{a}{p}\right\\}^{2}-\left\\{\frac{a}{p}\right\\}\right)+\frac{p}{2}(k+s_{p}+1)\\\
\cdot\left(\left\\{\frac{a(p-1)}{2p}\right\\}^{2}-\left\\{\frac{a(p-1)}{2p}\right\\}-\left\\{\frac{a(p-3)}{2p}\right\\}^{2}+\left\\{\frac{a(p-3)}{2p}\right\\}\right)$

where $\\{x\\}:=x-[x]$. It follows that

(4.23) $\operatorname{ord}_{\frac{a}{p}}h>-(k+s_{p}+3)p/8.$

Let us now consider $g_{v}$. By Corollary 2.7, we have

(4.24)
$\operatorname{ord}_{\frac{p}{c}}g_{v}=0,\quad\operatorname{ord}_{\frac{a}{p}}g_{v}\geq
0$

where $v>0$, $1\leq a,c\leq\frac{p-1}{2}$. For $g_{0}$, expanding the
definitions (i.e. Lemma 3.5, (2.3) and (1.10)) we find

$F(\tau)=-\frac{7}{12}+\sum_{n\geq 1}c_{n}q^{n/8}+\text{ ``nonholomorphic
part''}$

from which $\operatorname{ord}_{i\infty}F=0$. It follows that (4.24) still
holds for $v=0$. For $f_{j}$, Lemma 2.11 implies

$\operatorname{ord}_{\frac{p}{c}}U^{\prime}_{p,k}\left(\frac{\eta(p\tau)}{\eta(\tau)}\partial\widehat{A}_{3}\left(-\frac{j}{p}\right)\right)\geq\min_{0\leq
m<p}\frac{(m,p)^{2}}{p}\cdot\operatorname{ord}_{\frac{p+cm}{cp}}\frac{\eta(p\tau)}{\eta(\tau)}\partial\widehat{A}_{3}\left(-\frac{j}{p}\right).$

If $m=0$, then

$\frac{(m,p)^{2}}{p}\cdot\operatorname{ord}_{\frac{p+cm}{cp}}\frac{\eta(p\tau)}{\eta(\tau)}\partial\widehat{A}_{3}\left(-\frac{j}{p}\right)=-\frac{p-1}{24}+p\cdot\operatorname{ord}_{\frac{1}{c}}\partial\widehat{A}_{3}\left(-\frac{j}{p}\right)>-\frac{p-1}{24}$

according to Remark 2.12 and Corollary 2.5. Otherwise if $m>0$, then

$\frac{(m,p)^{2}}{p}\cdot\operatorname{ord}_{\frac{p+cm}{cp}}\frac{\eta(p\tau)}{\eta(\tau)}\partial\widehat{A}_{3}\left(-\frac{j}{p}\right)=\frac{1}{p}\cdot\operatorname{ord}_{\frac{cm}{p}}\frac{\eta(p\tau)}{\eta(\tau)}\partial\widehat{A}_{3}\left(-\frac{j}{p}\right)=\frac{p-1}{24p}$

since $p\cdot\frac{j}{p}\in\mathbb{Z}$ and hence
$\operatorname{ord}_{\frac{cm}{p}}\partial\widehat{A}_{3}\left(-\frac{j}{p}\right)=0$.
Therefore

$\operatorname{ord}_{\frac{p}{c}}U^{\prime}_{p,k}\left(\frac{\eta(p\tau)}{\eta(\tau)}\partial\widehat{A}_{3}\left(-\frac{j}{p}\right)\right)>-\frac{p-1}{24},$

and hence by Proposition 2.4

(4.25)
$\operatorname{ord}_{\frac{p}{c}}f_{j}>\frac{1-p}{24p}-\frac{p-1}{24}=-\frac{s_{p}}{p},\quad
1\leq j\leq p-1,\quad 1\leq c\leq\frac{p-1}{2}.$

Next we calculate $\operatorname{ord}_{\frac{a}{p}}f_{j}$ for $1\leq
a\leq\frac{p-1}{2}$. Again using Lemma 2.11 we find that

