# Shintani Lifts of Nearly Holomorphic Modular Forms

Yingkun Li, Shaul Zemel

###### Abstract.

In this paper, we compute the Fourier expansion of the Shintani lift of nearly
holomorphic modular forms. As an application, we deduce modularity properties
of generating series of cycle integrals of nearly holomorphic modular forms.

##  1\. Introduction

For a discriminant $d\in\mathbb{Z}$, let $\mathcal{Q}_{d}$ be the set of
binary quadratic forms of discriminant $d$, which is acted on by the group
$\Gamma:=\operatorname{SL}_{2}(\mathbb{Z})$ with finitely many orbits. When
$d<0$, each $\lambda\in\mathcal{Q}_{d}$ gives rise to a CM points
$z_{\lambda}$ in the upper-half plane $\mathcal{H}$. The values of the
$j$-function

$j(z):=\frac{1}{q}+744+196884q+\dots,\qquad q:=\mathbf{e}(z):=e^{2\pi iz}$

at such CM points are called _singular moduli_ , and they are algebraic
numbers generating abelian extensions of the imaginary quadratic field
$\mathbb{Q}(\sqrt{d})$ by the theory of complex multiplication. The paper [Za]
proved the surprising result that the $d$th trace of the normalized function
$J(z):=j(z)-744$ is the $|d|$th Fourier coefficient of a weakly holomorphic
modular form $g$ of weight $\frac{3}{2}$.

When $d>0$, each $\lambda=[A,B,C]\in\mathcal{Q}_{d}$ gives rise to a geodesic

$c_{\lambda}:=\\{z\in\mathcal{H}:A|z|^{2}+B\Re(z)+C=0\\}$

on $\mathcal{H}$. If $d$ is not a perfect square, then the stabilizer
$\Gamma_{\lambda}$ of $\lambda$ in $\Gamma$ is infinite and
$c(\lambda):=\Gamma_{\lambda}\backslash c_{\lambda}$ is a closed cycle on the
modular curve $Y=\Gamma\backslash\mathcal{H}$. Instead of values, one can
consider integrals of modular forms along these cycles. The non-holomorphic
Eisenstein series of weight 2, defined by

$E_{2}^{*}(z):=1-24\sum_{n\geq 1}\sigma_{1}(n)q^{n}-\frac{3}{\pi y},\qquad
y:=\Im z,$

offers an elegant example. For a fixed fundamental discriminant $\Delta<0$,
let $\chi_{\Delta}$ be the genus character from, e.g., Section 1.2 of [GKZ]
(with $N=1$), which takes $\lambda\in\mathbb{Z}^{3}$ to

(1.1)
$\chi_{\Delta}(\lambda):=\begin{cases}\big{(}\frac{\Delta}{n}\big{)},&\text{if
}\lambda\in\mathcal{Q}_{d}\text{ with }\Delta|d,\ (n,\Delta)=1,\text{ and
}\lambda\text{ represents }n,\\\ 0,&\text{otherwise}.\end{cases}$

Then for any fundamental discriminant $D<0$ co-prime to $\Delta$, we have the
formula

(1.2) $\sum_{\lambda\in\Gamma\backslash\mathcal{Q}_{\Delta
D}}\chi_{\Delta}(\lambda)\int_{c(\lambda)}E^{*}_{2}(z)dz=-12H(-\Delta)H(-D),$

where $H(n)$ is the Hurwitz class number considered in [HZ]. This twisted
cycle integral is also the $|D|$th Fourier coefficient of $12H(-\Delta)$ times
the weight $\frac{3}{2}$ mock modular form studied loc. cit. In fact, this
equality holds for any discriminant $D<0$ after suitably regularizing the left
hand side (see Corollary 1.12 of [ANS]111The different sign comes from the
opposite orientation that they use—compare the formula on page 14 with our
Equation (2.14) below.). Note that for a fundamental discriminant $D>0$, the
twisted trace of singular moduli

(1.3)
$A(D,-\Delta):=\frac{1}{\sqrt{D}}\sum_{\lambda\in\Gamma\backslash\mathcal{Q}_{\Delta
D},\ \lambda\gg
0}\frac{\chi_{\Delta}(\lambda)}{|\Gamma_{\lambda}|}J(z_{\lambda})$

is the $D$th Fourier coefficient of the weakly holomorphic modular form
$f_{-\Delta}=q^{\Delta}+O(q)$ of weight $\frac{1}{2}$ from [Za]. This
coefficient is the same with $J(z)$ replaced by $j(z)$ when $D\Delta$ is not a
square.

While searching for analogues of the algebraicity results above, Duke,
Imamoğlu and Tóth studied the generating series of untwisted cycle integrals
of the $j$-function in [DIT], and showed that it is a mock modular form of
weight $\frac{1}{2}$ whose shadow is the weight $\frac{3}{2}$ form $g$ from
[Za]. Furthermore, it is the first member of a family of mock modular forms
with weakly holomoprhic shadows of weight $\frac{3}{2}$. In addition, the
family also has a 0th member $\mathbf{Z}_{+}$, whose Fourier coefficients are
special values of Dirichlet $L$-functions of real quadratic fields (see
Theorem 4 of [DIT]). This function has a completion $\hat{\mathbf{Z}}_{+}$,
which essentially first appeared in [DI]. Its image under the weight
$\frac{1}{2}$ Laplacian is $-\frac{1}{8\pi}$ times the Jacobi theta function
$\theta(z):=\sum_{n\in\mathbb{Z}}q^{n^{2}}$. From these results, it is natural
to ask what can be said about the twisted cycle integrals of the modular form
$J(z)E_{2}^{*}(z)$. Our first result answers this question.

######  Theorem 1.1.

Let $\Delta<0$ be a fixed fundamental discriminant. For any discriminant
$D<0$, the twisted regularized cycle integral

$\operatorname{Tr}_{\Delta,D}\big{(}J\cdot
E_{2}^{*}\big{)}:=\sum_{\lambda\in\Gamma\backslash\mathcal{Q}_{\Delta
D}}\chi_{\Delta}(\lambda)\int_{c(\lambda)}^{\mathrm{reg}}J(z)E^{*}_{2}(z)dz,$

with the regularization defined as in Equation (4.6), is the $|D|$th Fourier
coefficient of the unique mock modular form $\tilde{f}_{-\Delta}$ of weight
$\frac{3}{2}$ and level 4 in Kohnen’s plus space which expands as
$48|\Delta|H(-\Delta)+O(q^{3})$ with shadow $\frac{3}{2\pi}f_{-\Delta}$.

The mock modular forms $\tilde{f}_{-\Delta}$ have previously been constructed
by Jeon, Kang and Kim in [JKK1] using Maass–Poincaré series. The sequel [JKK2]
expressed its Fourier coefficients, using the same approach as in [DIT], as
cycle integrals of sesqui-harmonic modular forms of weight zero. Using a theta
lift, Bruinier, Funke and Imamoğlu obtained another proof of the main result
of [DIT] in [BFI], which also gave a geometric interpretation of the Fourier
coefficients with square indices. This idea was used by Alfes-Neumann and
Schwagenscheidt in [ANS] to construct $\tilde{f}_{-\Delta}$ as the holomoprhic
part of the Shintani theta lift of a harmonic Maass form $\tilde{J}$ of weight
2, which expresses the Fourier coefficients of $\tilde{f}_{-\Delta}$ as the
twisted cycle integrals of $\tilde{J}$. As the form $J\cdot E_{2}^{*}$ is not
harmonic but just sesqui-harmonic of weight 2, it is not a priori clear its
twisted cycle integrals coincide with the ones in [JKK2] and [ANS].

To prove Theorem 1.1, we will follow the theta lift approach as in [BF1],
[BFI], and [ANS], and use the theta kernel with the same archimedean Schwartz
function as in [Sh]. We shall apply it to nearly holomorphic modular forms,
and compute the resulting Fourier expansions. Recall that a real-analytic
modular form $f$ on $\mathcal{H}$ with at most linear exponential growth near
the cusps is called _nearly holomorphic_ if it can be presented as

(1.4) $f(z)=\sum_{l=0}^{p}\frac{f_{l}(z)}{y^{l}}\qquad\text{with}\qquad
f_{l}:\mathcal{H}\to\mathbb{C}\text{ holomorphic for }0\leq l\leq p$

for some $p\in\mathbb{N}$, which is called the _depth_ of $f$ if $f_{p}$ is
not identically zero. In other words, it is annihilated by the operator
$L_{z}^{p+1}$, where

(1.5) $\begin{split}R_{\kappa}&:=2i\partial_{z}+\tfrac{\kappa}{y},\qquad
L=L_{z}:=-2iy^{2}\partial_{\overline{z}},\qquad\xi_{\kappa}=2iy^{\kappa}\overline{\partial_{\overline{z}}},\qquad\text{and}\\\
\Delta_{\kappa}&:=-R_{\kappa-2}L_{z}=-\xi_{2-\kappa}\xi_{\kappa}=-4y^{2}\partial_{z}+\partial_{\overline{z}}+2i\kappa
y\partial_{\overline{z}}=-y^{2}(\partial_{x}^{2}+\partial_{y}^{2})-\kappa
y(\partial_{y}-i\partial_{x})\end{split}$

are the weight raising operator of weight $\kappa$, the weight lowering
operator, the $\xi$-operator of weight $\kappa$ from [BF1], and the Laplacian
of weight $\kappa$. We denote the space of such modular forms of weight
$\kappa$ with respect to $\Gamma$ by $\widetilde{M}_{\kappa}^{!}$, and use the
superscript $\leq p$ to mean the subspace of forms with depth at most $p$.
Since these differential operators commute with the slash operators, the
condition of being nearly holomorphic is purely archimedean, and can be
defined for any weight, Fuchsian group, character, representation, or
multiplier system.

Nearly holomorphic modular forms of depth 0 are simply weakly holomorphic, and
the Fourier expansions of their Shintani lifts have been computed in [Sh],
[BFI], [BGK], and [ANS]. For arbitrary $p\in\mathbb{N}$, we give the complete
Fourier expansion of their Shintani lift in Theorem 4.3 below. In the
Introduction we will just state a special case for scalar-valued modular forms
of level 1, for which it is helpful to introduce a few notations. In the paper
itself we work with vector-valued theta kernels and Shintani lifts.

Given $\lambda=[A,B,C]\in\mathcal{Q}_{d}$, denote $\lambda(z):=Az^{2}+Bz+C$.
Suppose $f\in\widetilde{M}_{2k}^{!,\leq p}$ expands as

(1.6) $f(z)=\sum_{l=0}^{p}\sum_{n\in\mathbb{Z}}c(n,l)q^{n}y^{-l}.$

Given $d\in\mathbb{Z}$ that is not a square, we define, for $k$ even, the
trace

(1.7)
$\operatorname{Tr}_{d}(f):=\sum_{\lambda\in\Gamma\backslash\mathcal{Q}_{d}}\begin{cases}\frac{2}{|\Gamma_{\lambda}|}f(z_{\lambda}),&k=0,\
d<0,\\\ \int_{c(\lambda)}f(z)\lambda(z)^{k-1}dz,&d>0,\
\sqrt{d}\not\in\mathbb{Z}\end{cases}$

(we shall not use the negative $d$ case for $k\neq 0$). If $d=r^{2}>0$ with
$r\in\mathbb{N}$, then we have
$\Gamma\backslash\mathcal{Q}_{d}=\\{\pm[0,r,j]|0\leq j<r-1\\}$ and we define
the trace as

$\operatorname{Tr}_{d}(f):=2\lim_{T\to\infty}\sum_{j=0}^{r-1}\int_{\frac{(j,r)^{2}}{r^{2}}T^{-1}}^{T}f\big{(}-\tfrac{j}{r}+iy\big{)}(riy)^{k-1}idy+(2ir)^{k}\sum_{\begin{subarray}{c}0\leq
l\leq p\\\ n\leq 0,\ r|n\end{subarray}}c(n,l)\phi_{n}(k-l,T;2\pi),$

where the function $\phi_{n}$ is defined in Equation (4.4). Finally, for $d=0$
we set

(1.8) $\operatorname{Tr}_{0}(f):=c(0,0)\operatorname{CT}_{s=1-k}\zeta(s),$

where $\zeta(s)$ is the Riemann zeta function.

We can now state the Fourier expansion of the Shintani lift of an even power
of $E_{2}^{*}$.

######  Theorem 1.2.

Suppose $0<k\in\mathbb{N}$ is even. Consider the function taking
$z\in\mathcal{H}$ to

$\begin{split}&\sum_{\begin{subarray}{c}d\in\mathbb{N}\\\ d\equiv
0,1(\mathrm{mod\
}4)\end{subarray}}\sum_{b=0}^{k/2}\frac{\operatorname{Tr}_{d}\big{(}(\tfrac{\pi}{3}E_{2}^{*})^{k-2b}\big{)}}{(16\pi
y)^{b}(k-2b)!b!}q^{d}+2\sqrt{2\pi}\sum_{\begin{subarray}{c}0>d\in\mathbb{Z}\\\
d\equiv 0,1(\mathrm{mod\
}4)\end{subarray}}\frac{h_{k}\big{(}2\sqrt{2\pi|d|y}\big{)}}{\big{(}2\sqrt{2\pi|d|y}\big{)}^{k}}\cdot\frac{H(-d)q^{d}}{|d|^{\frac{1-k}{2}}}+\\\
-&\frac{\frac{\log(8\pi y)}{2}+C_{k}}{\big{(}\frac{k}{2}\big{)}!(16\pi
y)^{k/2}}+\frac{\sqrt{2\pi}Q_{k-1}(0)}{6(8\pi
y)^{(k-1)/2}}-2i^{k}\sum_{0<a\in\mathbb{N}}\frac{I_{k}\big{(}2a\sqrt{2\pi
y}\big{)}-\tilde{\Omega}_{k}\big{(}2a\sqrt{2\pi y}\big{)}}{\big{(}2a\sqrt{2\pi
y}\big{)}^{k}}a^{k}q^{a^{2}},\end{split}$

where $P_{k}$, $Q_{k-1}$, and $\tilde{\Omega}_{k}$ are the even rational
polynomials given in Corollary 3.7, Equation (3.9) and Remark 3.13
respectively, and the special functions $h_{k}$ and $I_{k}$ and the constant
$C_{k}$ are defined in Equations (3.17), (3.26) and (3.43) respectively. Then
this function is a real-analytic modular form of weight $k+\frac{1}{2}$ and
level 4 in Kohnen’s plus space. Furthermore, its image under $(-16\pi
L_{z})^{k/2}$ is
$2\pi\hat{\mathbf{Z}}_{+}(z)+\big{(}\frac{\gamma-\log\pi}{2}-2\log
2+H_{k}\big{)}\theta(z)$, where $\gamma$ is the Euler–Mascheroni constant and
$H_{k}$ is the $k$th harmonic number.

######  Remark 1.3.

For odd $k\in\mathbb{N}$, one can obtain a similar result for twisted cycle
integrals of $(E_{2}^{*})^{k}$ as in Equation (1.2).

In general, when $f$ has singularity at the cusp, its Shintani lift has the
following Fourier expansion, in which we recall that the power
$R_{\kappa}^{n}$ of the weight raising operator from Equation (1.5) means
$R_{\kappa+2n-2}\circ R_{\kappa+2n-4}\circ\dots\circ R_{\kappa+2}\circ
R_{\kappa}$.

######  Theorem 1.4.

Let $f\in\widetilde{M}_{2k}^{!,\leq p}$ have the expansion from Equation
(1.6), and suppose that $0<k\in\mathbb{N}$ is even and $c(0,k)=0$. Consider
the expansion

$\displaystyle\sum_{\begin{subarray}{c}d\in\mathbb{N}\\\ d\equiv
0,1(\mathrm{mod\ }4)\end{subarray}}\sum_{b=0}^{\lfloor
p/2\rfloor}\frac{\operatorname{Tr}_{d}(L_{z}^{2b}f)}{(16\pi
y)^{b}b!}q^{d}+\sqrt{2\pi}\sum_{\begin{subarray}{c}0>d\in\mathbb{Z}\\\ d\equiv
0,1(\mathrm{mod\
}4)\end{subarray}}\sum_{l=k}^{p}\frac{h_{l}\big{(}2\sqrt{2\pi|d|y}\big{)}}{\big{(}2\sqrt{2\pi|d|y}\big{)}^{l}}\cdot\frac{\operatorname{Tr}_{d}(R_{2k-2l}^{l-k}L_{z}^{l}f)}{2^{l-k}(l-k)!}\cdot\frac{q^{d}}{|d|^{\frac{1-k}{2}}}+$
$\displaystyle+\\!\sum_{\begin{subarray}{c}l=k-1\\\ l\mathrm{\
odd}\end{subarray}}^{p}\\!\frac{\big{(}\frac{l-1}{2}\big{)}!(-2)^{\frac{l-1}{2}}B_{l+1-k}(-1)^{\frac{k}{2}}c(0,l)}{(8\pi
y)^{l/2}(2\pi)^{k-l-\frac{1}{2}}(l+1-k)!}-\sqrt{8\pi}\\!\sum_{\begin{subarray}{c}0<r\in\mathbb{N}\\\
k\leq l\leq p\\\ n<0,\ r|n\end{subarray}}\\!\frac{J_{l}(2r\sqrt{2\pi
y})}{\big{(}2r\sqrt{2\pi y}\big{)}^{l}}\cdot\frac{(2\pi
n)^{l-k}}{(-1)^{\frac{k}{2}}}\cdot\frac{l!c(n,l)}{(l-k)!}r^{k}q^{r^{2}},$

where $B_{\kappa}$ is the $\kappa$th Bernoulli number, and the special
function $J_{l}$ is defined in Equation (3.31). Then this expansion defines a
real-analytic modular form of weight $k+\frac{1}{2}$ and level 4 in Kohnen’s
plus space, which is nearly holomorphic when $p<k$.

######  Remark 1.5.

Theorem 1.4 holds also for $k=0$, once one adds to the expansion $2\sqrt{y}$
times the constant

$\int_{Y}f(z)d\mu(z):=\lim_{T\to\infty}\int_{Y_{T}}f(z)d\mu(z).$

######  Remark 1.6.

In the setting of Theorem 1.4, the generating series
$\sum_{d}\operatorname{Tr}_{d}(f)q^{d}$ defines, when $p<k$, a quasi-modular
form of weight $k+\frac{1}{2}$ and depth
$\big{\lfloor}\frac{p}{2}\big{\rfloor}$. For $p\geq k$ this series can be
completed to a such a modular form using these special functions—see Remark
4.5.

The key ingredient to the calculation of the Fourier expansion is to find
rapidly decaying higher order anti-derivatives of the Schwartz function used
to construct the theta kernel. Such singular Schwartz functions are important
also in evaluating singular theta lifts and constructing Green currents for
special divisors on orthogonal and unitary Shimura varieties (see e.g. [FH]).
In our case, the first anti-derivative can be built from the error function
(see Equation (3.4)). Surprisingly, the other higher order derivatives turn
out to be constructed from linear combinations of the Gaussian and the error
function with polynomial coefficients. These polynomials, which are defined in
Equation (3.6), are closely related to the Hermite polynomials, and are of
independent interest.

The paper is organized as follows. After recalling some basic notions in
Section 2, we devote Section 3 to study properties of these polynomials and
related special functions, including their Fourier transforms, asymptotic
behaviors and certain lattice sum evaluations. Then in Section 4, we complete
the computations of the orbital integrals and the proof of the main theorem
4.3, as well as its implications for Theorems 1.1, 1.2 and 1.4.

Acknowledgement: The authors are grateful to D. Zagier for suggesting the
explicit value for the generating function $\Upsilon(\xi,t)$ in Equation (3.7)
below.

##  2\. Isotropic Lattices and Modular Forms

This section introduces the notions and notation that are required for the
rest of the paper. We follow the setup of [BF1], [BFI], [BFIL], [ANS], and
others.

###  2.1. Differential Operators on Modular Forms

For $\gamma=\big{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\big{)}\in\operatorname{SL}_{2}(\mathbb{R})$ and
$z\in\mathcal{H}$, denote $j(\gamma,z):=cz+d$. Let
$\operatorname{Mp}_{2}(\mathbb{R})$ denote the metaplectic double cover of
$\operatorname{SL}_{2}(\mathbb{R})$, and let
$\operatorname{Mp}_{2}(\mathbb{Z})$ be the inverse image of
$\operatorname{SL}_{2}(\mathbb{Z})$ in $\operatorname{Mp}_{2}(\mathbb{R})$.