$\displaystyle\operatorname{ord}_{\frac{a}{p}}U^{\prime}_{p,k}\left(\frac{\eta(p\tau)}{\eta(\tau)}\partial\widehat{A}_{3}\left(-\frac{j}{p}\right)\right)$
$\displaystyle\geq\min_{0\leq
m<p}\frac{1}{p}\cdot\operatorname{ord}_{\frac{a+pm}{p^{2}}}\frac{\eta(p\tau)}{\eta(\tau)}\partial\widehat{A}_{3}\left(-\frac{j}{p}\right)$
$\displaystyle=\frac{1}{p}\operatorname{ord}_{\frac{a}{p}}\frac{\eta(p\tau)}{\eta(\tau)}\partial\widehat{A}_{3}\left(-\frac{j}{p}\right)=\frac{p-1}{24p}.$

Thus

(4.26)
$\operatorname{ord}_{\frac{a}{p}}f_{j}\geq\frac{p-1}{24}+\frac{p-1}{24p}=\frac{s_{p}}{p},\quad
1\leq j\leq p-1,\quad 1\leq a\leq\frac{p-1}{2}.$

It follows from (4.21), (4.22), (4.24), (4.25) and Corollary 2.3 that
$\operatorname{ord}_{\frac{p}{c}}f>-\frac{s_{p}}{p}$. Since
$\mathrm{div}_{\frac{p}{c}}f=p\cdot\operatorname{ord}_{\frac{p}{c}}f$ the
first desired formula (4.18) follows. Similarly, the second desired formula
(4.19) follows from (4.21), (4.23), (4.24), (4.26), Corollary 2.3 and the fact
$\mathrm{div}_{\frac{a}{p}}f=\operatorname{ord}_{\frac{a}{p}}f$. Finally, the
third desired formula (4.20) holds since $f$ is holomorphic on $\mathscr{H}$.
∎

The above proposition allows us to check an identity of the form
$f=\overline{f}$, where $f$ is as above and $\overline{f}$ is any modular
function (with the trivial multiplier system) of the group $\Gamma_{1}(p)$
that is holomorphic on $\mathscr{H}$, with the aid of Frye and Garvan’s MAPLE
package [16]. First we shall estimate the orders of $\overline{f}$ at all
cusps except $1/p$: Assume that we have the lower bounds

(4.27) $\mathrm{div}_{\frac{p}{c}}\overline{f}\geq-
e_{\frac{p}{c}},\quad\mathrm{div}_{\frac{a}{p}}\overline{f}\geq-
e_{\frac{a}{p}},\quad 1\leq a,\,c\leq\frac{p-1}{2},\,a\neq 1$

where $e_{\frac{p}{c}},\,e_{\frac{a}{p}}\in\mathbb{Q}$.

######  Corollary 4.2.

Suppose

$f=\sum_{n_{0}\leq n\in\mathbb{Z}}a_{n}q^{n},\quad\overline{f}=\sum_{n_{0}\leq
n\in\mathbb{Z}}b_{n}q^{n}.$

If $a_{n}=b_{n}$ for all $n_{0}\leq n\leq n_{1}$ where

$n_{1}=\sum_{c=1}^{(p-1)/2}\max\\{s_{p},e_{\frac{p}{c}}\\}+\sum_{a=2}^{(p-1)/2}\max\left\\{\frac{(k+s_{p}+3)p}{8},e_{\frac{a}{p}}\right\\},$

then $f=\overline{f}$.