Given a representation $\rho$ of a finite index subgroup
$\Gamma\subseteq\operatorname{Mp}_{2}(\mathbb{Z})$ on a finite dimensional
complex vector space $V$, a function $f:\mathcal{H}\to V$ is called _modular
of weight $\kappa\in\frac{1}{2}\mathbb{Z}$ and representation $\rho$_ if the
functional equation

$f\mid_{\kappa}(M,\phi)(z):=\phi(z)^{-2\kappa}f(Mz)=\rho(M,\phi)f(z)$

holds for every element $(M,\phi)\in\Gamma$. Let
$\mathcal{A}_{\kappa}^{!}(\Gamma,\rho)$ (resp.
$\mathcal{A}_{\kappa}(\Gamma,\rho)$) denote the space of such functions that
are real-analytic with at most exponential (resp. polynomial) growth near the
cusps. It contains the subspaces
$\widetilde{M}_{\kappa}^{!}(\Gamma,\rho),M_{\kappa}^{!}(\Gamma,\rho)$,
$M_{\kappa}(\Gamma,\rho)$, and $S_{\kappa}(\Gamma,\rho)$ of nearly
holomorphic, weakly holomorphic, holomorphic, and cusp forms respectively. We
shall omit $\rho$ from the notation when it is trivial, as well as write
$\mathcal{A}_{\kappa,\rho}^{!}$, and similarly for its subspaces, when
$\Gamma=\operatorname{Mp}_{2}(\mathbb{Z})$.

The differential operators from Equation (1.5) preserves modularity, in the
sense that

$R_{\kappa}\mathcal{A}_{\kappa}^{!}(\Gamma,\rho)\subseteq\mathcal{A}_{\kappa+2}^{!}(\Gamma,\rho),\
L_{z}\mathcal{A}_{\kappa}^{!}(\Gamma,\rho)\subseteq\mathcal{A}_{\kappa-2}^{!}(\Gamma,\rho),\text{
and
}\Delta_{\kappa}\mathcal{A}_{\kappa}^{!}(\Gamma,\rho)\subseteq\mathcal{A}_{\kappa}^{!}(\Gamma,\rho).$

It is known that $L_{z}$, $R_{\kappa}$, and $\Delta_{\kappa}$ preserve near
holomorphicity, with $L_{z}$ decreasing the depth by 1, and $R_{\kappa}$ and
$\Delta_{\kappa}$ usually increasing it by 1. For more on these modular forms,
including their relations with quasi-modular forms and Shimura’s vector valued
modular forms, see [MR], [Ze3], and [Ze7].

Around a given point $w=s+it\in\mathcal{H}$, the natural local coordinate is

(2.1)
$\zeta=A_{w}(z):=\tfrac{z-w}{z-\overline{w}}\in\big{\\{}\zeta\in\mathbb{C}\big{|}|\zeta|<1\big{\\}},\qquad\mathrm{with}\qquad
1-A_{w}(z)=\tfrac{2it}{z-\overline{w}}$

for any $z\in\mathcal{H}$, which also satisfies

$|A_{\gamma w}(\gamma
z)|=\Bigg{|}\frac{\overline{j(\gamma,w)}A_{w}(z)}{j(\gamma,w)}\Bigg{|}=|A_{w}(z)|$

for every $z$ and $w$ in $\mathcal{H}$ and
$\gamma\in\operatorname{SL}_{2}(\mathbb{R})$. The expansion of a holomorphic
modular form $f$ of weight $\kappa\in\mathbb{Z}$ is given by Proposition 17 of
[BGHZ]222Note that there is a small typo there, where the expansion in $4\pi
yw$ should be in its additive inverse $-4\pi yw$. as

(2.2)
$f(z)=\bigg{(}\frac{2it}{z-\overline{w}}\bigg{)}^{\kappa}\sum_{n=0}^{\infty}R_{\kappa}^{n}f(w)\frac{t^{n}A_{w}(z)^{n}}{n!}=\big{(}1-A_{w}(z)\big{)}^{\kappa}\sum_{n=0}^{\infty}R_{\kappa}^{n}f(w)\frac{t^{n}A_{w}(z)^{n}}{n!}.$

We shall need a formula extending Equation (2.2) to nearly holomorphic modular
forms.

######  Lemma 2.1.

For $f\in\widetilde{M}^{!}_{\kappa}(\Gamma,\rho)$ and a point
$w=s+it\in\mathcal{H}$, we have the expansion

$f(z)=\big{(}1-A_{w}(z)\big{)}^{\kappa}\sum_{l=0}^{p}\frac{\big{(}1-\overline{A_{w}(z)}\big{)}^{l}}{t^{l}\big{(}1-\big{|}A_{w}(z)\big{|}^{2}\big{)}^{l}}\sum_{n=0}^{\infty}R_{\kappa-l}^{n}f_{l}(w)\frac{t^{n}A_{w}(z)^{n}}{n!}.$

###### Proof.

We write $f(z)$ as in Equation (1.4), and express each $f_{l}$ via Equation
(2.2), but with $\kappa$ replaced by $\kappa-l$. Recalling from Lemma 5.1 of
[Ze4] that $y$ equals $\frac{t(1-|A_{w}(z)|^{2})}{|1-A_{w}(z)|^{2}}$, we get

$f(z)=\sum_{l=0}^{p}\frac{\big{(}1-A_{w}(z)\big{)}^{\kappa-l}\sum_{n=0}^{\infty}R_{\kappa-l}^{n}f_{l}(w)\frac{t^{n}A_{w}(z)^{n}}{n!}}{t^{l}\big{(}1-\big{|}A_{w}(z)\big{|}^{2}\big{)}^{l}}\big{|}1-A_{w}(z)\big{|}^{2l}.$

Expanding $\big{|}1-A_{w}(z)\big{|}^{2}$ yields the desired result. This
proves the lemma. ∎

For any $\epsilon>0$ we will denote the pre-image of the ball of radius
$\epsilon$ in $\mathbb{C}$ under $A_{w}$ by $B_{\epsilon}(w)$ with the natural
orientation on its boundary. We shall later need the limit value of the
following integral, which is determined as follows.

######  Corollary 2.2.

Let $f$ and $w$ be as in Lemma 2.1, and take an integer $\mu$. Then

$\lim_{\epsilon\to 0}\int_{\partial
B_{\epsilon}(w)}\frac{f(z)}{(1-A_{w}(z))^{\kappa-2}}A_{w}(z)^{\mu}dz=\begin{cases}-\frac{4\pi
t^{|\mu|}}{(|\mu|-1)!}R_{\kappa}^{|\mu|-1}f(w),&\mu<0,\\\ 0,&\mu\geq
0.\end{cases}$

###### Proof.

The result follows from substituting in $\zeta=A_{w}(z)$, and thus
$dz=\frac{2it}{(1-\zeta)^{2}}d\zeta$, inside Lemma 2.1. This proves the
corollary. ∎

We will carry out some integrations of modular forms on $\mathcal{H}$, with
respect to the invariant measure $d\mu(z):=\frac{dz\wedge
d\overline{z}}{-2iy^{2}}=\frac{dxdy}{y^{2}}$. The following standard
consequence of Stokes’ theorem will be useful for evaluating some of these
integrals (see, e.g., Proposition 4.1.1 of [L]).

######  Lemma 2.3.

Let $\mathcal{R}$ be a connected domain in $\mathcal{H}$ whose boundary
$\partial\mathcal{R}$ is a piecewise smooth path in $\mathcal{H}$ (positively
oriented), and assume that $f$, $g$, and $G$ are real-analytic functions on
$\mathcal{R}$ such that $g=-L_{z}G$. Then we have the equality

$\int_{\mathcal{R}}f(z)g(z)d\mu(z)=\oint_{\partial\mathcal{R}}f(z)G(z)dz+\int_{\mathcal{R}}L_{z}f(z)G(z)d\mu(z).$

###  2.2. Lattices Producing Modular Curves

Let $V:=M_{2}(\mathbb{Q})^{0}$ be the signature $(2,1)$ quadratic space of
trace zero matrices over $\mathbb{Q}$ with quadratic form
$Q(\lambda):=-N\det\lambda$ for some $0<N\in\mathbb{Q}$. Then
$G:=\operatorname{Spin}(V)\cong\operatorname{SL}_{2}$, and the connected
component of the symmetric space of $G$ containing the line spanned by
$\big{(}\begin{smallmatrix}0&-1\\\ 1&0\end{smallmatrix}\big{)}$ is identified
with $\mathcal{H}$ via $z\mapsto\mathbb{R}Z^{\perp}(z)$ with

$Z^{\perp}(z):=\frac{1}{\sqrt{N}y}\bigg{(}\begin{array}[]{cc}x&-|z|^{2}\\\
1&-x\end{array}\bigg{)},\quad\mathrm{and\ we\ set}\quad
Z(z):=\frac{1}{\sqrt{N}}\bigg{(}\begin{array}[]{cc}z&-z^{2}\\\
1&-z\end{array}\bigg{)}=\frac{1}{\sqrt{N}}\binom{z}{1}(1\ \ -z).$

It is easy to check that $\gamma\cdot Z^{\perp}(z)=Z^{\perp}(\gamma z)$ and
$\gamma\cdot Z(z)=j(\gamma,z)^{2}Z(\gamma z)$ for every
$\gamma\in\operatorname{SL}_{2}(\mathbb{R})$.

Given $\lambda\in V_{\mathbb{R}}$ with $Q(\lambda)=-\xi^{2}<0$, we know that
$\lambda=\xi Z^{\perp}(z_{\lambda})$ for some
$z_{\lambda}=x_{\lambda}+iy_{\lambda}\in\mathcal{H}$, with
$\operatorname{sgn}(\xi)=-\operatorname{sgn}\big{(}\lambda,Z^{\perp}(z_{\lambda})\big{)}$.
In this case Lemma 4.2 of [Ze4] proves the equalities

(2.3) $\begin{split}\big{(}\lambda,Z^{\perp}(z)\big{)}&=-2\xi\cosh
d(z,z_{\lambda})=-2\xi\bigg{(}\frac{|z-{z_{\lambda}}|^{2}}{2yy_{\lambda}}+1\bigg{)}=-2\xi\frac{1+|A_{{z_{\lambda}}}(z)|^{2}}{1-|A_{{z_{\lambda}}}(z)|^{2}},\\\
\big{(}\lambda,Z(z)\big{)}&=-2\xi\frac{(z-{z_{\lambda}})(z-\overline{{z_{\lambda}}})}{2y_{\lambda}}=\frac{4\xi
y_{\lambda}A_{{z_{\lambda}}}(z)}{\big{(}1-A_{{z_{\lambda}}}(z)\big{)}^{2}},\end{split}$

where $d(z,z_{\lambda})$ is the hyperbolic distance between $z$ and
$z_{\lambda}$.

Fix an even, integral lattice $L\subseteq V$, with its dual
$L^{*}:=\operatorname{Hom}(L,\mathbb{Z})$ viewed as a subgroup of $V$
containing $L$, and $D_{L}:=L^{*}/L$ the associated finite quadratic module.
We denote $\Gamma=\Gamma_{L}\subseteq
G(\mathbb{Q})=\operatorname{SL}_{2}(\mathbb{Q})$ the inverse image of the
discriminant kernel333For convenience we shall henceforth assume that
$\Gamma\subseteq\operatorname{SL}_{2}(\mathbb{Z})$—see Remark 2.4 below. of
$L$, and set $Y:=Y_{L}:=\Gamma\backslash\mathcal{H}$ to be the associated
(open) modular curve, with the projection map $\pi:\mathcal{H}\to Y$. For
every $h\in D_{L}$ and $m\in\mathbb{Z}+Q(h)$, we denote

(2.4) $L_{m,h}:=\\{\lambda\in L+h|Q(\lambda)=m\\}.$

Typical examples can be found in [BO], [AE], [LZ], or [Ze6], e.g.,

(2.5) $L=\big{\\{}\big{(}\begin{smallmatrix}-B&C\\\
-A&B\end{smallmatrix}\big{)}\big{|}A,B,C\in\mathbb{Z}\big{\\}}\quad\mathrm{with}\quad
Q=-\det\quad\mathrm{and}\quad\Gamma=\operatorname{SL}_{2}(\mathbb{Z}).$

Also let $\rho_{L}$ be the Weil representation associated with $L$, in which
$\operatorname{Mp}_{2}(\mathbb{Z})$ operates on the vector space
$\mathbb{C}[D_{L}]$, with the canonical basis $\\{\mathfrak{e}_{h}\\}_{h\in
D_{L}}$ (see [Bo], [Sch], [Str], [Ze1], and others).

###  2.3. Cusps and Geodesics

The Baily–Borel completion $\mathcal{H}^{*}$ of $\mathcal{H}$ is obtained by
adding the set $\operatorname{Iso}(V)\cong\mathbb{P}^{1}(\mathbb{Q})$ of
isotropic lines in $V$. Let $\ell_{\infty}\in\operatorname{Iso}(V)$ be the
line spanned by $u_{\infty}:=\big{(}\begin{smallmatrix}0&1\\\
0&0\end{smallmatrix}\big{)}$, and given $\ell\in\operatorname{Iso}(V)$, we
take an element $\sigma_{\ell}\in\operatorname{SL}_{2}(\mathbb{Z})$ such that
$\ell=\sigma_{\ell}\ell_{\infty}$, and set
$u_{\ell}:=\sigma_{\ell}u_{\infty}$. If
$\Gamma_{\ell}\subseteq\Gamma\subseteq\operatorname{SL}_{2}(\mathbb{Z})$ is
the stabilizer of $\ell$, then there exists $\alpha_{\ell}\in\mathbb{N}$,
called the _width_ of the cusp $\ell$, such that

(2.6)
$\sigma_{\ell}^{-1}\Gamma_{\ell}\sigma_{\ell}=\big{\\{}\pm\big{(}\begin{smallmatrix}1&n\alpha_{\ell}\\\
0&1\end{smallmatrix}\big{)}\big{|}n\in\mathbb{Z}\big{\\}}.$

Let $0<\beta_{\ell}\in\mathbb{Q}$ be such that

(2.7) $L\cap\ell=\mathbb{Z}\beta_{\ell}u_{\ell},\qquad\text{and
set}\qquad\varepsilon_{\ell}:=\tfrac{\alpha_{\ell}}{\beta_{\ell}}.$

When $(L+h)\cap\ell\neq\emptyset$ we define $0\leq k_{\ell,h}<\beta_{\ell}$ to
be the unique number such that

(2.8)
$(L+h)\cap\ell(\mathbb{Z}\beta_{\ell}+k_{\ell,h})u_{\ell},\qquad\mathrm{and\
set\qquad}\omega_{\ell,h}:=\tfrac{k_{\ell,h}}{\beta_{\ell}}+\mathbb{Z}\in\mathbb{R}/\mathbb{Z}.$

All these parameters are constant on $\Gamma$-orbits.

Near the cusp associated with $\ell\in\operatorname{Iso}(V)$ we work with the
coordinates

(2.9) $z_{\ell}=x_{\ell}+iy_{\ell}:=\sigma_{\ell}^{-1}z\qquad\text{and}\qquad
q_{\ell}(z_{\ell}):=\mathbf{e}\big{(}\tfrac{z_{\ell}}{\alpha_{\ell}}\big{)}.$

For $\epsilon>0$ we define the neighborhood
$B_{\epsilon}(\ell):=\big{\\{}z\in\mathcal{H}\big{|}|q_{\ell}(z_{\ell})<\epsilon\\}$
of the cusp $\ell$. The set

(2.10)
$\mathcal{H}_{T}:=\mathcal{H}\setminus\bigcup_{\ell\in\operatorname{Iso}(V)}B_{e^{-2\pi
T}}(\ell),\qquad\mathrm{for}\quad T>1,$

is $\Gamma$-invariant, and $Y_{T}:={\Gamma}\backslash\mathcal{H}_{T}$ is a
truncated modular curve, with a fundamental domain444The fundamental domain
actually depends on a choice of representatives for
$\Gamma\backslash\operatorname{Iso}(V)$, but we suppress it from the notation
since this choice does not affect the results later.

(2.11)
$\mathcal{F}_{T}(L):=\bigcup_{\ell\in\Gamma\backslash\operatorname{Iso}(V)}\sigma_{\ell}\mathcal{F}_{T}^{\alpha_{\ell}},\qquad\mathrm{where}\qquad\mathcal{F}_{T}^{\alpha}:=\bigcup_{j=0}^{\alpha-1}\big{(}\begin{smallmatrix}1&j\\\
0&1\end{smallmatrix}\big{)}\mathcal{F}_{T}$

is composed, for $\alpha\in\mathbb{N}$, of $\alpha$ translations of

$\mathcal{F}_{T}:=\big{\\{}z=x+iy\in\mathcal{H}\big{|}|x|\leq\tfrac{1}{2},\
|z|\geq 1,\ y\leq T\\}.$

######  Remark 2.4.

The assumption $\Gamma=\Gamma_{L}\subseteq\operatorname{SL}_{2}(\mathbb{Z})$
is satisfied for the large family of lattices from [Ze6], but not for every
lattice $L$ in $V$. However, the only places where we use the assumption that
$\Gamma\subseteq\operatorname{SL}_{2}(\mathbb{Z})$ is in the form of the
fundamental domain from Equation (2.11), with $T>1$ being a sufficient bound,
and in the integrality of the parameter $\alpha_{\ell}$ from Equation (2.6).
Since none of these facts are used in any proof below, our results hold
equally well for more general lattices.

An element $\lambda\in V_{\mathbb{R}}$ with $Q(\lambda)>0$ defines a geodesic

(2.12)
$c_{\lambda}:=\big{\\{}z\in\mathcal{H}\big{|}\big{(}\lambda,Z^{\perp}(z)\big{)}=0\big{\\}}\subseteq\mathcal{H},\qquad\text{as
well as}\qquad c(\lambda):=\Gamma_{\lambda}\backslash c_{\lambda}\subseteq Y,$

where $\Gamma_{\lambda}$ is the stabilizer of $\lambda$ in $\Gamma$. For
$\lambda_{0}=\big{(}\begin{smallmatrix}1&0\\\ 0&-1\end{smallmatrix}\big{)}\in
V$, we orient the geodesic $c_{\lambda_{0}}=(0,i\infty)$ to go up, and
transfer this to an orientation on $c_{\lambda}$ and $c(\lambda)$ for each
such $\lambda\in V_{\mathbb{R}}$ via the action of
$\operatorname{SL}_{2}(\mathbb{R})$. We have the following well-known
dichotomy.

######  Lemma 2.5.

Let $\lambda\in V$ be such that $m=Q(\lambda)>0$. If $m\in
N\cdot(\mathbb{Q}^{\times})^{2}$ then $\Gamma_{\lambda}$ is the trivial
subgroup $\\{\pm I\\}$, and the geodesic $c_{\lambda}$ connects two cusps in
$\mathbb{P}^{1}(\mathbb{Q})$. Otherwise the image of $\Gamma_{\lambda}$ in
$\operatorname{SO}^{+}(V)\cong\operatorname{PSL}_{2}(\mathbb{Q})$ is infinite
cyclic.

In the first case in Lemma 2.5 we call $\lambda$ _split-hyperbolic_. For
$m\in\mathbb{Q}$ and for $\lambda\in V$ we then set, by a slight abuse of
notation,

(2.13) $\iota(m):=\begin{cases}1,&\text{if
}\sqrt{m/N}\in\mathbb{Q}^{\times},\\\
0,&\text{otherwise},\end{cases}\qquad\text{and}\qquad\iota(\lambda):=\begin{cases}1,&\text{if
}\lambda\text{ is split-hyperbolic},\\\ 0,&\text{otherwise}.\end{cases}$

If $\iota(\lambda)=1$, then $\lambda^{\perp}$ is spanned by $\ell_{\lambda}$
and $\ell_{-\lambda}$ in ${\mathrm{Iso}}(V)$, which correspond to where
$c_{\lambda}$ ends and begins respectively. If $Q(\lambda)=m$, then we have

(2.14)
$\sigma_{\ell_{\lambda}}^{-1}\lambda=\sqrt{\tfrac{m}{N}}\big{(}\begin{smallmatrix}1&-2r_{\lambda}\\\
0&-1\end{smallmatrix}\big{)}\quad\mathrm{for\ some}\quad
r_{\lambda}\in\mathbb{Q},\quad\mathrm{with}\quad
r_{\lambda}+\alpha_{\ell_{\lambda}}\mathbb{Z}\in\mathbb{Q}/\alpha_{\ell_{\lambda}}\mathbb{Z}\quad\mathrm{canonical}.$

The canonical image
$r_{\lambda}+\alpha_{\ell_{\lambda}}\mathbb{Z}\in\mathbb{Q}/\alpha_{\ell_{\lambda}}\mathbb{Z}$
from Equation (2.14), which we shall henceforth still denote by just
$r_{\lambda}$, is called the _real part_ of $c_{\lambda}$, and it is constant
on $\Gamma$-orbits.

For $\ell\in\operatorname{Iso}(V)$, $m\geq 0$, and $h\in D_{L}$ we set, for
later use, the symbol

(2.15) $\iota_{\ell}(m,h):=\begin{cases}1,&\text{there exists }\lambda\in
L_{m,h}\cap\ell^{\perp},\ \text{positively oriented if }m>0,\\\
0,&\text{otherwise}.\end{cases}$

In other words, for $m>0$ we have $\iota_{\ell}(m,h)=1$ if and only if
$\ell=\ell_{\lambda}$ for some $\lambda\in L_{m,h}$. The additive subgroup
$L\cap\ell$ acts on $L_{m,h}\cap\ell^{\perp}$, and we have the following
standard result.

######  Lemma 2.6 (Lemma 3.1 of [BFIL]).

For $\ell\in\operatorname{Iso}(V)$, $0<m\in\mathbb{Q}$, and $h\in D_{L}$ such
that $\iota_{\ell}(m,h)=1$, the natural map
$L_{m,h}\cap\ell^{\perp}\to(L_{m,h}\cap\ell^{\perp})/(L\cap\ell)$ factors
through $\Gamma_{\ell}\backslash(L_{m,h}\cap\ell^{\perp})$. For every
$\lambda\in L_{m,h}\cap\ell^{\perp}$ there are
$2\sqrt{\frac{m}{N}}\varepsilon_{\ell}$ pre-images of $\lambda+(L\cap\ell)$ in
$\Gamma_{\ell}\backslash(L_{m,h}\cap\ell^{\perp})$, namely the images of
$\\{\lambda+j\beta_{\ell}u_{\ell}|0\leq j\leq
2\sqrt{\frac{m}{N}}\varepsilon_{\ell}-1\\}$ modulo $\Gamma_{\ell}$.

######  Remark 2.7.

The number $2\sqrt{\frac{m}{N}}\varepsilon_{\ell}$ from Lemma 2.6 is therefore
integral. Moreover, for $\ell$, $m$, and $h$ as in Lemma 2.6, take some
positively oriented $\lambda\in L_{m,h}\cap\ell^{\perp}$, and let
$r_{\lambda}$ be as in Equation (2.14). Then we have

$\big{\\{}r_{\mu}\big{|}\mu\in L_{m,h}\cap\ell^{\perp}\text{ positively
oriented}\big{\\}}=r_{\lambda}+\tfrac{\beta_{\ell}}{2}\sqrt{\tfrac{N}{m}}\mathbb{Z}\Big{/}\alpha_{\ell_{\lambda}}\mathbb{Z}\subseteq\mathbb{Q}/\alpha_{\ell_{\lambda}}\mathbb{Z},$

a set of $2\sqrt{\frac{m}{N}}\varepsilon_{\ell}$ evenly spaced elements of
$\mathbb{Q}/\alpha_{\ell_{\lambda}}\mathbb{Z}$.

###  2.4. Schwartz Forms, Theta Functions, and Shintani Lifts

Given $k\in\mathbb{N}$, we can define the Schwartz function

(2.16)
$\tilde{\varphi}_{k}(\lambda;\tau,z):=\big{(}\lambda,Z(z)\big{)}^{k}\mathbf{e}\big{[}Q(\lambda)\tau+\big{(}\lambda,Z^{\perp}(z)\big{)}^{2}\tfrac{iv}{2}\big{]}$

for $\lambda\in V_{\mathbb{R}}$ and $\tau=u+iv\in\mathcal{H}$, and construct
the vector-valued theta function

(2.17) $\Theta_{k,L}(\tau,z):=\sum_{h\in
D_{L}}\Theta_{k,L,h}(\tau,z)\mathfrak{e}_{h},\quad\Theta_{k,L,h}(\tau,z):=\sqrt{v}\sum_{\lambda\in
L+h}\tilde{\varphi}_{k}(\lambda;\tau,z).$

Theorem 4.1 in [Bo] implies that for fixed $z\in\mathcal{H}$ we have
$\Theta_{k,L}(\tau,z)\in\mathcal{A}_{k+\frac{1}{2},\rho_{L}}$, whereas for
fixed $\tau\in\mathcal{H}$ it is easy to verify that
$\Theta_{k,L}(\tau,z)\in\mathcal{A}_{-2k}(\Gamma)$. After collecting terms, we
can use Equation (2.4) to rewrite

(2.18)
$\Theta_{k,L,h}(\tau,z)=\sqrt{v}\sum_{m\in\mathbb{Z}+Q(h)}\bigg{[}\sum_{\lambda\in
L_{m,h}}\big{(}\lambda,Z(z)\big{)}^{k}e^{-\pi
v(\lambda,Z^{\perp}(z))^{2}}\bigg{]}q_{\tau}^{m},\quad
q_{\tau}:=\mathbf{e}(\tau).$

Recall that the (probabilists’) Hermite polynomials are defined by

$\operatorname{He}_{n}(\xi):=(-1)^{n}e^{\xi^{2}/2}\big{(}\tfrac{d}{d\xi}\big{)}^{n}e^{-\xi^{2}/2}=\big{(}\xi-\tfrac{d}{d\xi}\big{)}^{n}\cdot
1=\sum_{b=0}^{\lfloor n/2\rfloor}\frac{(-1)^{b}n!}{b!(n-2b)!2^{b}}\xi^{n-2b}.$

Then for $\ell\in\operatorname{Iso}(V)$ and $k\in\mathbb{N}$ one defines the
unary theta function

(2.19)
$\begin{split}\Theta_{k,\ell}(\tau)&:=\sum_{\lambda\in(L^{*}\cap\ell^{\perp})/(L^{*}\cap\ell)}\frac{\operatorname{He}_{k}\big{(}\sqrt{2\pi
v}(\sigma_{\ell}^{-1}\lambda,\Im(Z(i)))\big{)}}{(2\pi
v)^{k/2}}q_{\tau}^{Q(\lambda)}\sum_{\begin{subarray}{c}h\in D_{L}\\\
h+(L^{*}\cap\ell)/(L\cap\ell)=\lambda\end{subarray}}\mathfrak{e}_{h}\\\
&=\sum_{h\in D_{L}}\sum_{\begin{subarray}{c}0\leq m\in\mathbb{Z}+Q(h)\\\
\iota(m)=1\end{subarray}}a(\Theta_{k,\ell},m,h,v)q_{\tau}^{m}\mathfrak{e}_{h}\in\mathcal{A}_{k+\frac{1}{2}}(\rho_{L}),\qquad\text{with}\\\
a&(\Theta_{k,\ell},m,h,v):=\frac{\operatorname{He}_{k}\big{(}2\sqrt{2\pi
mv}\big{)}}{(2\pi
v)^{k/2}}\begin{cases}\big{(}\iota_{\ell}(m,h)+(-1)^{k}\iota_{\ell}(m,-h)\big{)},&m>0,\\\
\iota_{\ell}(0,h),&m=0.\end{cases}\end{split}$

######  Remark 2.8.

The theta functions $\Theta_{Sh}(\tau,z)$ and $\Theta_{\ell,k}(\tau)$ from
Equations (4.1) and (4.2) of [ANS] correspond to
$(-\sqrt{N}/y^{2})^{k+1}\overline{\Theta_{k+1,L}(\tau,z)}$,
$(-i\sqrt{N})^{k}\overline{\Theta_{k,\ell}(\tau)}$, and Equation (2.19)
respectively in our setting.

The function $\Theta_{k,\ell}$ from Equation (2.19) appears in the asymptotic
expansion of the theta kernel $\Theta_{k,L}$ from (2.17), as is given in the
following result. It is essentially part (2) of Proposition 4.2 of [ANS],
which refers to Theorem 5.2 of [Bo] for the proof, and can also be proved
using properties of appropriate variants of the lattice sums from Equation
(3.34).

######  Lemma 2.9.

Given $\ell\in\operatorname{Iso}(V)$, there exists a constant $C_{\ell}>0$
such that

$(\Theta_{k,L}\mid_{2k,z}\sigma_{\ell})(\tau,z_{\ell})=\frac{i^{k}y_{\ell}^{k+1}}{\sqrt{N}\beta_{\ell}}\Theta_{k,\ell}(\tau)+O(e^{-C_{\ell}y_{\ell}^{2}})\quad\mathrm{as}\quad
y_{\ell}\to\infty.$

For $f\in\mathcal{A}_{2k}^{!}(\Gamma)$ we follow [ANS] and [BFI] (among
others) to define its _regularized Shintani lift_ , using the fundamental
domain from Equation (2.11), to be the theta integral

(2.20)
$\mathcal{I}_{k,L}(\tau,f):=\sum_{\ell\in\Gamma\backslash\operatorname{Iso}(V)}\operatorname{CT}_{s=0}\bigg{[}\lim_{T\to\infty}\int_{\mathcal{F}_{T}^{\alpha_{\ell}}}(f\mid_{2k}\sigma_{\ell})(z_{\ell})(\Theta_{k,L}\mid_{-2k}\sigma_{\ell})(\tau,z_{\ell})y_{\ell}^{-s}d\mu(z_{\ell})\bigg{]},$

which is an element of $\mathcal{A}_{k+\frac{1}{2},\rho_{L}}$. When the
constant term of $f$ at every cusp is zero, the integral converges absolutely
and no regularization is necessary (see Proposition 4.1 of [BF2]).

##  3\. Special Functions

In this section we define and study some special functions, which will be
useful in evaluating the Shintani lift of nearly holomorphic modular forms
below.

###  3.1. Familiar Functions

Let $\mathrm{g}(\xi)$ denote the Gaussian $e^{-\xi^{2}/2}$. For $\xi>0$ it has
the anti-derivative

(3.1)
$-\frac{\sqrt{\pi}}{\sqrt{2}}\cdot\operatorname{erfc}\big{(}\tfrac{\xi}{\sqrt{2}}\big{)}=-\int_{\xi}^{\infty}e^{-w^{2}/2}dw=-\int_{\xi^{2}/2}^{\infty}e^{-s}\frac{ds}{\sqrt{2s}}=-\frac{1}{\sqrt{2}}\Gamma\big{(}\tfrac{1}{2},\tfrac{\xi^{2}}{2}\big{)},$

where $\operatorname{erfc}$ is the complementary error function and
$\Gamma(\mu,t)$ is the _incomplete Gamma function_ defined by

$\Gamma(\mu,t):=\int_{t}^{\infty}e^{-s}s^{\mu}\frac{ds}{s}$

for $t>0$. If $0<\mu\in\mathbb{N}$ then this formula is well-defined for every
$t\in\mathbb{R}$, and for $0\leq\mu\in\mathbb{Z}$ it is meaningful for $t<0$
as follows: If $\mu=0$ then the integral is using the Cauchy principal value,
and for smaller $\mu$ we employ repeated integration by parts. The explicit
formulae are given by

(3.2)
$\Gamma(\mu,t)=\begin{cases}e^{-t}(\mu-1)!\sum_{a=0}^{\mu-1}\frac{t^{a}}{a!}=e^{-t}t^{\mu}\big{(}1-\tfrac{d}{dt}\big{)}^{\mu-1}\tfrac{1}{t},&\text{
when }0<\mu\in\mathbb{N}\text{ and }t\in\mathbb{R},\\\
\frac{(-1)^{\mu}}{|\mu|!}\bigg{(}\Gamma(0,t)+e^{-t}\sum_{a=0}^{|\mu|-1}\frac{a!}{(-t)^{a+1}}\bigg{)},&\text{
when }-\mu\in\mathbb{N},\text{ and }t\neq 0,\end{cases}$

$\Gamma(0,t)$ can also be written as $-\operatorname{Ei}(-t)$ using the
_exponential integral_
$\operatorname{Ei}(t):=-\int_{-t}^{\infty}e^{-w}\frac{dw}{w}$, and the
equality

(3.3) $\tfrac{d}{dt}\Gamma(\mu,t)=-e^{-t}t^{\mu-1}$

holds whenever $\Gamma(\mu,t)$ is defined.

Modifying the anti-derivative from Equation (3.1), we now define

(3.4)
$\mathrm{e}(\xi):=-\tfrac{\operatorname{sgn}(\xi)}{\sqrt{2}}\Gamma\big{(}\tfrac{1}{2},\tfrac{\xi^{2}}{2}\big{)}=-\operatorname{sgn}(\xi)\int_{|\xi|}^{\infty}e^{-w^{2}/2}dw\quad\mathrm{for}\quad\xi\neq
0.$

It decays rapidly as $|\xi|\to\infty$, but it is discontinuous at $\xi=0$ with
the jump

$\lim_{\xi\to 0^{+}}\mathrm{e}(\xi)-\lim_{\xi\to
0^{-}}\mathrm{e}(\xi)=-\sqrt{2}\Gamma\big{(}\tfrac{1}{2},0\big{)}=-\sqrt{2\pi}.$

We therefore have, as distributions on $\mathbb{R}$, the equality

(3.5)
$\tfrac{d}{d\xi}\mathrm{e}(\xi)=\mathrm{g}(\xi)-\sqrt{2\pi}\cdot\delta(\xi),$

where $\delta(\xi)$ is the Dirac delta distribution. Our goal next is to find
higher order anti-derivatives of $\mathrm{g}(\xi)$.

###  3.2. Two Families of Polynomials

First we will consider two sequences of polynomials in $\mathbb{Q}[\xi]$,
which we denote by $P_{\nu}$ and $Q_{\nu}$ with $\nu\in\mathbb{N}$ and defined
recursively as follows. Set

(3.6) $\begin{split}P_{0}(\xi)&=1\quad\text{and}\quad
Q_{0}(\xi)=0,\qquad\text{as well as}\\\
P_{\nu}^{\prime}(\xi)&=P_{\nu-1}(\xi)\quad\text{and}\quad
P_{\nu}(\xi)+Q_{\nu}^{\prime}(\xi)-\xi Q_{\nu}(\xi)=Q_{\nu-1}(\xi)\text{ for
}\nu\geq 1.\end{split}$

The fact that Equation (3.6) defines unique sequences of polynomials, and
their parity properties, are established via the following lemma.

######  Lemma 3.1.

Let $p$ and $q$ be two polynomials in $\mathbb{Q}[\xi]$. Then there is a
unique pair $(P,Q)$ of polynomials $P$ and $Q$ such that $P^{\prime}=p$ and
$P+Q^{\prime}-\xi Q=q$. Moreover, if $p$ and $q$ have opposite parities then
so do $P$ and $Q$, with that of $P$ (resp. $Q$) coinciding with that of $q$
(resp. $p$). In addition, if $p\neq 0$ has the leading coefficient $r$ and
$\deg q\leq\deg p$, then $\deg P=\deg Q+1=\deg p+1$ and the leading
coefficients of both $P$ and $Q$ are $\frac{r}{\deg P}$.

###### Proof.

For each $\mu\geq 1$, there exists a unique $p_{\mu}\in\mathbb{Q}[\xi]$ of
degree $\mu$ and leading coefficient $-1$ such that
$\int\mathrm{g}(\xi)p_{\mu}(\xi)d\xi=\xi^{\mu-1}\mathrm{g}(\xi)$, with
$\mathrm{g}$ the Gaussian. Therefore there exist a unique $c\in\mathbb{Q}$ and
$Q\in\mathbb{Q}[\xi]$ such that
$\int\mathrm{g}(\xi)\big{(}q(\xi)-\tilde{P}(\xi)+c\big{)}d\xi=\mathrm{g}(\xi)Q(\xi)$,
where $\tilde{P}\in\mathbb{Q}[\xi]$ satisfies $\tilde{P}(0)=0$ and
$\tilde{P}^{\prime}=p$. Setting $P:=\tilde{P}-c$ proves the first assertion.
Integrating from $-\infty$ to $\infty$ shows that $c=0$ when $p$ is even and
$q$ is odd. The last two assertions can now be checked from this construction.
This proves the lemma. ∎

######  Corollary 3.2.

Equation (3.6) defines unique sequences $\\{P_{\nu}\\}_{\nu=0}^{\infty}$ and
$\\{Q_{\nu}\\}_{\nu=0}^{\infty}$ of polynomials in $\xi$. Moreover, $P_{\nu}$
is a polynomial of degree $n$ with leading coefficient $\frac{1}{\nu!}$ and
parity $(-1)^{\nu}$ for any $n\geq 0$, and $Q_{\nu}$ is a polynomial of degree
$\nu-1$ with leading coefficient $\frac{1}{\nu!}$ and parity $(-1)^{\nu-1}$
for any $n\geq 1$.

###### Proof.

All the statements hold for $\nu=0$ by Equation (3.6), and once they hold for
$\nu-1$, taking $p=P_{\nu-1}$ and $q=Q_{\nu-1}$ in Lemma 3.1 determines
$P_{\nu}$ and $Q_{\nu}$ as $P$ and $Q$ respectively, with the required
properties. This proves the corollary. ∎

It will be useful for us to consider the (ordinary) generating series

(3.7)
$\Psi(\xi,t):=\sum_{\nu=0}^{\infty}P_{\nu}(\xi)t^{\nu}\qquad\text{and}\qquad\Upsilon(\xi,t):=\sum_{\nu=0}^{\infty}Q_{\nu}(\xi)t^{\nu}.$

They converge for every $t$ and $\xi$ because of the degrees and leading
coefficients of $P_{\nu}$ and $Q_{\nu}$ and can be characterized as follows.

######  Proposition 3.3.

The functions $\Psi=\Psi(\xi,t)$ and $\Upsilon=\Upsilon(\xi,t)$ are the only
functions whose Taylor expansions in $t$ is based on polynomials in $\xi$ and
which satisfy the equalities $\Psi(\xi,0)=1$ and $\Upsilon(\xi,0)=0$ for every
$\xi$ and the differential equations

$\partial_{\xi}\Psi(\xi,t)=t\cdot\Psi(\xi,t)\quad\mathrm{and}\quad\Psi(\xi,t)+\partial_{\xi}\Upsilon(\xi,t)-\xi\cdot\Upsilon(\xi,t)=1+t\cdot\Upsilon(\xi,t).$

###### Proof.

Assume that $\Psi$ and $\Upsilon$ satisfies these properties, and write their
Taylor series in $t$ as in Equation (3.7), with polynomial coefficients
$\tilde{P}_{\nu}(\xi)$ and $\tilde{Q}_{\nu}(\xi)$ respectively. Now, the
equalities with $t=0$ imply that $\tilde{P}_{\nu}$ and $\tilde{Q}_{\nu}$
satisfy the condition for $\nu=0$ in Equation (3.6), and also explain the
existence of the term 1 in the second differential equation (set $t=0$ there).
In addition, comparing the coefficient of $t^{\nu}$ with $\nu\geq 1$ in the
series resulting from substituting these expansions into the differential
equations yields the other part of Equation (3.6). Hence
$\\{\tilde{P}_{\nu}\\}_{\nu=0}^{\infty}$ and
$\\{\tilde{Q}_{\nu}\\}_{\nu=0}^{\infty}$ are sequences satisfying that
equation, so that $\tilde{P}_{\nu}=P_{\nu}$ and $\tilde{Q}_{\nu}=Q_{\nu}$ for
every $\nu\in\mathbb{N}$ by Corollary 3.2. This proves the proposition. ∎

Concerning the polynomiality of the coefficients, see Remark 3.5 below.
Proposition 3.3 allows us to determine the series $\Psi$ and $\Upsilon$
explicitly.