###### Proof.

Assume by contradiction that $f\neq\overline{f}$. According to the assumption
(4.27) and Proposition 4.1 we have

$\displaystyle\mathrm{div}_{\frac{p}{c}}(f-\overline{f})$
$\displaystyle\geq\min\\{-s_{p},-e_{\frac{p}{c}}\\},\quad$ for $\displaystyle
1\leq c\leq\frac{p-1}{2},$
$\displaystyle\mathrm{div}_{\frac{a}{p}}(f-\overline{f})$
$\displaystyle\geq\min\left\\{-\frac{(k+s_{p}+3)p}{8},-e_{\frac{a}{p}}\right\\},\quad$
for $\displaystyle 2\leq a\leq\frac{p-1}{2},$
$\displaystyle\mathrm{div}_{\tau}(f-\overline{f})$ $\displaystyle\geq 0,\quad$
for $\displaystyle\tau\in\mathscr{H}.$

Since $a_{n}=b_{n}$ for $n\leq n_{1}$,
$\mathrm{div}_{\frac{1}{p}}(f-\overline{f})\geq[n_{1}]+1$. Applying the
valence formula (4.17) to $f-\overline{f}$ we obtain

$\displaystyle 0$
$\displaystyle=\sum_{c=1}^{(p-1)/2}\mathrm{div}_{\frac{p}{c}}(f-\overline{f})+\sum_{a=2}^{(p-1)/2}\mathrm{div}_{\frac{a}{p}}(f-\overline{f})+\mathrm{div}_{\frac{1}{p}}(f-\overline{f})+\sum_{\tau\in\Gamma_{1}(p)\backslash\mathscr{H}}\mathrm{div}_{\tau}(f-\overline{f})$
$\displaystyle\geq-\sum_{c=1}^{(p-1)/2}\max\\{s_{p},e_{\frac{p}{c}}\\}-\sum_{a=2}^{(p-1)/2}\max\left\\{\frac{(k+s_{p}+3)p}{8},e_{\frac{a}{p}}\right\\}+([n_{1}]+1)+0$
$\displaystyle>0,$

which is a contradiction. Therefore $f=\overline{f}$. ∎

######  Remark 4.3.

Proposition 4.1 and Corollary 4.2 hold as well for $f$ being (3.17) instead of
(3.15). For the proof, we proceed just as above, but with all the occurrences
of $\widehat{A}_{3}$ replaced by $\widehat{A}_{1}$ and $g_{v}(\tau)$ redefined
by $g_{1,v/p}(p\tau)$. Since our estimates of orders of $f_{j}$ and $g_{v}$
are independent of the level $\ell$, the proofs above also work for $f$ being
(3.17). Moreover, it is possible to obtain more accurate lower bounds of
$\mathrm{div}_{\frac{p}{c}}f$ and $\mathrm{div}_{\frac{a}{p}}f$ where $f$ is
(3.17) or (3.15) using the exact formula in Corollary 2.5, but the ones we
have given suffice.

In Appendices A and B we represent (3.15) and (3.17) for $p=5$, $7$ as
polynomials of $t$ and $t^{-1}$ where $t$, depending on $p$, is a generalized
Dedekind eta quotient. To prove each of these identities, we let $f$ be the
left-hand side, $\overline{f}$ be the right-hand side and apply Corollary 4.2
or its $M_{\omega}$-version. Thus we just need to check the Fourier
coefficients of $f$ and $\overline{f}$ up to the $q^{n_{1}}$-term. Below we
give an estimate of $n_{1}$.

For $N_{5}(s,k)$ in Appendix A, we have

$s_{p}=\frac{p^{2}-1}{24}=1,\qquad\frac{(k+s_{p}+3)p}{8}\leq\frac{5\cdot(4+1+3)}{8}=5.$

Note that $\overline{f}$ is a linear combination of $t^{-2}$, $t^{-1}$ and
$1$, where

$t=q\frac{J_{5,1}^{5}}{J_{5,2}^{5}}=\frac{\eta_{5,1}^{5}}{\eta_{5,2}^{5}}.$

By (2.27) we have for $a=1,2$ and $c=1,2$,

$\displaystyle\mathrm{div}_{\frac{5}{c}}(\eta_{5,1})=5\cdot\operatorname{ord}_{\frac{5}{c}}(\eta_{5,1})=\frac{1}{12},\quad$
$\displaystyle\mathrm{div}_{\frac{a}{5}}(\eta_{5,1})=\operatorname{ord}_{\frac{a}{5}}(\eta_{5,1})=\frac{a^{2}}{10}-\frac{a}{2}+\frac{5}{12},$
$\displaystyle\mathrm{div}_{\frac{5}{c}}(\eta_{5,2})=5\cdot\operatorname{ord}_{\frac{5}{c}}(\eta_{5,2})=\frac{1}{12},\quad$
$\displaystyle\mathrm{div}_{\frac{a}{5}}(\eta_{5,2})=\operatorname{ord}_{\frac{a}{5}}(\eta_{5,2})=\frac{2a^{2}}{5}-a+\frac{5}{12}.$