######  Theorem 3.4.

The functions $\Psi$ and $\Upsilon$ from Equation (3.7) are given by

$\Psi(\xi,t)=e^{\xi
t+t^{2}/2}\qquad\mathrm{and}\qquad\Upsilon(\xi,t)=e^{(\xi+t)^{2}/2}\int_{\xi}^{\xi+t}e^{-w^{2}/2}dw=e^{\xi
t+t^{2}/2}\int_{0}^{t}e^{-\xi w-w^{2}/2}dw.$

They also satisfy the differential equations

$(\partial_{t}-\xi)\Psi(\xi,t)=t\Psi(\xi,t)\qquad\mathrm{and}\qquad(\partial_{t}-\xi)\Upsilon(\xi,t)=t\Upsilon(\xi,t)+1.$

###### Proof.

It suffices to show that the asserted series have the properties from
Proposition 3.3. The equalities with $t=0$ are immediate, and it is easy to
see that $\Psi(\xi,t)$ satisfies the first differential equation there and
that its expansion in $t$ involves polynomials in $\xi$ as its Taylor
coefficients. In addition, simple differentiation shows that

$\partial_{\xi}\Upsilon(\xi,t)=(\xi+t)\Upsilon(\xi,t)+e^{(\xi+t)^{2}/2}\big{[}e^{-(\xi+t)^{2}/2}-e^{-\xi^{2}/2}\big{]}=(\xi+t)\Upsilon(\xi,t)+1-\Psi(\xi,t),$

from which the second differential equation from that proposition quickly
follows.

It remains to verify that the Taylor expansion of $\Upsilon(\xi,t)$ in $t$
consists of polynomials in $\xi$. This is equivalent to
$\partial_{t}^{n}\Upsilon(\xi,t)\big{|}_{t=0}$ being a polynomial in $\xi$ for
every $n$. To see this, we first evaluate $\partial_{t}\Upsilon(\xi,t)$ as
$1+(\xi+t)\Upsilon(\xi,t)$, yielding the remaining differential equation as
well. Simple induction now shows that for every $n$ the derivative
$\partial_{t}^{n}\Upsilon(\xi,t)$ is a polynomial in $\xi+t$ plus another
polynomial in $\xi+t$ times $\Upsilon(\xi,t)$. After substituting $t=0$, the
fact that $\Upsilon(\xi,0)=0$ yields the desired assertion. This proves the
theorem. ∎

######  Remark 3.5.

Solving the differential equation from Proposition 3.3 implies that there
exist functions $\psi(t)$ and $\phi(t)$ such that $\Psi(\xi,t)$ and
$\Upsilon(\xi,t)$ are

$\psi(t)e^{\xi t+t^{2}/2}\quad\mathrm{and}\quad
e^{(\xi+t)^{2}/2}\Bigg{[}\phi(t)+\int_{\xi}^{\xi+t}e^{-w^{2}/2}dw+\big{(}\psi(t)-1\big{)}\int_{\xi}^{\infty}e^{-w^{2}/2}dw\Bigg{]}$

respectively, and that $\psi(0)=1$ and $\phi(0)=0$. Showing that the Taylor
expansions involve only polynomials in $\xi$ if and only if $\psi$ and $\phi$
are the constant functions with the respective values, seems, however, not
very straightforward.

######  Remark 3.6.

The differential equations from Theorem 3.4 also imply that

$\nu P_{\nu}(\xi)-\xi P_{\nu-1}(\xi)=P_{\nu-2}(\xi)\quad\mathrm{and}\quad\nu
Q_{\nu}(\xi)-\xi Q_{\nu-1}(\xi)=Q_{\nu-2}(\xi)\quad\mathrm{for\
all}\quad\nu\geq 2.$

For the explicit expressions for the polynomials from Equation (3.6), recall
that

(3.8) $e^{\xi
t-t^{2}/2}=\sum_{\nu=0}^{\infty}\operatorname{He}_{\nu}(\xi)\frac{t^{\nu}}{\nu!}.$

Theorem 3.4 then implies the following result.

######  Corollary 3.7.

For every $\nu\in\mathbb{N}$ we have

$P_{\nu}(\xi)=\frac{(-i)^{\nu}}{\nu!}\operatorname{He}_{\nu}(i\xi):=\frac{e^{-\xi^{2}/2}}{\nu!}\tfrac{d^{\nu}}{d\xi^{\nu}}e^{\xi^{2}/2}=\big{(}\xi+\tfrac{d}{d\xi}\big{)}^{\nu}\cdot\frac{1}{\nu!}=\sum_{a=0}^{\lfloor\nu/2\rfloor}\frac{\xi^{\nu-2a}}{a!(\nu-2a)!2^{a}}.$

In particular, $P_{\nu}$ is a polynomial of degree $\nu$ and parity
$(-1)^{\nu}$ such that $\nu!P_{\nu}\in\mathbb{Z}[\xi]$ and $P_{\nu}(0)$ is
$\frac{1}{2^{\nu/2}(\nu/2)!}$ for even $\nu$ and 0 for odd $\nu$.

Note that the proof of Theorem 3.4 yields the equality
$\partial_{t}\Upsilon(\xi,t)=\partial_{\xi}\Upsilon(\xi,t)+\Psi(\xi,t)$, from
which we deduce that
$Q_{\nu+1}(\xi)=\frac{Q_{\nu}^{\prime}(\xi)+P_{\nu}(\xi)}{\nu+1}$, and using
Equation (3.6) we get

(3.9)
$Q_{\nu}(\xi)=\sum_{a=0}^{\nu-1}\frac{(\nu-1-a)!}{\nu!}\tfrac{d^{a}}{d\xi^{a}}P_{\nu-1-a}(\xi)=\sum_{a=0}^{\lfloor(\nu-1)/2\rfloor}\frac{(\nu-1-a)!}{\nu!}P_{\nu-1-2a}(\xi),$

with $P_{\mu}(\xi)$ given in Corollary 3.7. It is thus indeed a polynomial of
degree $\nu-1$ and parity $(-1)^{\nu-1}$ such that
$\nu!Q_{\nu}\in\mathbb{Z}[\xi]$. It follows that $Q_{\nu}(0)=0$ for even
$\nu$, while if $\nu$ is odd then

(3.10)
$Q_{\nu}(0)=\frac{2^{\frac{\nu-1}{2}}\big{(}\frac{\nu-1}{2}\big{)}!}{\nu!}=\prod_{j=0}^{(\nu-1)/2}\frac{1}{2j+1}.$

######  Remark 3.8.

The recursion (3.6) extends naturally to $\nu\in\mathbb{Z}$, in which case
$P_{\nu}(\xi)=0$ and

$Q_{\nu}(\xi)=(-1)^{1-\nu}\operatorname{He}_{-1-\nu}(\xi)$

for $\nu\leq-1$. We take these as the definitions of $P_{\nu}$ and $Q_{\nu}$
in these cases. Then Remark 3.6 extends to all $\nu\in\mathbb{Z}$.

The polynomials $\operatorname{He}_{\nu}$ form an _Appell sequence_ , and
Corollary 3.7 shows that the same applies also for the polynomials
$\nu!P_{\nu}$. This means explicitly that the equalities

(3.11)
$\operatorname{He}_{\nu}(\xi+\eta)=\sum_{j=0}^{\nu}\binom{\nu}{j}\eta^{j}\operatorname{He}_{\nu-j}(\xi)\quad\text{and}\quad
P_{\nu}(\xi+\eta)=\sum_{j=0}^{\nu}\frac{\eta^{j}P_{\nu-j}(\xi)}{j!}$

hold. This either follows from the generating function $\Psi(\xi,t)$ (as well
as the exponential generating function $e^{\xi t-t^{2}/2}$) being $e^{\xi t}$
times a function of $\xi$ alone (see, e.g., [Ze5], even though this was known
much earlier), or by simple computations using the explicit formulae. A change
of variable in Equation (3.11) produces, for every $l\in\mathbb{N}$, the
equality

(3.12)
$\sum_{\nu=0}^{l}\frac{(-1)^{\nu}}{(l-\nu)!}(\xi+\zeta)^{l-\nu}P_{\nu}(\xi)=P_{l}(\zeta)\in\mathbb{Q}[\xi,\zeta].$

###  3.3. Auxiliary Polynomials

We now define a few other families of polynomials, which will appear later in
the Fourier expansion of the Shintani lift.

######  Lemma 3.9.

For any $l\in\mathbb{N}$, the polynomial

$\Pi_{l}(\xi,\zeta):=\sum_{\nu=0}^{l}\frac{(-1)^{\nu}}{(l-\nu)!}(\xi+\zeta)^{l-\nu}Q_{\nu}(\xi)\in\mathbb{Q}[\xi,\zeta]$

has degree $l-1$ in $\xi$, and it satisfies the equality
$\Pi_{l}(-\zeta,\zeta)=-Q_{l}(\zeta)$. We also have

$\partial_{\zeta}\Pi_{l}(\xi,\zeta)=\Pi_{l-1}(\xi,\zeta)\qquad\text{and}\qquad\partial_{\xi}\big{(}\Pi_{l}(\xi,\zeta)\mathrm{g}(\xi)\big{)}=\bigg{(}\frac{(\xi+\zeta)^{l}}{l!}-P_{l}(\zeta)\bigg{)}\mathrm{g}(\xi)$

for every $l\in\mathbb{N}$, and the generating series

$\sum_{l=0}^{\infty}\Pi_{l}(\xi,\zeta)t^{l}=-e^{t^{2}/2+\zeta
t}\int_{0}^{t}e^{\xi w-w^{2}/2}dw.$

###### Proof.

The degree in $\xi$ and the value of $\Pi_{l}(-\zeta,\zeta)$ are immediate
from the definition. Substituting the definition of $\Pi_{l}(\xi,\zeta)$
inside the generating series and setting produces

$\sum_{l=0}^{\infty}\sum_{\nu=0}^{l}\frac{(-1)^{\nu}t^{l}}{(l-\nu)!}(\xi+\zeta)^{l-\nu}Q_{\nu}(\xi)=\sum_{\nu=0}^{\infty}\sum_{\mu=0}^{\infty}\frac{t^{\mu}}{\mu!}(\xi+\zeta)^{\mu}(-t)^{\nu}Q_{\nu}(\xi)=e^{t(\xi+\zeta)}\Upsilon(\xi,-t)$

(with $\mu=l-\nu\geq 0$), where we have substituted Equation (3.7). The value
of the series, which we denote by $\Phi(\xi,\zeta,t)$, now follows from
Theorem 3.4. One checks directly that this series satisfies the differential
equations

$\partial_{\zeta}\Phi(\xi,\zeta,t)=t\Phi(\xi,\zeta,t)\text{ and
}(\partial_{\xi}-\xi)\Phi(\xi,\zeta,t)=e^{t^{2}/2+\zeta
t}\\!\int_{0}^{t}\\!(\xi-w)e^{\xi w-w^{2}/2}dw=e^{(\xi+\zeta)t}-\Psi(\zeta,t)$

(using Theorem 3.4 again), from which the two required equalities follow for
every $l$ after expanding everything in $t$. This proves the lemma. ∎

######  Remark 3.10.

Write the sum $\Pi_{l}(\xi,\zeta)+Q_{l}(\zeta)$ as
$\tilde{\Pi}_{l}(\xi+\zeta,\zeta)$ for some polynomial $\tilde{\Pi}_{l}$. The
equality $\Pi_{l}(-\zeta,\zeta)=-Q_{l}(\zeta)$ from Lemma 3.9 implies that
$\tilde{\Pi}_{l}(\omega,\zeta)\in\omega\mathbb{Q}[\omega,\zeta]$.

Using the polynomial $\tilde{\Pi}_{l}$ from Remark 3.10, we define

(3.13)
$E_{l}(\zeta):=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{\tilde{\Pi}_{l}(\xi+\zeta,\zeta)}{\xi+\zeta}\mathrm{g}(\xi)d\xi\in\mathbb{R}[\zeta].$

We shall need their following properties.

######  Lemma 3.11.

We have $E_{0}=E_{1}=0$, $E_{l}(-\zeta)=(-1)^{l}E_{l}(\zeta)$, and
$E_{l}(\zeta)\in\mathbb{Q}[\zeta]$ for the polynomials from Equation (3.13).
Moreover, if $H_{n}:=\sum_{a=1}^{n}\frac{1}{a}$ denotes the $n$th harmonic
number, then we have

$E_{l}(0)=-P_{l}(0)\sum_{a=1,\ 2\nmid
a}^{l}\frac{1}{a}=-\frac{P_{l}(0)}{2}(2H_{l}-H_{\lfloor
l/2\rfloor})\qquad\mathrm{for}\qquad l\in\mathbb{N}.$

###### Proof.

Theorem 3.4 and Lemma 3.9 evaluate the generating series

$\sum_{l=0}^{\infty}E_{l}(\zeta)t^{l}=\sum_{l=0}^{\infty}\\!\int_{-\infty}^{\infty}\\!\\!g(\xi)\frac{\Pi_{l}(\xi,\zeta)+Q_{l}(\zeta)}{\xi+\zeta}t^{l}d\xi=\\!\int_{0}^{t}\\!\\!e^{\zeta(t-w)+(t^{2}-w^{2})/2}\int_{-\infty}^{\infty}g(\xi)\frac{1-e^{(\xi+\zeta)w}}{\xi+\zeta}d\xi
dw.$

We now claim that for every real $\zeta$, $w$, and $h$ we have

(3.14)
$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{e^{(\xi+\zeta)w}-1}{\xi+\zeta}\mathrm{g}(\xi+h)d\xi=\int_{0}^{w}e^{(\zeta-h)s+s^{2}/2}ds.$

Indeed, both sides vanish for $r=0$, and their derivative with respect to $r$
coincide because $\int_{-\infty}^{\infty}\mathrm{g}(\xi)d\xi=\sqrt{2\pi}$.
Substituting Equation (3.14) into our generating series yields

$\sum_{l=0}^{\infty}E_{l}(\zeta)t^{l}=-e^{\zeta
t+t^{2}/2}\int_{0}^{t}e^{-\zeta w-w^{2}/2}\int_{0}^{w}e^{\zeta
s+s^{2}/2}dsdw,$

yielding the vanishing of $E_{0}$ and $E_{1}$. We apply the operator
$\zeta+t-\partial_{t}$ to both sides, and using Theorem 3.4 again, with
Equation (3.7), we get

$\sum_{l=1}^{\infty}\big{(}\zeta
E_{l}(\zeta)+E_{l-1}(\zeta)-(l+1)E_{l+1}(\zeta)\big{)}t^{l}=\int_{0}^{t}e^{\zeta
s+s^{2}/2}ds=\int_{0}^{t}\Psi(\zeta,s)ds=\sum_{l=1}^{\infty}\frac{P_{l-1}(\zeta)}{l}t^{l}.$

The resulting recurrence relation establishes the rationality and the parity,
and as the asserted values for the constant terms satisfy the resulting
relation for $\zeta=0$, these values follow as well. This proves the lemma. ∎

Given $l\in\mathbb{N}$, Equation (3.12) and the definition of the polynomials
$\Pi_{l}$ in Lemma 3.9 show that
$\frac{P_{l}(\xi)-(-1)^{l}P_{l}(\zeta)}{\xi+\zeta}$ and
$\frac{Q_{l}(\xi)-(-1)^{l}\Pi_{l}(\xi,\zeta)}{\xi+\zeta}$ are polynomials in
$\mathbb{Q}[\xi,\zeta]$. We can then define

(3.15)
$\Omega_{l}(\zeta):=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{\big{(}P_{l}(\xi)-(-1)^{l}P_{l}(\zeta)\big{)}\mathrm{e}(\xi)+\big{(}Q_{l}(\xi)-(-1)^{l}\Pi_{l}(\xi,\zeta)\big{)}\mathrm{g}(\xi)}{\xi+\zeta}d\xi,$

and deduce the following properties.

######  Lemma 3.12.

The generating series of the polynomials from Equation (3.15) is

$\sum_{l=0}^{\infty}\Omega_{l}(\zeta)t^{l}=\int_{0}^{t}e^{-\zeta
s}\frac{e^{t^{2}/2}-e^{s^{2}/2}}{t-s}ds.$

###### Proof.

By applying Theorem 3.4 and the generating series from Lemma 3.9, we can write

$\sum_{l=0}^{\infty}\Omega_{l}(\zeta)t^{l}=\frac{e^{-\zeta
t+t^{2}/2}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{e^{(\zeta+\xi)t}-1}{\zeta+\xi}\bigg{(}\mathrm{e}(\xi)+\mathrm{g}(\xi)\int_{0}^{t}e^{-\xi
w-w^{2}/2}dw\bigg{)}d\xi,$

where the second term in the parentheses is $\int_{0}^{t}\mathrm{g}(\xi+w)dw$.
We now claim that

(3.16)
$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{e^{(\zeta+\xi)t}-1}{\zeta+\xi}\mathrm{e}(\xi)d\xi=\int_{0}^{t}e^{\zeta
s}\frac{1-e^{s^{2}/2}}{s}ds$

Indeed, as with Equation (3.14), both sides vanish for $t=0$, and for
comparing their derivatives with respect to $t$, we employ integration by
parts and use Equation (3.5) and the equality
$\int_{-\infty}^{\infty}\mathrm{g}(\xi)d\xi=\sqrt{2\pi}$ again. Substituting
Equations (3.14) and (3.16) transforms our expression for the generating
series into

$e^{-\zeta t+t^{2}/2}\bigg{(}\int_{0}^{t}e^{\zeta
s}\frac{1-e^{s^{2}/2}}{s}ds+\int_{0}^{t}\int_{0}^{t}e^{(\zeta-w)s+s^{2}/2}dsdw\bigg{)}=e^{-\zeta
t+t^{2}/2}\int_{0}^{t}e^{\zeta s}\frac{1-e^{s^{2}/2-ts}}{s}ds,$

from which the desired formula follows by taking $s\mapsto t-s$. This proves
the lemma. ∎

######  Remark 3.13.

Expanding all the exponents in the generating series from Lemma 3.12 and
integrating yields the series
$\sum_{m=0}^{\infty}\sum_{b=1}^{\infty}\sum_{r=0}^{2b-1}\frac{t^{2b+m}(-\zeta)^{m}}{2^{b}b!m!(2b+m-r)}$,
where the internal sum over $r$ can be written as the difference
$H_{2b+m}-H_{m}$ between two harmonic numbers. Writing $\frac{1}{2^{b}b!}$ as
$P_{2b}(0)$ as well as $P_{2b-1}(0)=0$ using Corollary 3.7, we deduce that

$\Omega_{l}(\zeta)=(-1)^{l}\sum_{\nu=1}^{l}P_{\nu}(0)(H_{l}-H_{l-\nu})\frac{\zeta^{l-\nu}}{(l-\nu)!}=(-1)^{l}(l-\tfrac{1}{2})\frac{\zeta^{l-2}}{l!}+O(\zeta^{l-4})$

is a rational polynomial of degree $l-2$ and parity $(-1)^{l}$ every
$l\in\mathbb{N}$. We shall later also need, for $k\in\mathbb{N}$, the
rational, $(-1)^{k}$-symmetric, degree $k-2$ polynomial

$\tilde{\Omega}_{k}(\eta):=(-i)^{k}\Omega_{k}(i\eta)=(-1)^{k}\sum_{\nu=1}^{k}\frac{\operatorname{He}_{\nu}(0)}{\nu!}(H_{k}-H_{k-\nu})\frac{\eta^{k-\nu}}{(k-\nu)!}.$

###  3.4. Singular Schwartz Functions

For every $\nu\in\mathbb{Z}$ and $\xi\in\mathbb{R}$, we shall now define

(3.17)
$h_{\nu}(\xi):=P_{\nu}(\xi)\mathrm{e}(\xi)+Q_{\nu}(\xi)\mathrm{g}(\xi),$

which has the following property generalizing Equation (3.5).

######  Proposition 3.14.

For any $\nu\in\mathbb{Z}$ the function $h_{\nu}$ has parity $(-1)^{\nu-1}$,
and we have

$\tfrac{d}{d\xi}h_{\nu}(\xi)=h_{\nu-1}(\xi)-\sqrt{2\pi}\cdot
P_{\nu}(0)\cdot\delta(\xi).$

###### Proof.

The parity follows from that of the polynomials $P_{\nu}$ and $Q_{\nu}$ from
Corollary 3.2. The second claim follows from Equation (3.5), the equality
$\mathrm{g}^{\prime}(\xi)=-\xi\mathrm{g}(\xi)$, and the relation from Equation
(3.6). This proves the proposition. ∎

######  Remark 3.15.

The decay of $\mathrm{g}$ and $\mathrm{e}$ from Equations (3.1) and (3.4)
imply that

$|{h}_{\nu}(\xi)|=o_{\epsilon,\nu}\big{(}e^{-(1-\epsilon)\xi^{2}/2}\big{)}\quad\text{as}\quad|\xi|\to\infty\qquad\mathrm{for}\quad\nu\in\mathbb{Z}\quad\mathrm{and}\quad\epsilon>0.$

######  Lemma 3.16.

For any $\nu\in\mathbb{Z}$ and $\xi>0$ we have the equalities $\nu
h_{\nu}(\xi)-\xi h_{\nu-1}(\xi)=h_{\nu-2}(\xi)$ and
$\xi^{3}\frac{d}{d\xi}\frac{h_{\nu}(\xi)}{\xi^{\nu}}=-\frac{h_{\nu-2}(\xi)}{\xi^{\nu-2}}$.