Hence

$\mathrm{div}_{\frac{5}{1}}(t)=0,\quad\mathrm{div}_{\frac{5}{2}}(t)=0,\quad\mathrm{div}_{\frac{1}{5}}(t)=1,\quad\mathrm{div}_{\frac{2}{5}}(t)=-1.$

It follows that in (4.27) we can choose
$e_{\frac{5}{1}}=e_{\frac{5}{2}}=e_{\frac{2}{5}}=0$ and hence

$n_{1}=\max\\{s_{5},e_{\frac{5}{1}}\\}+\max\\{s_{5},e_{\frac{5}{2}}\\}+\max\left\\{\frac{5(k+s_{5}+3)}{8},e_{\frac{2}{5}}\right\\}\leq
7.$

This means if the Fourier coefficients of $f$ and $\overline{f}$ coincide up
to the $q^{7}$-term, then $f=\overline{f}$ for $N_{5}(s,k)$.

The proof for $M_{5}(s,k)$ is the same as for $N_{5}(s,k)$: we need to check
the Fourier coefficients up to the $q^{7}$-term as well.

Finally, the proofs for identities in Appendix B that deal with $N_{7}(s,k)$
and $M_{7}(s,k)$ are similar so we omit the details. We have in this case

$\displaystyle\mathrm{div}_{\frac{7}{1}}(t)=0,\quad\mathrm{div}_{\frac{7}{2}}(t)=0,\quad\mathrm{div}_{\frac{7}{3}}(t)=0,$
$\displaystyle\mathrm{div}_{\frac{1}{7}}(t)=1,\quad\mathrm{div}_{\frac{2}{7}}(t)=0,\quad\mathrm{div}_{\frac{3}{7}}(t)=-1,$
$\displaystyle
e_{\frac{7}{1}}=e_{\frac{7}{2}}=e_{\frac{7}{3}}=e_{\frac{2}{7}}=0,\quad
e_{\frac{3}{7}}=4,$ $\displaystyle
s_{p}=2,\quad\frac{(k+s_{p}+3)p}{8}\leq\frac{77}{8},\quad
n_{1}\leq\frac{101}{4}.$

Thus to prove each identity in Appendix B, one need just to check the Fourier
coefficients of both sides up to the $q^{25}$-term.