###### Proof.

The first equality is a direct consequence of Remarks 3.6 and 3.8, and the
second one follows the first via Proposition 3.14. This proves the lemma. ∎

Since $h_{\nu}$ are in $L^{1}$ with exponential decay, their Fourier
transforms $\widehat{h_{\nu}}$ should be bounded and $C^{\infty}$. However,
for $\nu\geq 0$, $h_{\nu}$ is not $C^{\infty}$ hence not a Schwartz function,
and thus $\widehat{h_{\nu}}$ need not be in $L^{1}$. The explicit formula is
given in the following result.

######  Proposition 3.17.

For every $\nu\geq-1$ and $t\in\mathbb{R}$ we have the equality

$\widehat{h_{\nu}}(t):=\int_{-\infty}^{\infty}h_{\nu}(\xi)\mathbf{e}(-\xi
t)d\xi=\sqrt{2\pi}\bigg{(}\frac{\mathrm{g}(2\pi t)}{(2\pi
it)^{\nu+1}}-\sum_{r=0}^{\nu}\frac{P_{r}(0)}{(2\pi
it)^{\nu-r+1}}\bigg{)}=\sqrt{2\pi}\sum_{r=\nu+1}^{\infty}\frac{P_{r}(0)}{(2\pi
it)^{\nu-r+1}}.$

In particular, $\widehat{h_{\nu}}$ is bounded and $C^{\infty}$.

###### Proof.

The case $\nu=-1$ is just the Fourier transform of the Gaussian, combining
with Equation (3.8) and Corollary 3.7. Now, applying the Fourier transform to
Proposition 3.14 yields the equality $2\pi
it\cdot\widehat{h_{\nu}}(t)=\widehat{h_{\nu-1}}(t)-\sqrt{2\pi}\cdot
P_{\nu}(0)$ for every $\nu\in\mathbb{Z}$. The general formula follows by
induction on $\nu$ (to both sides), and implies the boundedness and
$C^{\infty}$ properties. This proves the proposition. ∎

For two indices $\kappa$ and $\nu$ in $\mathbb{Z}$, we define the function

(3.18)
$\varphi_{\kappa,\nu}(\lambda,z):=\frac{(\lambda,Z(z))^{\kappa}}{(2\pi)^{(\nu+1)/2}}h_{\nu}\big{(}\sqrt{2\pi}\big{(}\lambda,Z^{\perp}(z)\big{)}\big{)},$

with $\lambda\in V_{\mathbb{R}}$ and $z\in\mathcal{H}$ such that
$\big{(}\lambda,Z^{\perp}(z)\big{)}\neq 0$. For $\kappa<0$ we also impose the
condition $\big{(}\lambda,Z(z)\big{)}\neq 0$. These functions decay like
Schwartz functions by Remark 3.15, But near points where
$\big{(}\lambda,Z^{\perp}(z)\big{)}$ or $\big{(}\lambda,Z(z)\big{)}$ vanishes,
they may become discontinuous. They are therefore “singular” Schwartz
functions.

The parity from Proposition 3.14 implies that

(3.19)
$\varphi_{\kappa,\nu}(-\lambda,z)=(-1)^{\kappa+\nu+1}\varphi_{\kappa,\nu}(\lambda,z)\quad\text{for\
every}\quad\lambda\in V_{\mathbb{R}}\quad\text{and}\quad z\in\mathcal{H},$

and the behavior of $Z(z)$ and $Z^{\perp}(z)$ under the action of
$\operatorname{SL}_{2}(\mathbb{R})$ implies that the function
$\varphi_{\kappa,\nu}(\lambda,z)$ from Equation (3.18) has the modularity
property

(3.20) $\varphi_{\kappa,\nu}(\gamma\lambda,\gamma
z)=j(\gamma,z)^{-2\kappa}\varphi_{\kappa,\nu}(\lambda,z)\quad\text{for}\quad\lambda\in
V_{\mathbb{R}},\ z\in\mathcal{H},\text{ and
}\gamma\in\operatorname{SL}_{2}(\mathbb{R}).$

Note that for all $k\in\mathbb{N}$ the functions from Equations (2.16) and
(3.18), and the expansions from Equations (2.17) and (2.18), are related by

(3.21)
$\tilde{\varphi}_{k}(\lambda;\tau,z)=v^{-k/2}q_{\tau}^{Q(\lambda)}\varphi_{k,-1}\big{(}\sqrt{v}\lambda,z\big{)}\mathrm{\
and\
}\Theta_{k,L,h}(\tau,z)=v^{\frac{1-k}{2}}\\!\\!\sum_{m\in\mathbb{Z}+Q(h)}\\!\sum_{\lambda\in
L_{m,h}}\\!\varphi_{k,-1}\big{(}\sqrt{v}\lambda,z\big{)}q_{\tau}^{m},$

since ${h}_{-1}$ is just the Gaussian $\mathrm{g}$. Moreover, since
$Z(z)-yZ^{\perp}(z)=\frac{iy}{\sqrt{N}}\big{(}\begin{smallmatrix}1&-2z\\\
0&1\end{smallmatrix}\big{)}$ we can write

(3.22)
$\varphi_{\kappa,\nu}(\lambda,z)=\frac{y^{\kappa}(\xi+i\eta)^{\kappa}}{(2\pi)^{(\kappa+\nu+1)/2}}h_{\nu}(\xi)\quad\text{with}\quad\xi=\sqrt{2\pi}\big{(}\lambda,Z^{\perp}(z)\big{)}\quad\text{and}\quad\eta=\sqrt{\tfrac{2\pi}{N}}\Big{(}\lambda,\big{(}\begin{smallmatrix}1&-2z\\\
0&1\end{smallmatrix}\big{)}\Big{)},$

and prove the following result.

######  Proposition 3.18.

Take $\kappa$ and $\nu$ in $\mathbb{Z}$ as well as an element $0\neq\lambda\in
V_{\mathbb{R}}$. Then we have

$-L_{z}\varphi_{\kappa,\nu}(\lambda,z)=\varphi_{\kappa+1,\nu-1}(\lambda,z)$

at every point $z\in\mathcal{H}$ such that
$\big{(}\lambda,Z^{\perp}(z)\big{)}\neq 0$, where if $\kappa<0$ then we assume
that $z$ must also satisfy $\big{(}\lambda,Z(z)\big{)}\neq 0$.

###### Proof.

Write $\varphi_{\kappa,\nu}(\lambda,z)$ as in Equation (3.22), and then simple
calculations give that $L_{z}y=y^{2}$, $L_{z}\xi=-y(\xi+i\eta)$, and
$L_{z}\eta=0$ in these parameters. Combining this with Proposition 3.14
produces the desired result. This proves the proposition. ∎

Note that the assumption $\big{(}\lambda,Z(z)\big{)}\neq 0$ is always
satisfied when $Q(\lambda)\geq 0$ (and $\lambda\neq 0$), and if $Q(\lambda)<0$
then it holds for every $z\in\mathcal{H}$ except a single point, in which
$Z^{\perp}(z)$ is a scalar multiple of $\lambda$.

###  3.5. Fourier Transforms

We shall also need the Fourier expansion of the following generalization of
the functions $h_{\nu}$. Take $\kappa\in\mathbb{Z}$, and then for
$\xi\in\mathbb{R}$ and $\eta\in\mathbb{R}$ we define

(3.23)
$\begin{split}\mathrm{e}_{\kappa}(\xi;\eta)&:=(\xi+i\eta)^{\kappa-1}h_{0}(\xi)=(\xi+i\eta)^{\kappa-1}\mathrm{e}(\xi)\qquad\mathrm{and}\\\
\mathrm{g}_{\kappa}(\xi;\eta)&:=(\xi+i\eta)^{\kappa-1}h_{-1}(\xi)=(\xi+i\eta)^{\kappa-1}\mathrm{g}(\xi).\end{split}$

For any $l\in\mathbb{N}$ we use Equation (3.12) and the polynomials from Lemma
3.9 and define

(3.24)
$\mathbf{g}_{\kappa,l}(\xi;\eta):=P_{l}(i\eta)\mathrm{e}_{\kappa}(\xi;\eta)+\Pi_{l}(\xi,i\eta)\mathrm{g}_{\kappa}(\xi;\eta)=\sum_{\nu=0}^{l}\frac{(-1)^{\nu}}{(l-\nu)!}(\xi+i\eta)^{\kappa+l-\nu-1}h_{\nu}(\xi).$

By applying Equation (3.5) and the derivative formula from that lemma (or
Proposition 3.14 with Equation (3.12)), we get

(3.25)
$\partial_{\xi}\mathbf{g}_{\kappa+1,l}(\xi;\eta)=\kappa\mathbf{g}_{\kappa,l}(\xi;\eta)+\frac{\mathrm{g}_{\kappa+l+1}(\xi;\eta)}{l!}-\sqrt{2\pi}(i\eta)^{\kappa}P_{l}(i\eta)\delta(\xi).$

We assume that $\eta\neq 0$ in case $\kappa\leq 0$, and then the expression
from Equations (3.23) and (3.24) are $L^{1}$ as functions of $\xi$. Let thus
$\widehat{\mathrm{e}_{\kappa}}$, $\widehat{\mathrm{g}_{\kappa}}$ and
$\widehat{\mathbf{g}_{\kappa,l}}$ denote the Fourier transform of
$\mathrm{e}_{\kappa},\mathrm{g}_{\kappa}$ and $\mathbf{g}_{\kappa,l}$ in $\xi$
respectively. Result for $\eta<0$ can be obtained from those with $\eta>0$
because we have

$\mathbf{g}_{\kappa,l}(\xi;-\eta)=(-1)^{\kappa+l}\mathbf{g}_{\kappa,l}(-\xi;\eta)\quad\mathrm{and\
hence}\quad\widehat{\mathbf{g}_{\kappa,l}}(t;-\eta)=(-1)^{\kappa+l}\widehat{\mathbf{g}_{\kappa,l}}(-t;\eta).$

For analyzing the behavior of $\widehat{\mathbf{g}_{\kappa,l}}(t;\eta)$, we
shall need additional special functions. For every $\nu\in\mathbb{Z}$,
$j\in\mathbb{N}$, $t\in\mathbb{R}$, and $\eta>0$ we define

(3.26) $I_{\nu,j}(\eta,t):=\int_{-t}^{\infty}\frac{(w+t)^{j}}{j!}{e^{-\eta
w}}\bigg{(}e^{-w^{2}/2}-\sum_{\mu=0}^{\nu}\frac{\operatorname{He}_{\mu}(0)}{\mu!}w^{\mu}\bigg{)}\frac{dw}{w^{\nu+1}},\quad
I_{\nu}(\eta):=I_{\nu,0}(\eta,0).$

For $\nu\leq-1$ this is defined for all $\eta\in\mathbb{R}$. It is easy to
check that

(3.27)
$\partial_{\eta}I_{\nu,j}(\eta,t)=\frac{\operatorname{He}_{\nu}(0)}{\nu!\eta^{j+1}}e^{\eta
t}-I_{\nu-1,j}(\eta,t)\qquad\mathrm{and\ thus}\qquad
I_{\nu}^{\prime}(\eta)=\frac{\operatorname{He}_{\nu}(0)}{\nu!\eta}-I_{\nu-1}(\eta)$

for all $\nu$ and $j$, while Equation (3.1) and simple differentiation give

(3.28)
$I_{-1}(\eta)=-e^{\eta^{2}/2}\mathrm{e}(\eta)\qquad\mathrm{and}\qquad\partial_{t}I_{\nu,j}(\eta,t)=\begin{cases}I_{\nu,j-1}(\eta,t),&\text{
if }j\geq 1,\\\ e^{\eta t}\mathrm{g}(t),&\text{ if }j=0\text{ and
}\nu=-1.\end{cases}$

Using the functions from Equation (3.26), we can now evaluate the Fourier
transforms $\widehat{\mathrm{e}_{\kappa}}$ and $\widehat{\mathrm{g}_{\kappa}}$
of the functions from Equation (3.23) as follows.

######  Lemma 3.19.

For $\kappa\geq 1$ we have the equalities

$\begin{split}\widehat{\mathrm{e}_{\kappa}}(t;\eta)&=\sqrt{2\pi}\sum_{\mu=0}^{\kappa-1}\binom{\kappa-1}{\mu}\frac{(i\eta)^{\kappa-1-\mu}}{(-2\pi
i)^{\mu}}\cdot\bigg{(}\frac{d}{dt}\bigg{)}^{\mu}\bigg{(}\frac{\mathrm{g}(2\pi
t)-1}{2\pi it}\bigg{)}\qquad\mathrm{and}\\\
\widehat{\mathrm{g}_{\kappa}}(t;\eta)&=\sqrt{2\pi}(-i)^{\kappa-1}\operatorname{He}_{\kappa-1}(2\pi
t-\eta)\mathrm{g}(2\pi t).\end{split}$

On the other hand, when $\kappa\leq 0$ and $\eta>0$ we get

$\widehat{\mathrm{e}_{\kappa}}(t;\eta)=\sqrt{2\pi}\cdot i^{\kappa}e^{-2\pi\eta
t}I_{0,-\kappa}(\eta,2\pi
t)\quad\mathrm{and}\quad\widehat{\mathrm{g}_{\kappa}}(t;\eta)=\sqrt{2\pi}\cdot
i^{\kappa-1}e^{-2\pi\eta t}I_{-1,-\kappa}(\eta,2\pi t).$

###### Proof.

The results for $\kappa\geq 1$ follow from applying the differential operator
$\big{(}i\eta-\frac{\partial_{t}}{2\pi i}\big{)}^{\kappa-1}$ to the cases
$\nu=0$ or $\nu=-1$ of Proposition 3.17 respectively, combined with Equation
(3.11) for the latter. For the case $\kappa\leq 0$ we recall that

$\int_{-\infty}^{\infty}(\xi+i\eta)^{\kappa-1}\mathbf{e}(-\xi
t)d\xi=\begin{cases}2\pi\cdot i^{\kappa-1}\frac{(2\pi
t)^{|\kappa|}e^{-2\pi\eta t}}{|\kappa|!},&\text{ if }t>0,\\\ 0,&\text{ if
}t<0,\end{cases}$

and we can expressing the Fourier transform of the product as the convolution
of the Fourier transforms. This proves the lemma. ∎

######  Remark 3.20.

The proof of Lemma 3.19 can evaluate, using the full Proposition 3.17 and
Corollary 3.7, the Fourier transform the generalization
$\xi\mapsto(\xi+i\eta)^{\kappa-1}h_{\nu}(\xi)$ of the functions from Equation
(3.23) to any $\nu$. When $\kappa\leq 0$ this gives just
$\sqrt{2\pi}i^{\kappa+\nu}I_{\nu,\kappa}(\eta,t)$. For $\kappa=0$ and $t=0$
this reduces to the equality
$\int_{-\infty}^{\infty}\frac{h_{\nu}(\xi)}{\xi+i\eta}d\xi=\sqrt{2\pi}i^{\nu}I_{\nu}(\eta)$.

We can now deduce the following useful property.

######  Lemma 3.21.

For every $\nu\in\mathbb{Z}$ and $\eta>0$ we have the equality
$\eta^{3}\frac{d}{d\eta}\frac{I_{\nu}(\eta)}{\eta^{\nu}}=\frac{I_{\nu-2}(\eta)}{\eta^{\nu-2}}$.

###### Proof.

Using Equation (3.27), Remark 3.20, and the first equality in Lemma 3.16 we
get

$\displaystyle\eta^{1+\nu}\frac{d}{d\eta}\frac{I_{\nu}(\eta)}{\eta^{\nu}}$
$\displaystyle=\frac{\operatorname{He}_{\nu}(0)}{\nu!}-(\eta
I_{\nu-1}(\eta)+\nu
I_{\nu}(\eta))=\frac{\operatorname{He}_{\nu}(0)}{\nu!}-\frac{(-i)^{\nu}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{\nu
h_{\nu}(\xi)+i\eta h_{\nu-1}(\xi)}{\xi+i\eta}d\xi$
$\displaystyle=\frac{\operatorname{He}_{\nu}(0)}{\nu!}-\frac{(-i)^{\nu}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\bigg{(}\frac{h_{\nu-2}(\xi)}{\xi+i\eta}+h_{\nu-1}(\xi)\bigg{)}d\xi.$

The first integral produces the desired result $I_{\nu-2}(\eta)$ by Remark
3.20 again, and integrating Proposition 3.14 evaluates, via Remark 3.15, the
second one as $-(-1)^{\nu}P_{\nu}(0)$. As this cancels with the first term by
Corollary 3.7, this proves the lemma. ∎

One can also show that $I_{\nu}$, using the polynomials from Equation (3.30)
below, that

(3.29)
$I_{\nu}=\widetilde{P}_{\nu}I_{0}-\widetilde{Q}_{\nu}I_{-1}+\hat{\Omega}_{\nu},\quad\text{with
a polynomial }\hat{\Omega}_{\nu}\text{ of degree }|\nu|-2\text{ and parity
}(-1)^{\nu}.$

We can also evaluate the Fourier transform of $\mathbf{g}_{\kappa,l}$ at
$t=0$.

######  Proposition 3.22.

Given any $\kappa\in\mathbb{Z}$, $l\in\mathbb{N}$, and $\eta>0$ we have

$\widehat{\mathbf{g}_{\kappa,l}}(0;\eta)=\begin{cases}-\frac{\sqrt{2\pi}i^{\kappa+l}}{\kappa\cdot
l!}\bigg{(}\operatorname{He}_{\kappa+l}(\eta)-\eta^{\kappa}\operatorname{He}_{l}(\eta)\bigg{)},&\text{
if }\kappa\neq 0\text{ and }\kappa+l\geq 0,\\\
\sqrt{2\pi}\cdot(-i)^{l}\big{(}I_{l}(\eta)-\tilde{\Omega}_{l}(\eta)\big{)},&\text{
if }\kappa=0,\end{cases}$

where $I_{\nu,j}$ and $\tilde{\Omega}_{k}$ are defined in Equation (3.26) and
Remark 3.13 respectively.

###### Proof.

When $\kappa\neq 0$ and $\eta>0$ we can integrate Equation (3.25) (using the
fact that Remark 3.15 extends to the functions $\mathbf{g}_{\kappa,l}$) to
obtain

$\displaystyle\widehat{\mathbf{g}_{\kappa,l}}(0;\eta)$
$\displaystyle=\int_{-\infty}^{\infty}\mathbf{g}_{\kappa,l}(\xi;\eta)d\xi=\frac{1}{\kappa}\int_{-\infty}^{\infty}\bigg{(}-\frac{\mathrm{g}_{\kappa+l+1}(\xi;\eta)}{l!}+\sqrt{2\pi}(i\eta)^{\kappa}P_{l}(i\eta)\delta(\xi)\bigg{)}d\xi$
$\displaystyle=\frac{1}{\kappa}\bigg{(}-\frac{\widehat{\mathrm{g}_{\kappa+l+1}}(0;\eta)}{l!}+\sqrt{2\pi}P_{l}(i\eta)(i\eta)^{\kappa}\bigg{)},$

which gives the desired value by Lemma 3.19 and Corollary 3.7.