##  Appendix A Representations of $N_{5}(s,k)$ and $M_{5}(s,k)$

$\displaystyle G:=\frac{J_{5,2}}{J_{5,1}}\qquad
P:=\frac{J_{5,1}^{2}}{qJ_{5}^{5}}\qquad t:=q\frac{J_{5,1}^{5}}{J_{5,2}^{5}}.$
$\displaystyle P\cdot G^{2}\cdot
N_{5}(1,0)=-\frac{3}{10}t^{-1}-\frac{1}{10},\qquad$ $\displaystyle P\cdot
G^{2}\cdot N_{5}(2,0)=-\frac{1}{10}t^{-1}-\frac{17}{10},$ $\displaystyle
P\cdot G^{3}\cdot N_{5}(1,1)=-\frac{3}{10}t^{-1}-\frac{11}{10},\qquad$
$\displaystyle P\cdot G^{3}\cdot
N_{5}(2,1)=-\frac{1}{10}t^{-1}+\frac{13}{10},$ $\displaystyle P\cdot
G^{4}\cdot N_{5}(1,2)=-\frac{11}{10}t^{-1}+\frac{3}{10},\qquad$ $\displaystyle
P\cdot G^{4}\cdot N_{5}(2,2)=-\frac{1}{5}t^{-1}-\frac{2}{5},$ $\displaystyle
P\cdot G^{5}\cdot N_{5}(1,3)=\frac{1}{10}t^{-1}-\frac{3}{10},\qquad$
$\displaystyle P\cdot G^{5}\cdot N_{5}(2,3)=-\frac{9}{5}t^{-1}+\frac{2}{5},$
$\displaystyle P\cdot G^{6}\cdot N_{5}(1,4)=-t^{-1},\qquad$ $\displaystyle
P\cdot G^{6}\cdot
N_{5}(2,4)=\frac{5}{12}t^{-2}+\frac{1}{2}t^{-1}+\frac{5}{12},$ $\displaystyle
P\cdot G^{2}\cdot M_{5}(1,0)=-\frac{3}{10}t^{-1}-\frac{1}{10},\qquad$
$\displaystyle P\cdot G^{2}\cdot
M_{5}(2,0)=-\frac{1}{10}t^{-1}+\frac{33}{10},$ $\displaystyle P\cdot
G^{3}\cdot M_{5}(1,1)=-\frac{4}{5}t^{-1}+\frac{2}{5},\qquad$ $\displaystyle
P\cdot G^{3}\cdot M_{5}(2,1)=-\frac{1}{10}t^{-1}+\frac{13}{10},$
$\displaystyle P\cdot G^{4}\cdot
M_{5}(1,2)=\frac{2}{5}t^{-1}+\frac{4}{5},\qquad$ $\displaystyle P\cdot
G^{4}\cdot M_{5}(2,2)=-\frac{17}{10}t^{-1}-\frac{9}{10},$ $\displaystyle
P\cdot G^{5}\cdot M_{5}(1,3)=\frac{1}{10}t^{-1}-\frac{3}{10},\qquad$
$\displaystyle P\cdot G^{5}\cdot M_{5}(2,3)=\frac{16}{5}t^{-1}+\frac{2}{5},$
$\displaystyle P\cdot G^{6}\cdot M_{5}(1,4)=4t^{-1},\qquad$ $\displaystyle
P\cdot G^{6}\cdot M_{5}(2,4)=-2t^{-1}.$