When $\kappa=0$ we evaluate directly, where the formula from Remark 3.20 gives

$\widehat{\mathbf{g}_{0,l}}(0;\eta)=\sqrt{2\pi}(-i)^{l}I_{l}(\eta)-(-1)^{l}\int_{-\infty}^{\infty}\frac{h_{l}(\xi)-(-1)^{l}(P_{l}(i\eta)\mathrm{e}(\xi)+\Pi_{l}(\xi,i\eta)\mathrm{g}(\xi))}{\xi+i\eta}d\xi.$

As Equation (3.17) transforms the latter integral into
$(-1)^{l}\sqrt{2\pi}\Omega_{l}(i\eta)$ via Equation (3.15), we get the desired
value by Remark 3.13. This proves the proposition. ∎

###  3.6. Asymptotic Estimates

For determining the asymptotic behavior of the functions $I_{\nu,j}$ and the
Fourier transforms $\widehat{\mathbf{g}_{\kappa,l}}$, we shall also need the
following special functions. Like the definition of $\tilde{\Omega}_{k}$ in
Remark 3.13, the polynomials $P_{\nu}$ and $Q_{\nu}$ for $\nu\in\mathbb{Z}$
given in Equation (3.6) and Remark 3.8 have the modifications

(3.30)
$\widetilde{P}_{\nu}(\eta):=i^{\nu}P_{\nu}(i\eta)\qquad\mathrm{and}\qquad\widetilde{Q}_{\nu}(\eta):=i^{\nu-1}Q_{\nu}(i\eta),$

of the same parities as $P_{\nu}$ and $Q_{\nu}$. Using these we define
functions

(3.31) $\begin{split}&J_{-1}(\eta):=\mathrm{g}(i\eta)=e^{\eta^{2}/2},\quad
J_{0}(\eta):=-\int_{0}^{\eta}e^{r^{2}/2}dr=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}\frac{e^{-\eta
s}-e^{\eta s}}{s}\mathrm{g}(s)ds\\\ &\mathrm{and}\qquad
J_{\nu}(\eta):=\tilde{P}_{\nu}(\eta)J_{0}(\eta)-\tilde{Q}_{\nu}(\eta)J_{-1}(\eta)\qquad\mathrm{for}\qquad\nu\in\mathbb{Z}\end{split}$

(the two expressions for $J_{0}(\eta)$ are the same because they vanish at
$\eta=0$ and have the same derivative, and since
$\widetilde{P}_{0}=\widetilde{Q}_{-1}=1$ and
$\widetilde{P}_{-1}=\widetilde{Q}_{0}=0$ the two definitions for $\nu=0$ and
for $\nu=-1$ coincide). Since the polynomials from Equation (3.30) satisfy
identities analogous to those from Equation (3.6) and Remark 3.6, the latter
of which yields the equality $\eta J_{\nu}(\eta)+\eta
J_{\nu-1}(\eta)=-J_{\nu-2}(\eta)$ for every $\eta$ and $\nu$, evaluating the
derivatives of $J_{0}$ and $J_{-1}$ implies, for all $\nu\in\mathbb{Z}$, the
relations

(3.32) $J_{\nu}^{\prime}=-J_{\nu-1},\quad
J_{\nu}(-\eta)=(-1)^{\nu-1}J_{\nu}(\eta),\quad\mathrm{and}\quad\eta^{3}\frac{d}{d\eta}\frac{J_{\nu}(\eta)}{\eta^{\nu}}=\frac{J_{\nu-2}(\eta)}{\eta^{\nu-2}}.$

The following estimates will be helpful when we evaluate sums of Fourier
transforms later.

######  Lemma 3.23.

For any $\epsilon>0$ and fixed $\eta\in\mathbb{R}$ we have the asymptotic
growth

$I_{-1,0}(\eta,t)=\begin{cases}\sqrt{2\pi}J_{-1}(\eta)+o_{\eta,\epsilon}(e^{-(1-\epsilon)t^{2}/2}),&t\to\infty,\\\
o_{\eta,\epsilon}(e^{-(1-\epsilon)t^{2}/2}),&t\to-\infty,\end{cases}$

with $J_{-1}(\eta)$ from Equation (3.31). More generally, given
$j\in\mathbb{N}$ we have

$I_{-1,j}(\eta,t)=P_{j}(t-\eta)I_{-1,0}(\eta,t)+o_{j,\eta,\epsilon}(e^{-(1-\epsilon)t^{2}/2})\qquad\mathrm{as}\qquad
t\to\pm\infty.$

Moreover, given $\eta>0$ and such $\epsilon$, and with $J_{0}$ from Equation
(3.31), we get

$\frac{j!}{t^{j}}I_{0,j}(\eta,t)=-(-1)^{j}j!\Gamma(-j,-\eta
t)+\begin{cases}\sqrt{2\pi}J_{0}(\eta)+O_{j,\eta}\big{(}\frac{1}{t}\big{)},&t\to\infty,\\\
o_{j,\eta,\epsilon}(e^{-(1-\epsilon)t^{2}/2}),&t\to-\infty.\end{cases}$

###### Proof.

Using Equation (3.4) and the value $\sqrt{2\pi}$ of
$\int_{-\infty}^{\infty}g(\xi)d\xi$, the first equation follows from

$I_{-1,0}(\eta,t)=\int_{-t}^{\infty}e^{-\eta
w-w^{2}/2}dw=e^{\eta^{2}/2}\int_{\eta-t}^{\infty}\mathrm{g}(s)ds=\begin{cases}e^{\eta^{2}/2}\big{(}\sqrt{2\pi}-\mathrm{e}(t-\eta)\big{)},&t>\eta,\\\
-e^{\eta^{2}/2}\mathrm{e}(\eta-t),&t<\eta,\end{cases}$

and Remark 3.15. Next, using the definition in Equation (3.26) and simple
algebra we get, for every $\eta\in\mathbb{R}$ and $j\in\mathbb{N}$, the
equality

$\delta_{j,0}e^{\eta
t-t^{2}/2}=-\int_{-t}^{\infty}\frac{d}{dw}\big{(}\tfrac{(w+t)^{j}}{j!}e^{-\eta
w-w^{2}/2}\big{)}dw=(j+1)I_{-1,j+1}-(t-\eta)I_{-1,j}-(1-\delta_{j,0})I_{-1,j-1}.$

Since the left hand side times any polynomial is
$o_{\epsilon}(e^{-(1-\epsilon)t^{2}/2})$ as $t\to\pm\infty$, a simple
induction on $j$ combines with Remark 3.6 to prove the second relation.

Now suppose $t\neq 0$ and $\eta>0$. For $j=0$ we evaluate

$I_{0,0}(\eta,t)=\int_{-t}^{\infty}e^{-\eta
w}\frac{\mathrm{g}(w)-1}{w}dw=\int_{-t}^{|t|}e^{-\eta
w}\frac{\mathrm{g}(w)}{w}dw+\int_{|t|}^{\infty}e^{-\eta
w}\frac{\mathrm{g}(w)}{w}dw-\Gamma(0,-\eta t).$

As $t\to-\infty$, the first term above vanishes and the second term is
$o_{\eta,\epsilon}(e^{-(1-\epsilon)t^{2}/2})$ for any fixed
$\eta\in\mathbb{R}$. As $t\to\infty$, the second term behaves the same,
whereas using Equation (3.31) we see that the first term contributes

$\int_{-t}^{t}\\!\\!e^{-\eta
w}\frac{\mathrm{g}(w)}{w}dw\\!=\\!\\!\int_{0}^{\infty}\\!\frac{e^{-\eta
w}\\!-\\!e^{\eta w}}{w}\mathrm{g}(w)dw-\\!\int_{t}^{\infty}\\!\frac{e^{-\eta
w}\\!-\\!e^{\eta
w}}{w}\mathrm{g}(w)dw\\!=\\!\sqrt{2\pi}J_{0}(\eta)+o_{\eta,\epsilon}(e^{-(1-\epsilon)t^{2}/2}).$

This proves the third equality for $j=0$. When $j\geq 1$ we can write
$\frac{j!}{t^{j}}I_{0,j}(\eta,t)$ as

$\frac{1}{t^{j}}\int_{-t}^{\infty}\frac{(w+t)^{j}-t^{j}}{w}e^{-\eta
w-w^{2}/2}dw-\frac{e^{\eta
t}}{t^{j}}\int_{0}^{\infty}\frac{s^{j}-t^{j}}{s-t}e^{-\eta
s}ds+\int_{-t}^{\infty}e^{-\eta w}\frac{\mathrm{g}(w)-1}{w}dw.$

After expanding $(w+t)^{j}$ binomially, the first term is seen to be
$o_{j,\eta,\epsilon}(e^{-(1-\epsilon)t^{2}/2})$ as $t\to-\infty$ and
$O_{j,\eta}\big{(}\frac{1}{t}\big{)}$ as $t\to\infty$. Expanding
$\frac{s^{j}-s^{j}}{s-t}$ and Equation (3.2) evaluate the second term as
$-(-1)^{j}j!\Gamma(-j,-\eta t)+\Gamma(0,-\eta t)$, and the third term is just
the expression for $j=0$. Putting everything together proves the last desired
equality. This proves the lemma. ∎

In fact, the first term in the last equation in the proof of Lemma 3.23 can
easily be evaluated explicitly up to
$o_{j,\eta,\epsilon}(e^{-(1-\epsilon)t^{2}/2})$ also when $t\to\infty$, but
the estimate $O_{j,\eta}\big{(}\frac{1}{t}\big{)}$ will be sufficient for our
purposes.

######  Remark 3.24.

The generating series $\sum_{j=0}^{\infty}I_{-1,j}(0,0)X^{j}$ of the constants
$\big{\\{}I_{-1,j}(0,0)\\}_{j=0}^{\infty}$ is

$\int_{0}^{\infty}e^{wX-w^{2}/2}dw=e^{X^{2}/2}\int_{0}^{\infty}e^{-(w-X)^{2}/2}dw=e^{X^{2}/2}\bigg{(}\int_{0}^{\infty}e^{-w^{2}/2}dw+\int_{0}^{X}e^{-w^{2}/2}dw\bigg{)}$

(by symmetry), which equals $\Upsilon(0,X)+\sqrt{\pi/2}\Psi(0,X)$ by Theorem
3.4. It therefore follows from Equations (3.7) and (3.17) and the parity from
Proposition 3.14 that

$I_{-1,j}(0,0)=Q_{j}(0)+\sqrt{\pi/2}P_{j}(0)=\lim_{\xi\to
0^{-}}h_{j}(\xi)=(-1)^{j-1}\lim_{\xi\to 0^{+}}h_{j}(\xi)\quad\mathrm{for}\quad
j\in\mathbb{N}.$

We can now state the asymptotic expansion of
$\widehat{\mathbf{g}_{\kappa,l}}(t;\eta)$.

######  Proposition 3.25.

Take $\kappa\in\mathbb{Z}$, $l\in\mathbb{N}$, and $\eta\in\mathbb{R}$, with
$\eta>0$ in case $\kappa\leq 0$. We then have

$\widehat{\mathbf{g}_{\kappa,l}}(t;\eta)=-\sqrt{2\pi}\frac{i^{\kappa+l}\operatorname{He}_{l}(\eta)}{l!(-2\pi
t)^{\kappa}}e^{-2\pi\eta t}\Gamma(\kappa,-2\pi\eta
t)+o_{\kappa,l,\eta,\epsilon}(e^{-2\pi^{2}(1-\epsilon)t^{2}}),$

an expression holding as $t\to-\infty$ as well as in the limit $t\to\infty$
when $\kappa\geq 1$. On the other hand, if $\kappa\leq 0$ and $\eta>0$ then
the limit as $t\to\infty$ is given by the same formula plus the term

$\frac{2\pi(-i)^{\kappa+l}}{|\kappa|!(-2\pi t)^{\kappa}}e^{-2\pi\eta
t}\big{[}J_{l}(\eta)+O_{\kappa,l,\eta}\big{(}\tfrac{1}{t}\big{)}\big{]},$

where $J_{l}(\eta)$ is the function defined in Equation (3.31).

###### Proof.

Taking the Fourier transform of Equation (3.24) allows us to write

$\widehat{\mathbf{g}_{\kappa,l}}(t;\eta)=P_{l}(i\eta)\widehat{\mathrm{e}_{\kappa}}(t;\eta)+\Pi_{l}\big{(}\tfrac{\partial_{t}}{-2\pi
i},i\eta\big{)}\widehat{\mathrm{g}_{\kappa}}(t;\eta).$

When $\kappa\geq 1$ we apply Lemma 3.19 and Equation (3.2) to obtain that

$\widehat{\mathrm{e}_{\kappa}}(t;\eta)=-\sqrt{2\pi}\frac{e^{-2\pi\eta
t}\Gamma(\kappa,-2\pi\eta t)}{(2\pi
it)^{\kappa}}+o_{\kappa,l,\epsilon}(\eta^{\kappa-1}e^{-(1-\epsilon)t^{2}/2})\qquad\mathrm{as}\qquad
t\to\pm\infty$

and that $\Pi_{l}\big{(}\frac{\partial_{t}}{-2\pi
i},i\eta\big{)}\widehat{\mathrm{g}_{\kappa}}(t;\eta)$ is bounded by the same
error term. This establishes, via Corollary 3.7, the desired formula in this
case. On the other hand, for $\kappa\leq 0$ and $\eta>0$ we deduce from Lemma
3.19 that

$\widehat{\mathbf{g}_{\kappa,l}}(t;\eta)=\sqrt{2\pi}\cdot
i^{\kappa}e^{-2\pi\eta t}P_{l}(i\eta)I_{0,-\kappa}(\eta,2\pi
t)+\sqrt{2\pi}\cdot i^{\kappa-1}\Pi_{l}\big{(}\tfrac{\partial_{t}}{-2\pi
i},i\eta\big{)}e^{-2\pi\eta t}I_{-1,-\kappa}(\eta,2\pi t).$

Lemma 3.23 now shows that in the limit $t\to-\infty$ the first summand is

$-\sqrt{2\pi}\cdot i^{\kappa}P_{l}(i\eta)\frac{e^{-2\pi\eta
t}\Gamma(\kappa,-2\pi\eta t)}{(-2\pi
t)^{\kappa}}+o_{\kappa,l,\epsilon}(e^{-2\pi^{2}(1-\epsilon)t^{2}})$

and the second one goes into the error term, which yields the required formula
also here by another application of Corollary 3.7. In the limit $t\to\infty$
we get the same contribution, but we need to consider the additional terms
from Lemma 3.23 in that limit. The term arising from the first summand is

(3.33) $\frac{2\pi\cdot i^{\kappa}e^{-2\pi\eta t}}{|\kappa|!(2\pi
t)^{\kappa}}\big{[}P_{l}(i\eta)J_{0}(\eta)+O_{\kappa,l,\eta}\big{(}\tfrac{1}{t}\big{)}\big{]}.$

For the one from the second summand, Remark 3.10 allows us to write

$\displaystyle\Pi_{l}\big{(}\tfrac{\partial_{t}}{-2\pi i},i\eta\big{)}$
$\displaystyle e^{-2\pi\eta t}I_{-1,-\kappa}(\eta,2\pi
t)=\big{[}-Q_{l}(i\eta)+\tilde{\Pi}_{l}\big{(}\tfrac{\partial_{t}}{-2\pi
i}+i\eta,i\eta\big{)}\big{]}e^{-2\pi\eta t}I_{-1,-\kappa}(\eta,2\pi t)$
$\displaystyle=e^{-2\pi\eta t}\big{[}-Q_{l}(i\eta)I_{-1,-\kappa}(\eta,2\pi
t)+\tilde{\Pi}_{l}\big{(}\tfrac{\partial_{t}}{-2\pi
i},i\eta\big{)}I_{-1,-\kappa}(\eta,2\pi t)\big{]}$

(by the action on that exponent). Lemma 3.23 and the fact that Corollary 3.2
gives the estimate $P_{|\kappa|}(2\pi t-\eta)=\frac{(2\pi
t)^{|\kappa|}}{|\kappa|!}\big{[}1+O_{\kappa,\eta}\big{(}\frac{1}{t}\big{)}\big{]}$
show that $\sqrt{2\pi}\cdot i^{\kappa-1}$ times the first summand here is

$-\frac{2\pi\cdot i^{\kappa-1}e^{-2\pi\eta t}}{|\kappa|!(2\pi
t)^{\kappa}}\big{[}Q_{l}(i\eta)J_{-1}(\eta)+O_{\kappa,l,\eta}\big{(}\tfrac{1}{t}\big{)}\big{]}.$

Equations (3.30) and (3.31) now show that this expression combines with the
one from Equation (3.33) to the asserted extra term, up to the required error
term. Finally, Equation (3.28) and the property of $\tilde{\Pi}_{l}$ from
Remark 3.10 imply, via Lemma 3.23 again, that the expression involving that
polynomial also goes into the error term. This proves the proposition. ∎

###  3.7. Lattice Sums

To evaluate the constant term of the Shintani lift, we need to calculate
certain lattice sums involving the function $\mathbf{g}_{\kappa,l}$ defined in
Equation (3.24). For $\kappa\in\mathbb{Z}$, $l\in\mathbb{N}$,
$\eta\in\mathbb{R}$, a real number $\upsilon>0$, and an element
$\omega\in\mathbb{R}/\mathbb{Z}$, we consider the sum

(3.34)
$\mathbf{G}_{\kappa,l}(\omega;\upsilon,\eta):=\sum_{0\neq\xi\in\mathbb{Z}+\omega}\mathbf{g}_{\kappa,l}(\upsilon\xi;\eta),$

which converges absolutely by Remark 3.15 and defines a continuous function in
$\eta$. We will be interested in its value at $\eta=0$, denoted by

(3.35)
$\mathbf{G}_{\kappa,l}(\omega;\upsilon):=\mathbf{G}_{\kappa,l}(\omega;\upsilon,0)=\lim_{\eta\to
0^{+}}\mathbf{G}_{\kappa,l}(\omega;\upsilon,\eta).$

and its asymptotic expansion as $\upsilon\to 0^{+}$.

For analyzing it, we first need to recall a few familiar functions. Let
$\\{B_{\mu}(\omega)\\}_{\mu=0}^{\infty}$ (for $\omega\in\mathbb{R}$) be the
_Bernoulli polynomials_ defined by

$\frac{te^{\omega
t}}{e^{t}-1}=\sum_{\mu=0}^{\infty}B_{\mu}(\omega)\frac{t^{\mu}}{\mu!},\quad\mathrm{so\
that\ in\ particular}\quad B_{1}(\omega)=\omega-\tfrac{1}{2},$

and then $B_{\mu}:=B_{\mu}(0)$ are the _Bernoulli numbers_. Moreover, one
defines $\mathbb{B}_{\mu}$ to be the 1-periodic function that coincides with
$B_{\mu}$ on the interval $(0,1)$, and whose value on the integers is 0 in
case $\mu=1$ and $B_{\mu}$ otherwise. Then $\mathbb{B}_{\mu}$ with $\mu\geq 2$
is continuous on $\mathbb{R}$ (in particular, $\mathbb{B}_{0}$ is the constant
function 1), and we have

(3.36) $\mathbb{B}_{1}(0)=0=\lim_{\omega\to
0+}\mathbb{B}_{1}(\omega)+\tfrac{1}{2}=\lim_{\omega\to
0-}\mathbb{B}_{1}(\omega)-\tfrac{1}{2}\qquad\mathrm{and}\qquad\mathbb{B}_{\mu}(\omega)=-\sum_{0\neq
m\in\mathbb{Z}}\frac{\mu!\mathbf{e}(m\omega)}{(2\pi im)^{\mu}},$

the latter Fourier expansion being valid for every
$\omega\in\mathbb{R}/\mathbb{Z}$ and $\mu\in\mathbb{N}$ (this is essentially
Equations (13), (14), and (15) in Section 1.13 of [EMOT]).

We also recall from Section 1.11 of [EMOT] the function

(3.37) $F(q,s)=\sum_{m=1}^{\infty}\frac{q^{m}}{m^{s}}\qquad\mathrm{for}\qquad
s\in\mathbb{C}\quad\mathrm{and}\quad
q\in\mathbb{C}\quad\mathrm{with}\quad|q|<1.$

Since we shall only use this function when $s=-j$ for $j\in\mathbb{N}$, where
$F(q,-j)$ is a polynomial in $q$ divided by $(1-q)^{j+1}$, the analytic
continuation to any $q\in\mathbb{C}\setminus[1,\infty)$, and even to any
$1\neq q\in\mathbb{C}$, is immediate. Writing $q=\mathbf{e}(\omega)$ for
$\omega\in\mathbb{R}$, the fact that $F(q,0)=\frac{q}{1-q}$ combines with
Equation (15) of Section 1.11 of [EMOT] (for $j\geq 1$) to gives, for all
$j\in\mathbb{N}$, the expansion

(3.38) $F\big{(}\mathbf{e}(\omega),-j\big{)}=\frac{j!}{(-2\pi
i\cdot\omega)^{j+1}}-\frac{B_{j+1}+\delta_{j,0}}{j+1}+O(\omega).$

Another function to recall is the _polygamma function_ , defined for
$m\in\mathbb{N}$ and $z\in\mathbb{C}$ by

(3.39)
$\psi^{(m)}(z):=\frac{d^{m+1}}{dz^{m+1}}\log\Gamma(z)=-\gamma\delta_{m,0}+(-1)^{m}m!\sum_{a=0}^{\infty}\bigg{(}\frac{\delta_{m,0}}{(a+1)^{m+1}}-\frac{1}{(a+z)^{m+1}}\bigg{)},$

where $\delta_{m,0}$ is the Kronecker $\delta$-symbol again.