##  Appendix B Representations of $N_{7}(s,k)$ and $M_{7}(s,k)$

$\displaystyle G:=\frac{J_{7,3}}{J_{7,2}}\qquad
P:=\frac{J_{7,1}^{2}}{qJ_{7}^{5}}\qquad
t:=q\frac{J_{7,1}^{3}}{J_{7,2}^{2}J_{7,3}}.$ $\displaystyle P\cdot G^{3}\cdot
N_{7}(1,0)=-\frac{6}{7}t^{-1}+\frac{10}{7}-t+\frac{3}{7}t^{2}$ $\displaystyle
P\cdot G^{4}\cdot
N_{7}(1,1)=-\frac{5}{14}t^{-1}-\frac{11}{7}+2t-\frac{1}{14}t^{2},$
$\displaystyle P\cdot G^{5}\cdot
N_{7}(1,2)=-\frac{17}{14}t^{-1}+\frac{6}{7}+3t-\frac{37}{14}t^{2},$
$\displaystyle P\cdot G^{6}\cdot
N_{7}(1,3)=-\frac{1}{14}t^{-1}+\frac{19}{14}-\frac{16}{7}t+\frac{11}{14}t^{2}+\frac{3}{14}t^{3},$
$\displaystyle P\cdot G^{7}\cdot
N_{7}(1,4)=-\frac{9}{7}t^{-1}+\frac{38}{7}-\frac{64}{7}t+\frac{50}{7}t^{2}-\frac{15}{7}t^{3},$
$\displaystyle P\cdot G^{8}\cdot N_{7}(1,5)=-t^{-1}+1+3t-5t^{2}+2t^{3},$
$\displaystyle P\cdot G^{9}\cdot
N_{7}(1,6)=-\frac{27}{14}t^{-1}+\frac{67}{7}-\frac{243}{14}t+\frac{195}{14}t^{2}-\frac{31}{7}t^{3}+\frac{3}{14}t^{4},$
$\displaystyle P\cdot G^{3}\cdot
N_{7}(2,0)=-\frac{3}{14}t^{-1}-\frac{22}{7}+5t-\frac{23}{14}t^{2},$
$\displaystyle P\cdot G^{4}\cdot
N_{7}(2,1)=-\frac{3}{14}t^{-1}+\frac{13}{7}-3t+\frac{19}{14}t^{2},$
$\displaystyle P\cdot G^{5}\cdot
N_{7}(2,2)=-\frac{3}{7}t^{-1}+\frac{5}{7}-t+\frac{5}{7}t^{2},$ $\displaystyle
P\cdot G^{6}\cdot
N_{7}(2,3)=-\frac{15}{7}t^{-1}+\frac{40}{7}-\frac{39}{7}t+\frac{18}{7}t^{2}-\frac{4}{7}t^{3},$
$\displaystyle P\cdot G^{7}\cdot
N_{7}(2,4)=-\frac{1}{14}t^{-1}-\frac{23}{14}+\frac{40}{7}t-\frac{87}{14}t^{2}+\frac{31}{14}t^{3},$
$\displaystyle P\cdot G^{8}\cdot N_{7}(2,5)=-2t^{-1}+9-15t+11t^{2}-3t^{3},$
$\displaystyle P\cdot G^{9}\cdot
N_{7}(2,6)=\frac{37}{14}t^{-1}-\frac{90}{7}+\frac{333}{14}t-\frac{289}{14}t^{2}+\frac{57}{7}t^{3}-\frac{15}{14}t^{4},$
$\displaystyle P\cdot G^{3}\cdot
N_{7}(3,0)=-\frac{1}{14}t^{-1}+\frac{9}{7}-3t+\frac{25}{14}t^{2},$
$\displaystyle P\cdot G^{4}\cdot
N_{7}(3,1)=-\frac{1}{14}t^{-1}-\frac{12}{7}+6t-\frac{59}{14}t^{2},$
$\displaystyle P\cdot G^{5}\cdot
N_{7}(3,2)=-\frac{1}{7}t^{-1}+\frac{18}{7}-5t+\frac{18}{7}t^{2},$
$\displaystyle P\cdot G^{6}\cdot
N_{7}(3,3)=-\frac{3}{14}t^{-1}-\frac{41}{14}+\frac{50}{7}t-\frac{65}{14}t^{2}+\frac{9}{14}t^{3},$
$\displaystyle P\cdot G^{7}\cdot
N_{7}(3,4)=-\frac{20}{7}t^{-1}+\frac{79}{7}-\frac{115}{7}t+\frac{73}{7}t^{2}-\frac{17}{7}t^{3},$
$\displaystyle P\cdot G^{8}\cdot
N_{7}(3,5)=\frac{7}{12}t^{-2}-\frac{2}{3}t^{-1}-\frac{5}{3}+\frac{19}{6}t-t^{2}-t^{3}+\frac{7}{12}t^{4},$
$\displaystyle P\cdot G^{9}\cdot
N_{7}(3,6)=-\frac{23}{7}t^{-1}+\frac{96}{7}-\frac{158}{7}t+\frac{128}{7}t^{2}-\frac{51}{7}t^{3}+\frac{8}{14}t^{4},$
$\displaystyle P\cdot G^{3}\cdot
M_{7}(1,0)=-\frac{5}{14}t^{-1}-\frac{4}{7}-t+\frac{27}{14}t^{2},$
$\displaystyle P\cdot G^{4}\cdot
M_{7}(1,1)=-\frac{6}{7}t^{-1}+\frac{3}{7}+2t-\frac{11}{7}t^{2},$
$\displaystyle P\cdot G^{5}\cdot
M_{7}(1,2)=\frac{2}{7}t^{-1}+\frac{13}{7}-4t+\frac{13}{7}t^{2},$
$\displaystyle P\cdot G^{6}\cdot
M_{7}(1,3)=-\frac{1}{14}t^{-1}+\frac{19}{14}-\frac{16}{7}t+\frac{11}{14}t^{2}+\frac{3}{14}t^{3},$
$\displaystyle P\cdot G^{7}\cdot
M_{7}(1,4)=\frac{3}{14}t^{-1}-\frac{29}{14}+\frac{27}{7}t-\frac{33}{14}t^{2}+\frac{5}{14}t^{3},$
$\displaystyle P\cdot G^{8}\cdot M_{7}(1,5)=-t^{-1}+8-18t+16t^{2}-5t^{3},$
$\displaystyle P\cdot G^{9}\cdot
M_{7}(1,6)=\frac{71}{14}t^{-1}-\frac{129}{7}+\frac{345}{14}t-\frac{197}{14}t^{2}+\frac{18}{7}t^{3}+\frac{3}{14}t^{4},$
$\displaystyle P\cdot G^{3}\cdot
M_{7}(2,0)=-\frac{3}{14}t^{-1}+\frac{27}{7}-2t-\frac{23}{14}t^{2},$
$\displaystyle P\cdot G^{4}\cdot
M_{7}(2,1)=-\frac{3}{14}t^{-1}+\frac{13}{7}-3t+\frac{19}{14}t^{2},$
$\displaystyle P\cdot G^{5}\cdot
M_{7}(2,2)=-\frac{27}{14}t^{-1}+\frac{47}{7}-8t+\frac{45}{14}t^{2},$
$\displaystyle P\cdot G^{6}\cdot
M_{7}(2,3)=\frac{5}{14}t^{-1}+\frac{3}{14}-\frac{18}{7}t+\frac{43}{14}t^{2}-\frac{15}{14}t^{3},$
$\displaystyle P\cdot G^{7}\cdot
M_{7}(2,4)=-\frac{11}{8}t^{-1}+\frac{41}{7}-\frac{51}{7}t+\frac{23}{7}t^{2}-\frac{2}{7}t^{3},$
$\displaystyle P\cdot G^{8}\cdot M_{7}(2,5)=5t^{-1}-19+27t-17t^{2}+4t^{3},$
$\displaystyle P\cdot G^{9}\cdot
M_{7}(2,6)=-\frac{13}{7}t^{-1}+\frac{50}{7}-\frac{68}{7}t+\frac{34}{7}t^{2}+\frac{1}{7}t^{3}-\frac{4}{7}t^{4},$
$\displaystyle P\cdot G^{3}\cdot
M_{7}(3,0)=-\frac{1}{14}t^{-1}+\frac{9}{7}-3t+\frac{25}{14}t^{2},$
$\displaystyle P\cdot G^{4}\cdot
M_{7}(3,1)=-\frac{1}{14}t^{-1}+\frac{37}{7}-8t+\frac{39}{14}t^{2},$
$\displaystyle P\cdot G^{5}\cdot
M_{7}(3,2)=-\frac{1}{7}t^{-1}-\frac{31}{7}+9t-\frac{31}{7}t^{2},$
$\displaystyle P\cdot G^{6}\cdot
M_{7}(3,3)=-\frac{19}{7}t^{-1}+\frac{67}{7}-\frac{69}{7}t+\frac{13}{7}t^{2}+\frac{8}{7}t^{3},$
$\displaystyle P\cdot G^{7}\cdot
M_{7}(3,4)=\frac{29}{7}t^{-1}-\frac{68}{7}+\frac{32}{7}t+\frac{24}{7}t^{2}-\frac{17}{7}t^{3},$
$\displaystyle P\cdot G^{8}\cdot M_{7}(3,5)=-3t^{-1}+10-12t+6t^{2}-t^{3},$
$\displaystyle P\cdot G^{9}\cdot
M_{7}(3,6)=\frac{17}{14}t^{-1}-\frac{44}{7}+\frac{153}{7}t-\frac{101}{14}t^{2}+\frac{5}{7}t^{3}-\frac{9}{14}t^{4}.$

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