Let $\tilde{\psi}^{(m)}$ be the 1-periodic function that coincides with
$\psi^{(m)}$ from Equation (3.39) on $(0,1]$. Then for $\kappa\in\mathbb{Z}$
and $\omega\in\mathbb{R}/\mathbb{Z}$ we define

(3.40)
$\Phi_{\kappa}(\omega):=\begin{cases}-\mathbb{B}_{\kappa}(\omega)/\kappa,&\kappa\geq
1,\\\
-\big{[}\tilde{\psi}^{(|\kappa|)}(-\omega)+(-1)^{\kappa}\tilde{\psi}^{(|\kappa|)}(\omega)\big{]}\big{/}2|\kappa|!,&\kappa\leq
0,\end{cases}$

as well as

(3.41) $\Xi_{\kappa}(\omega):=\frac{(-2\pi
i)^{1-\kappa}}{\sqrt{2\pi}}\begin{cases}F(\mathbf{e}(\omega),\kappa)/|\kappa|!+\delta_{\kappa,0}/2,&\kappa\leq
0\text{ and }\omega\not\in\mathbb{Z},\\\
-\mathbb{B}_{1-\kappa}(0)/(1-\kappa)!,&\kappa\leq 1\text{ and
}\omega\in\mathbb{Z},\\\ 0,&\text{otherwise.}\end{cases}$

Note that for all $\kappa\in 2\mathbb{Z}$ we have the equality

(3.42) $\Phi_{\kappa}(1)=\operatorname{CT}_{s=1-\kappa}\zeta(s).$

Recalling the notation $H_{n}$ for the $n$th harmonic number, we shall also
need the constant

(3.43) $C_{l}:=\frac{\gamma+\log 2-2H_{l}+H_{\lfloor
l/2\rfloor}}{2}=\frac{\gamma+\log 2}{2}-\sum_{a=1,\ 2\nmid a}^{l}\frac{1}{a}.$

We remark that it is easy to see, via the asymptotic $H_{n}=\log
n+\gamma+o(1)$ as $n\to\infty$, that $C_{l}$ from Equation (3.43) grows as
$-\frac{\log l}{2}+o(1)$ as $l\to\infty$.

The evaluation of the expression that we need is now carried out as follows.

######  Proposition 3.26.

Take $\kappa\in\mathbb{Z}$, $l\in\mathbb{N}$, $\eta\in\mathbb{R}$,
$\upsilon>0$, and $\omega\in\mathbb{R}/\mathbb{Z}$. Then the value of the
expression $\mathbf{G}_{\kappa,l}(\omega;\upsilon)$ from Equation (3.35) is

$-\sqrt{2\pi}\upsilon^{\kappa-1}\big{[}P_{l}(0)\Phi_{\kappa}(\omega)+Q_{l}(0)\Xi_{\kappa}(\omega)\big{]}+\begin{cases}-\frac{\sqrt{2\pi}\cdot
i^{\kappa+l}\operatorname{He}_{\kappa+l}(0)}{\upsilon\kappa\cdot
l!}+o_{\kappa,l,\epsilon}(e^{-2\pi^{2}(1-\epsilon)/\upsilon^{2}}),&\kappa\geq
1,\\\
\frac{\sqrt{2\pi}}{\upsilon}\delta_{\kappa,0}P_{l}(0)(\log\upsilon+C_{l})+O_{\kappa,l,\omega}(\upsilon^{\kappa}),&\kappa\leq
0.\end{cases}$

###### Proof.

Consider first the case where $\kappa\geq 1$. For $\omega\neq 0$ we can apply
the Poisson Summation Formula, after which Propositions 3.22 and 3.25 yield

$\displaystyle\mathbf{G}_{\kappa,l}(\omega;\upsilon)$
$\displaystyle=\frac{1}{\upsilon}\bigg{(}\widehat{\mathbf{g}_{\kappa,l}}(0;0)+\lim_{\eta\to
0}\sum_{0\neq
m\in\mathbb{Z}}\mathbf{e}(m\omega)\widehat{\mathbf{g}_{\kappa,l}}\big{(}\tfrac{m}{\upsilon};\eta\big{)}\bigg{)}$
$\displaystyle=-\frac{\sqrt{2\pi}i^{\kappa+l}}{\upsilon\kappa}\bigg{(}\frac{\operatorname{He}_{\kappa+l}(0)}{l!}+\frac{\kappa\operatorname{He}_{l}(0)}{l!}\lim_{\eta\to
0}\sum_{0\neq
m\in\mathbb{Z}}\frac{\mathbf{e}(m\omega)\upsilon^{\kappa}}{(-2\pi
m)^{\kappa}}e^{-2\pi\eta m/\upsilon}\Gamma\big{(}\kappa,-\tfrac{2\pi\eta
m}{\upsilon}\big{)}\bigg{)}$
$\displaystyle=\frac{\sqrt{2\pi}}{\upsilon\kappa}\bigg{(}\frac{-i^{\kappa+l}\operatorname{He}_{\kappa+l}(0)}{l!}-\kappa!P_{l}(0)\sum_{0\neq
m\in\mathbb{Z}}\frac{\mathbf{e}(m\omega)\upsilon^{\kappa}}{(2\pi
im)^{\kappa}}\bigg{)}$
$\displaystyle=\frac{\sqrt{2\pi}}{\upsilon\kappa}\bigg{(}\frac{-i^{\kappa+l}\operatorname{He}_{\kappa+l}(0)}{l!}+P_{l}(0)\upsilon^{\kappa}\mathbb{B}_{\kappa}(\omega)\bigg{)},$

via Corollary 3.7 and Equation (3.36), up to an error term of
$o_{\kappa,l,\epsilon}(e^{-2\pi^{2}(1-\epsilon)/\upsilon^{2}})$. From the
definition, it is also clear that

(3.44) $\mathbf{G}_{\kappa,l}(0;\upsilon)=\lim_{\omega\to
0^{+}}\big{(}\mathbf{G}_{\kappa,l}(\omega+\mathbb{Z};\upsilon)-\mathbf{g}_{\kappa,l}(\upsilon\omega;0)\big{)}.$

Equation (3.24) implies that $-\lim_{\xi\to 0}\mathbf{g}_{\kappa,l}(\xi;0)$
vanishes for $\kappa>1$, and we have

$-\lim_{\xi\to 0^{+}}\mathbf{g}_{1,l}(\xi;0)=\lim_{\xi\to
0^{+}}(-1)^{l+1}h_{l}(\xi)=I_{-1,l}(0,0)=P_{l}(0)\sqrt{\pi/2}+Q_{l}(0)$

by Remark 3.24. Substituting these into Equation (3.44), and applying Equation
(3.36) for $\kappa=1$, completes the proof for $\kappa\geq 1$.

For $\kappa\leq 0$, we will first consider the case where $l=1$, in which

$\mathbf{g}_{\kappa,1}(\xi;0)=\xi^{\kappa}h_{0}(\xi)-\xi^{\kappa-1}h_{1}(\xi)=-\xi^{\kappa-1}\mathrm{g}(\xi)\quad\mathrm{hence}\quad\mathbf{G}_{\kappa,1}(\omega;\upsilon)=-\lim_{\eta\to
0^{+}}\sum_{0\neq\xi\in\mathbb{Z}+\omega}\mathrm{g_{\kappa}}(\upsilon\xi;\eta).$

Assuming that $\omega\neq 0$, we apply Equation (3.35), the Poisson summation
formula again, Lemma 3.19, Lemma 3.23, Equation (3.11), and Equation (3.37)
(with its analytic continuation), which compares
$\mathbf{G}_{\kappa,1}(\omega;\upsilon)$ with

$\displaystyle-\frac{\sqrt{2\pi}i^{\kappa-1}}{\upsilon}\bigg{(}\lim_{\eta\to
0^{+}}I_{-1,-\kappa}(\eta,0)+\lim_{\eta\to 0^{+}}\sum_{0\neq
m\in\mathbb{Z}}e^{-2\pi\eta m/\upsilon}I_{-1,-\kappa}\big{(}\eta,\tfrac{2\pi
m}{\upsilon}\big{)}\mathbf{e}(m\omega)\bigg{)}$
$\displaystyle=-\frac{\sqrt{2\pi}i^{\kappa-1}}{\upsilon}\bigg{(}I_{-1,-\kappa}(0,0)+\sum_{m=1}^{\infty}\Big{(}P_{|\kappa|}\big{(}\tfrac{2\pi
m}{\upsilon}\big{)}+o_{\kappa,\epsilon}(e^{-2\pi^{2}(1-\epsilon)m^{2}/\upsilon^{2}})\Big{)}\sqrt{2\pi}\mathbf{e}(m\omega)\bigg{)}$
(3.45)
$\displaystyle=-\frac{\sqrt{2\pi}i^{\kappa-1}}{\upsilon}\bigg{(}I_{-1,-\kappa}(0,0)+\sqrt{2\pi}\sum_{j=0}^{|\kappa|}\big{(}\tfrac{2\pi}{\upsilon}\big{)}^{j}\frac{P_{|\kappa|-j}(0)}{j!}F(\mathbf{e}(\omega),-j)\bigg{)}+o_{\kappa,\epsilon}(e^{-2\pi^{2}(1-\epsilon)/\upsilon^{2}}).$

Now, the summands with $j<|\kappa|$ give
$O_{\kappa,\omega}(\upsilon^{\kappa})$, and the same applies for the first
term when $\kappa\leq-1$. Since $I_{-1,0}(0,0)=\sqrt{\frac{\pi}{2}}$ by Remark
3.24, this is indeed the desired value, since $P_{1}(0)=0$. For $\omega=0$ we
apply Equation (3.44), where we have seen that the second term there is now
$+\frac{\mathrm{g}(\upsilon\omega)}{(\upsilon\omega)^{|\kappa|+1}}$. We expand
the term $F\big{(}\mathbf{e}(\omega),\kappa\big{)}$ from Equation (3.7) as in
Equation (3.38), and observe that the singularities in $\omega$ cancel with
those of the Laurent expansion of
$\frac{\mathrm{g}(\upsilon\omega)}{(\upsilon\omega)^{|\kappa|+1}}$, which is
$\sum_{\nu=0}^{\infty}i^{\nu}P_{\nu}(0)(\upsilon\omega)^{\nu+\kappa-1}$ by
Equation (3.8) and Corollary 3.7. Substituting into the limit from Equation
(3.44) yields

$-\frac{\sqrt{2\pi}i^{\kappa-1}}{\upsilon}\bigg{(}I_{-1,-\kappa}(0,0)-\sqrt{2\pi}\sum_{j=0}^{|\kappa|}\big{(}\tfrac{2\pi}{\upsilon}\big{)}^{j}\frac{P_{|\kappa|-j}(0)}{(j+1)!}(B_{j+1}+\delta_{j,0})\bigg{)}+1+o_{\kappa,\epsilon}(e^{-2\pi^{2}(1-\epsilon)/\upsilon^{2}}),$

where again the same terms (and the 1) go into the error term. Since for
$\kappa=0$ the two terms cancel, the result follows also in this case.

We now consider the case $l=0$, where Equations (3.34), (3.35), and (3.24) and
the trick from the proof of Lemma 8.5 of [BFI] evaluate
$\mathbf{G}_{\kappa,0}(\omega;\upsilon)$ as

(3.46)
$\sum_{0\neq\xi\in\mathbb{Z}+\omega}\frac{\mathrm{e}(\upsilon\xi)}{(\upsilon\xi)^{|\kappa|+1}}=-\\!\\!\sum_{0\neq\xi\in\mathbb{Z}+\omega}\frac{\operatorname{sgn}(\xi)}{(\upsilon\xi)^{|\kappa|+1}}\operatorname{CT}_{s=0}(\upsilon|\xi|)^{-s}\bigg{(}\int_{0}^{\infty}e^{-w^{2}/2}w^{s}dw-\int_{0}^{\upsilon|\xi|}e^{-w^{2}/2}w^{s}dw\bigg{)}.$

Recalling the Hurwitz zeta function
$\zeta(s,z):=\sum_{n=1}^{\infty}\frac{1}{(n+z)^{s}}$, the first term in
Equation (3.46) is the constant term at $s=0$ of
$-\frac{2^{(s-1)/2}}{\upsilon^{|\kappa|+1+s}}\Gamma\big{(}\frac{s+1}{2}\big{)}$
times

$\sum_{0<\xi\in\mathbb{Z}+\omega}\frac{1}{\xi^{|\kappa|+1+s}}+\sum_{0<\xi\in\mathbb{Z}-\omega}\frac{(-1)^{\kappa}}{\xi^{|\kappa|+1+s}}=\zeta(|\kappa|+1+s,\omega)+(-1)^{\kappa}\zeta(|\kappa|+1+s,-\omega),$

where $\pm\omega$ here means the corresponding representatives in $(0,1]$.
Since $\zeta(m+1+s,z)$ with $m\in\mathbb{N}$ expands as
$\frac{\delta_{m,0}}{s}-\frac{(-1)^{m}\psi^{(m)}(z)}{m!}+O(s)$ (see Equation
(9) on Section 1.10 of [EMOT] for $m=0$ and just Equation (3.39) for $m>0$),
the Taylor expansion of the remaining functions and the value $-\gamma-2\log
2$ of $\psi\big{(}\frac{1}{2}\big{)}$ produce the constant term

$-\sqrt{2\pi}\upsilon^{\kappa-1}\bigg{(}-\frac{\tilde{\psi}^{(|\kappa|)}(-\omega)+(-1)^{\kappa}\tilde{\psi}^{(|\kappa|)}(\omega)}{2|\kappa|!}-\delta_{\kappa,0}\frac{\gamma+\log
2+2\log\upsilon}{2}\bigg{)},$

which is the desired expression since $C_{0}=\frac{\gamma+\log 2}{2}$ by
Equation (3.43). The second term in Equation (3.46) becomes, after a simple
substitution,

$\operatorname{CT}_{s=0}\sum_{0\neq\xi\in\mathbb{Z}+\omega}\int_{0}^{1}\frac{\mathrm{g}(\upsilon\rho\xi)}{(\upsilon\xi)^{|\kappa|}}\rho^{s}d\rho=-\operatorname{CT}_{s=0}\int_{0}^{1}\mathbf{G}_{\kappa+1,1}(\omega;\upsilon\rho)\rho^{s-\kappa}d\rho.$

For $\kappa\leq-1$ our expression for
$\mathbf{G}_{\kappa+1,1}(\omega;\upsilon\rho)$ is
$O_{\kappa}\big{(}(\upsilon\rho)^{\kappa}\big{)}$, hence the integral
converges at $s=0$ and is $O_{\kappa}(\upsilon^{\kappa})$. When $\kappa=0$ we
have
$\mathbf{G}_{1,1}(\omega;\upsilon\rho)=-\frac{\sqrt{2\pi}}{\upsilon\rho}-\sqrt{2\pi}\Xi_{1}(\omega)$
up to rapidly decreasing functions, so that the integral is
$-\frac{\sqrt{2\pi}}{\upsilon s}$ (with no constant term), again plus
$O(1)=O_{\kappa}(\upsilon^{\kappa})$. This proves the result for $l=0$ as
well.

For general $l\in\mathbb{N}$, Equation (3.24) and Remark 3.10 allow us to
write

$\displaystyle\mathbf{G}_{\kappa,l}(\omega;\upsilon)$
$\displaystyle=\sum_{0\neq\xi\in\mathbb{Z}+\omega}\mathbf{g}_{\kappa,l}(\upsilon\xi;0)=P_{l}(0)\sum_{0\neq\xi\in\mathbb{Z}+\omega}\frac{\mathrm{e}(\upsilon\xi)}{(\upsilon\xi)^{|\kappa|+1}}+\sum_{0\neq\xi\in\mathbb{Z}+\omega}\Pi_{l}(\upsilon\xi,0)\frac{\mathrm{g}(\upsilon\xi)}{(\upsilon\xi)^{|\kappa|+1}}$
$\displaystyle=P_{l}(0)\mathbf{G}_{\kappa,0}(\omega;\upsilon)+Q_{l}(0)\mathbf{G}_{\kappa,1}(\omega;\upsilon)+\sum_{0\neq\xi\in\mathbb{Z}+\omega}\frac{\tilde{\Pi}_{l}(\upsilon\xi,0)}{\upsilon\xi}\frac{\mathrm{g}(\upsilon\xi)}{(\upsilon\xi)^{|\kappa|}},$

where $\frac{\tilde{\Pi}_{l}(\upsilon\xi)}{\upsilon\xi}$ is a polynomial in
$\upsilon\xi$. The first two terms now give the desired formula, up to
$\frac{\sqrt{2\pi}}{\upsilon}P_{l}(0)(C_{l}-C_{0})$ in case $\kappa=0$. When
$\kappa\leq-1$ it suffices to view the third term as a linear combination of
$\mathbf{G}_{j,1}(\omega;\upsilon)$ with $j\geq\kappa+1$, all of which are
$O(\upsilon^{j-1})$ when $j\leq 0$ and $O\big{(}\frac{1}{\upsilon}\big{)}$ in
case $j>0$, since these are all $O(\upsilon^{\kappa})$. For $\kappa=0$ we
evaluate
$\sum_{\xi\in\mathbb{Z}+\omega}\frac{\tilde{\Pi}_{l}(\upsilon\xi,0)}{\upsilon\xi}\mathrm{g}(\upsilon\xi)$
using the Poisson Summation Formula, where all the Fourier terms with $m\neq
0$ give $o_{l,\epsilon}(e^{-2\pi^{2}(1-\epsilon)/\upsilon^{2}})$ once again.
Finally, $\frac{1}{\upsilon}$ times the 0th Fourier term is
$\frac{\sqrt{2\pi}}{\upsilon}E_{l}(0)$ by Equation (3.13), which is precisely
the required expression by Lemma 3.11 and Equation (3.43). This completes the
proof of the proposition. ∎

##  4\. Nearly Holomorphic Modular Forms

In this section we shall prove our result in the most general case, evaluating
the Shintani lift $\mathcal{I}_{k,L}(\tau,f)$ from Equation (2.20) for nearly
holomorphic modular forms $f\in\widetilde{M}_{2k}^{!}(\Gamma)$. For comparing
our formulae with those from [ANS], consult Remark 2.8.

###  4.1. Traces and Regularizations

Recall that if $\lambda\in L^{*}$ satisfies $Q(\lambda)<0$ then the stabilizer
$\Gamma_{\lambda}$ of $\lambda$ in $\Gamma$ is finite, and $\lambda$ is a
multiple of $Z^{\perp}(z_{\lambda})$ for a unique $z_{\lambda}\in\mathcal{H}$.
We then define, for every $k\in\mathbb{Z}$ and
$f\in\mathcal{A}_{0}^{!}(\Gamma)$, the _trace_

(4.1)
$\operatorname{Tr}_{\lambda}^{(k)}(f):=\frac{\big{[}-\operatorname{sgn}\big{(}\lambda,Z^{\perp}(z_{\lambda})\big{)}\big{]}^{k}}{|\Gamma_{\lambda}|}f(z_{\lambda}).$

If $Q(\lambda)>0$ then we recall the geodesic
$c_{\lambda}\subseteq\mathcal{H}$ and its image $c(\lambda)\subseteq X$ from
Equation (2.12), and that when $\lambda$ is not split-hyperbolic, i.e., when
$\iota(\lambda)=0$ in the notation of Equation (2.13), the latter is a closed
geodesic inside the open modular curve $Y$. We can then define, for every
$g\in\mathcal{A}_{2k}^{!}(\Gamma)$, the _trace_

(4.2)
$\operatorname{Tr}_{\lambda}(g):=\oint_{c(\lambda)}g(z)\big{(}\lambda,Z(z)\big{)}^{k-1}dz.$

On the other hand, when $\lambda$ is split-hyperbolic, i.e., when
$\iota(\lambda)=1$, the image $c(\lambda)$ of $c_{\lambda}$ in $Y$ is not
compact, and if $g$ grows towards the cusps then the integral corresponding to
that from Equation (4.2) does not converge. We shall regularize it only for
nearly holomorphic modular forms, i.e., $g\in\widetilde{M}_{2k}^{!}(\Gamma)$.
Then its Fourier expansion near the cusp associated with some
$\ell\in\operatorname{Iso}(V)$ is given, in the coordinates from Equation
(2.9), by

(4.3)
$g_{\ell}(z_{\ell}):=(g\mid_{2k}\sigma_{\ell})(z_{\ell})=\sum_{l=0}^{p}\frac{g_{\ell,l}(z_{\ell})}{y_{\ell}^{l}}=\sum_{l=0}^{p}\sum_{n\in\mathbb{Z}}\frac{c_{\ell}(n,l)q_{\ell}^{n}}{y_{\ell}^{l}}=\sum_{l=0}^{p}\sum_{n\leq
0}\frac{c_{\ell}(n,l)q_{\ell}^{n}}{y_{\ell}^{l}}+g_{\ell}^{0}(z_{\ell}),$

where $p$ is the depth of $g$, $c_{\ell}(n,l)=0$ for all $0\leq l\leq p$ when
$n\ll 0$, and the latter decomposition is into the (finite) principal part and
the cuspidal part $g_{\ell}^{0}$. Recalling our extension of the incomplete
Gamma function in (3.2), and the truncated modular curve $Y_{T}$ from Equation
(2.10) for any $T>1$. We also set for $n$ and $\kappa$ from $\mathbb{Z}$,
positive reals $c$ and $T$, split-hyperbolic $\lambda\in V$, and
$g\in\widetilde{M}_{2k}^{!}(\Gamma)$ with expansion as in Equation (4.3) for
$\ell=\ell_{\lambda}$, the quantities

(4.4)
$\begin{split}\phi_{n}(\kappa,T;r)&:=\begin{cases}\Gamma(\kappa,rnT)/(rn)^{\kappa},&n\neq
0,\\\ -T^{\kappa}/\kappa,&n=0\mathrm{\ and\ }\kappa\neq 0,\\\ -\log T,&n=0{\
and\ }\kappa=0\end{cases}\qquad\mathrm{and}\\\
\operatorname{Sing}_{\lambda}(g,T)&:=i^{k}(2\sqrt{Q(\lambda)})^{k-1}\sum_{l=0}^{p}\sum_{n\in\mathbb{Z}}c_{\ell_{\lambda}}(n,l)\mathbf{e}\bigg{(}\frac{nr_{\lambda}}{\alpha_{\ell_{\lambda}}}\bigg{)}\phi_{n}\bigg{(}k-l,T;\frac{2\pi}{\alpha_{\ell_{\lambda}}}\bigg{)}.\end{split}$

Note that when $n=0$, $\phi_{0}$ is independent of $r$, and we can then omit
it from the notation. In addition, assuming that
$f\in\widetilde{M}_{2k}^{!}(\Gamma)$ expands as in Equation (4.3), the weight
lowering property of the operator $L_{z}$ implies that for every
$\nu\in\mathbb{N}$ we can write $(L_{z}^{\nu}f)_{\ell}(z_{\ell})$ as

(4.5)
$(L_{z}^{\nu}f\mid_{2k-2\nu}\sigma_{\ell})(z_{\ell})=L_{z_{\ell}}^{\nu}f_{\ell}(z_{\ell})=(-1)^{\nu}\sum_{l=\nu}^{p}\frac{l!f_{\ell,l}(z_{\ell})}{(l-\nu)!y_{\ell}^{l}}=(-1)^{\nu}\sum_{n\in\mathbb{Z}}\sum_{l=\nu}^{p}\frac{l!c_{\ell}(n,l)q_{\ell}^{n}}{(l-\nu)!y_{\ell}^{l-\nu}}.$

We can now define the _trace_ to be

(4.6)
$\operatorname{Tr}_{\lambda}(g):=\lim_{T\to\infty}\Bigg{(}\int_{c(\lambda)\cap
Y_{T}}g(z)\big{(}\lambda,Z(z)\big{)}^{k-1}dz+\operatorname{Sing}_{\lambda}(g,T)+(-1)^{k}\operatorname{Sing}_{-\lambda}(g,T)\Bigg{)}.$

We now prove that this is a regularization of the required trace.

######  Proposition 4.1.

The quantity in the limit defining $\operatorname{Tr}_{\lambda}(g)$ in
Equation (4.6) exists and is independent of the choice of $T$ when it is
sufficiently large.

###### Proof.

Since $c_{\lambda}$ only intersects the cusps $\ell_{\pm\lambda}$, there
exists some $R>1$ such that for all $T>R$, we find that $c(\lambda)\cap
Y_{T}\cong c_{\lambda}\cap\mathcal{H}_{T}$ is contained in
$\mathcal{H}_{R}\cup\sigma_{\ell_{\lambda}}\mathcal{F}_{T}^{\alpha_{\ell_{\lambda}}}\cup\sigma_{\ell_{-\lambda}}\mathcal{F}_{T}^{\alpha_{\ell_{-\lambda}}}$.
We thus obtain

$\int_{c(\lambda)\cap
Y_{T}}g(z)\big{(}\lambda,Z(z)\big{)}^{k-1}dz=\int_{c_{\lambda}\cap\mathcal{H}_{R}}\tilde{g}(z)dz+\int_{c_{\lambda}\cap\sigma_{\ell_{\lambda}}\mathcal{F}_{T}^{\alpha_{\ell_{\lambda}}}\setminus\mathcal{H}_{R}}\tilde{g}(z)dz+\int_{c_{\lambda}\cap\sigma_{\ell_{-\lambda}}\mathcal{F}_{T}^{\alpha_{\ell_{-\lambda}}}\setminus\mathcal{H}_{R}}\tilde{g}(z)dz$

for every $T>R$, where we wrote $\tilde{g}(z)$ for
$g(z)\big{(}\lambda,Z(z)\big{)}^{k-1}$. The first term is independent of $T$,
and if we change, in the integral corresponding to $\pm\lambda$, the variable
to $z_{\ell}$ for $\ell=\ell_{\pm\lambda}$ from Equation (2.9), then it
becomes

$(\pm
1)^{k}\int_{r_{\pm\lambda}+iR}^{r_{\pm\lambda}+iT}g_{\ell}(z_{\ell})\big{(}\sigma_{\ell}^{-1}(\pm\lambda),Z(z_{\ell})\big{)}^{k-1}dz_{\ell}=(\pm
i)^{k}(2\sqrt{Q(\lambda)})^{k-1}\int_{R}^{T}g_{\ell}(r_{\pm\lambda}+iy_{\ell})y_{\ell}^{k-1}dy_{\ell}$

via Equation (2.14). This expression is a differentiable function of $T$, and
Equations (4.3) and (3.3) show that its derivative is minus that of
$\operatorname{Sing}_{\lambda}(g,T)$ from Equation (4.4). Hence the expression
from Equation (4.6) is independent of $T$ as long as $T>R$, and in particular
$\operatorname{Tr}_{\lambda}(g)$ exists. This proves the proposition. ∎

Note that $\lim_{T\to\infty}\phi_{n}(\kappa,T;r)=0$ for any $r>0$ when $n>0$,
and the integral of the part $g_{\ell_{\pm\lambda}}^{0}$ from Equation (4.3)
converges as $T\to\infty$. Hence the regularization from Proposition 4.1 is
essentially only of the integral of the principal part.

Following [ANS] and [BFI], we now give an equivalent expression for the
regularized theta lift $\mathcal{I}_{k,L}(\tau,f)$ of $f$ from Equation
(2.20).

######  Proposition 4.2.

For $f\in\widetilde{M}^{!}_{2k}(\Gamma)$ with asymptotic expansion at the cusp
$\ell$ as in Equation (4.3), the regularized theta lift
$\mathcal{I}_{k,L}(\tau,f)$ of $f$ from Equation (2.20) can be written as

$\lim_{T\to\infty}\Bigg{(}\int_{Y_{T}}f(z)\Theta_{k,L}(\tau,z)d\mu(z)+\sum_{\ell\in\Gamma\backslash\operatorname{Iso}(V)}i^{k}\frac{\varepsilon_{\ell}}{\sqrt{N}}\Theta_{k,\ell}(\tau)\sum_{l=0}^{p}c_{\ell}(0,l)\phi_{0}(k-l,T)\Bigg{)},$

where $\phi_{0}$ is defined in Equation (4.4)555Note that $c_{\ell}(0,l)$ is
well-defined for $\ell\in\Gamma\backslash\operatorname{Iso}(V)$ by the
modularity of $f$..

###### Proof.

The proof uses Lemma 2.9 and proceeds in the same way as that of Proposition
5.2 of [ANS], with the only difference being that now we have

$\int_{1}^{\infty}\int_{0}^{\alpha_{\ell}}(f\mid_{2k}\sigma_{\ell})(z_{\ell})y_{\ell}^{k-1-s}dx_{\ell}dy_{\ell}=\alpha_{\ell}\sum_{l=0}^{p}c_{\ell}(0,l)\int_{1}^{\infty}y_{\ell}^{k-s-1-l}dy_{\ell}=\alpha_{\ell}\sum_{l=0}^{p}\frac{c_{\ell}(0,l)}{k-l-s}.$

∎

For $k$, $f$, and $g$ as above and $m\neq 0$, we can define the combinations

(4.7) $\operatorname{Tr}_{m,h}^{(k)}(f):=\sum_{\lambda\in\Gamma\backslash
L_{m,h}}\operatorname{Tr}_{\lambda}^{(k)}(f)\text{ for
}m<0,\quad\operatorname{Tr}_{m,h}(g):=\sum_{\lambda\in\Gamma\backslash
L_{m,h}}\operatorname{Tr}_{\lambda}(g)\text{ for }m>0.$

Note that $\Gamma\backslash L_{m,h}$ is finite when $m\neq 0$, so that there
is no question of convergence in Equation (4.7). On the other hand, when
$Q(\lambda)=m=0$, we will define the trace to be

(4.8)
$\operatorname{Tr}_{0,h}(g):=\sum_{\ell\in\Gamma\backslash\operatorname{Iso}(V)}\frac{\varepsilon_{\ell}}{\sqrt{N}}\iota_{\ell}(0,h)c_{\ell}(0,0)(\sqrt{N}\beta_{\ell})^{k}\Phi_{k}(\omega_{\ell,h}),$

where $\iota_{\ell}(m,h)$, $\omega_{\ell,h}$, and $\Phi_{\kappa}$ are defined
in Equations (2.15), (2.8), and (3.40) respectively.

The main term of the Shintani lift from Theorem 4.3 below will have the traces
from Equations (4.7) and (4.8) as coefficients. However, for the terms with
$\iota(m)=1$ we need to define some corrections. Recall that when $m>0$ and
$\iota_{\ell}(m,h)=1$, Remark 2.7 implies that the numbers $r_{\lambda}$ for
oriented $\lambda\in L_{m,h}\cap\ell^{\perp}$ are all the same modulo
$\frac{\beta_{\ell}}{2}\sqrt{\frac{N}{m}}\mathbb{Z}$. We can thus define, for
our element $f\in\widetilde{M}^{!}_{2k}(\Gamma)$ of depth $p$ expanded as in
Equation (4.3) near each cusp $\ell\in\Gamma\backslash\operatorname{Iso}(V)$,
the _complementary trace_

(4.9)
$\begin{split}&\operatorname{Tr}^{\mathrm{c}}_{m,h}(f,v):=-(-2i\sqrt{m})^{k}\cdot\sqrt{2\pi}\sum_{\ell\in\Gamma\backslash\operatorname{Iso}(V)}\big{(}\iota_{\ell}(m,h)+(-1)^{k}\iota_{\ell}(m,-h)\big{)}\frac{\varepsilon_{\ell}}{\sqrt{N}}\\\
&\times\sum_{\begin{subarray}{c}0>n\in\mathbb{Z}\\\ n\equiv 0\bmod
2\varepsilon_{\ell}\sqrt{m/N}\end{subarray}}\mathbf{e}\bigg{(}\frac{nr_{\lambda}}{\alpha_{\ell}}\bigg{)}\sum_{l=k}^{p}\bigg{(}\frac{2\pi
n}{\alpha_{\ell}}\bigg{)}^{l-k}\frac{l!c_{\ell}(n,l)}{(l-k)!}\cdot\frac{J_{l}\big{(}2\sqrt{2\pi
mv}\big{)}}{\big{(}2\sqrt{2\pi mv}\big{)}^{l}},\end{split}$

with $J_{l}(\eta)$ from Equation (3.31). Note that the sum over $n$ in
Equation (4.9) is finite, and is empty for all but finitely many values of
$m$.

We also define the _complementary trace from constants_

(4.10)
$\begin{split}&\operatorname{Tr}^{\mathrm{cc}}_{m,h}(f,v):=-(-2i\sqrt{m})^{k}\sum_{\ell\in\Gamma\backslash\operatorname{Iso}(V)}\big{(}\iota_{\ell}(m,h)+(-1)^{k}\iota_{\ell}(m,-h)\big{)}\frac{\varepsilon_{\ell}}{\sqrt{N}}\\\
&\times k!c_{\ell}(0,k)\frac{I_{k}\big{(}2\sqrt{2\pi
mv}\big{)}-\tilde{\Omega}_{k}\big{(}2\sqrt{2\pi mv}\big{)}}{\big{(}2\sqrt{2\pi
mv}\big{)}^{k}},\end{split}$

where $I_{k}$ and $\tilde{\Omega}_{k}$ are defined in Equations (3.26) and
Remark 3.13 respectively.

When $m=0$, there is only a complementary trace from constants, which is
defined by

(4.11)
$\begin{split}&\operatorname{Tr}^{\mathrm{cc}}_{0,h}(f,v):=\delta_{k,0}\delta_{h,0}\sqrt{v}\int_{Y}^{{\mathrm{reg}}}f(z)d\mu(z)+\sum_{\ell\in\Gamma\backslash\operatorname{Iso}(V)}\iota_{\ell}(0,h)\frac{\varepsilon_{\ell}}{\sqrt{N}}\times\\\
&\Bigg{(}-c_{\ell}(0,k)k!P_{k}(0)\frac{\log\big{(}\sqrt{2\pi
Nv}\beta_{\ell}\big{)}+C_{k}}{(2\pi
v)^{k/2}}+\sum_{l=0}^{p}l!c_{\ell}(0,l)Q_{l}(0)\frac{(\sqrt{N}\beta_{\ell})^{k-l}\Xi_{k-l}(\omega_{\ell,h})}{(2\pi
v)^{l/2}}\Bigg{)},\end{split}$

where $C_{k}$ and $\Xi_{\kappa}$ are defined in Equations (3.43) and (3.41)
respectively, and for $f\in\widetilde{M}^{!}_{2k}(\Gamma)$ the regularized
integral $\int_{Y}^{{\mathrm{reg}}}f(z)d\mu(z)$ is the (convergent) limit
$\lim_{T\to\infty}\int_{Y_{T}}f(z)d\mu(z)$.

###  4.2. Main Theorem and Proof

We can now state and prove our main theorem. Given $k\in\mathbb{N}$ and an
element $f\in\widetilde{M}_{2k}^{!}(\Gamma)$, we gather the traces from
Equations (4.7) and (4.8) and define

(4.12) $\mathcal{I}_{k,L,h}^{\mathrm{nh}}(\tau,f):=\sum_{b=0}^{\lfloor
p/2\rfloor}\sum_{0\leq
m\in\mathbb{Z}+Q(h)}\frac{\operatorname{Tr}_{m,h}(L_{z}^{2b}f)}{(4\pi
v)^{b}b!}q_{\tau}^{m},$

which is a nearly holomorphic function of depth
$\big{\lfloor}\frac{p}{2}\big{\rfloor}$ on $\mathcal{H}$ that is bounded at
$\infty$. Using the negative index case of Equation (4.7), we also define

(4.13)
$\mathcal{I}_{k,L,h}^{\mathrm{neg}}(\tau,f):=\sum_{0>m\in\mathbb{Z}+Q(h)}\sum_{l=k}^{p}\frac{4^{k}\sqrt{\pi}|m|^{\frac{k-1}{2}}h_{l}\big{(}2\sqrt{2\pi|m|v}\big{)}}{\sqrt{2}\big{(}4\sqrt{2\pi|m|v}\big{)}^{l}(l-k)!}\operatorname{Tr}_{m,h}^{(k)}(R_{2k-2l}^{l-k}L_{z}^{l}f)q_{\tau}^{m},$

which resembles the non-holomorphic part of a harmonic weak Maass form with
holomorphic $\xi$-image. We also gather the traces from Equations (4.9),
(4.10), and (4.11), and set

(4.14)
$\mathcal{I}_{k,L,h}^{\mathrm{c}}(\tau,f):=\sum_{\begin{subarray}{c}0<m\in\mathbb{Z}+Q(h)\\\
\iota(m)=1\end{subarray}}\operatorname{Tr}^{\mathrm{c}}_{m,h}(f,v)q_{\tau}^{m}\quad\mathrm{and}\quad\mathcal{I}_{k,L,h}^{\mathrm{cc}}(\tau,f):=\sum_{\begin{subarray}{c}0\leq
m\in\mathbb{Z}+Q(h)\\\
\iota(m)=1\end{subarray}}\operatorname{Tr}^{\mathrm{cc}}_{m,h}(f,v)q_{\tau}^{m},$

where the former is a finite sum of increasing terms, and the second one is
infinite but converges.

The main result, which evaluates the regularized Shintani lift of $f$, now
reads as follows.

######  Theorem 4.3.

Write the vector-valued Shintani lift $\mathcal{I}_{k,L}(\tau,f)$ of
$f\in\widetilde{M}_{2k}^{!}(\Gamma)$, defined in Equation (2.20), as
$\sum_{h\in D_{L}}\mathcal{I}_{k,L,h}(\tau,f)\mathfrak{e}_{h}$. Then the
scalar-valued coefficient associated with $h\in D_{L}$ is given by

$\mathcal{I}_{k,L,h}(\tau,f)=\mathcal{I}_{k,L,h}^{\mathrm{nh}}(\tau,f)+\mathcal{I}_{k,L,h}^{\mathrm{neg}}(\tau,f)+\mathcal{I}_{k,L,h}^{\mathrm{c}}(\tau,f)+\mathcal{I}_{k,L,h}^{\mathrm{cc}}(\tau,f),$

where the terms are defined in Equations (4.12), (4.13), and (4.14).

###### Proof.

We apply Proposition 4.2, and expand $\Theta_{k,L}(\tau,z)$ and
$\Theta_{k,\ell}(\tau)$ using Equations (3.21) and (2.19) respectively, which
gives

(4.15)
$\mathcal{I}_{k,L,h}(\tau,f)=\lim_{T\to\infty}\left(\begin{array}[]{c}\displaystyle
v^{\frac{1-k}{2}}\int_{Y_{T}}f(z)\sum_{m\in\mathbb{Z}+Q(h)}q_{\tau}^{m}\sum_{\lambda\in
L_{m,h}}\varphi_{k,-1}\big{(}\sqrt{v}\lambda,z\big{)}d\mu(z)+\\\
\displaystyle\sum_{\begin{subarray}{c}0\leq m\in\mathbb{Z}+Q(h)\\\
\iota(m)=1\end{subarray}}q_{\tau}^{m}\sum_{\ell\in\Gamma\backslash\operatorname{Iso}(V)}i^{k}\frac{\varepsilon_{\ell}}{\sqrt{N}}a(\Theta_{k,\ell},m,h,v)\sum_{l=0}^{p}c_{\ell}(0,l)\phi_{0}(k-l,T)\end{array}\right).$

We may interchange the order of integration and summation as both are
absolutely convergent for fixed $T$. Propositions 4.8, 4.9, 4.10, and 4.11
below now evaluate the coefficient of $q_{\tau}^{m}$ to be the one implied by
the asserted sum. This proves the theorem. ∎

A much simpler but interesting special case is the one where the depth $p<k$.

######  Corollary 4.4.

For $f\in\widetilde{M}_{2k}^{!,\leq p}(\Gamma)$ with $p<k$, we have

$\mathcal{I}_{k,L}(\tau,f)=\sum_{h\in
D_{L}}\mathcal{I}^{\mathrm{nh}}_{k,L,h}(\tau,f)\mathfrak{e}_{h}\in\widetilde{M}_{k+\frac{1}{2}}^{\leq\lfloor
p/2\rfloor}(\rho_{L}).$

For $p=0$, we recover the part of Theorem 6.1 in [ANS] with weakly holomorphic
input.

###### Proof.

If $p<k$, then $\mathcal{I}^{\mathrm{neg}}_{k,L,h}$ and
$\mathcal{I}^{\mathrm{c}}_{k,L,h}$ vanishes identically, and Equations (4.10)
and (4.11) imply that $\mathcal{I}^{\mathrm{cc}}_{k,L,h}$ consists only of the
constant term

$\operatorname{Tr}^{\mathrm{cc}}_{0,h}(f,v)=\sum_{\ell\in\Gamma\backslash\operatorname{Iso}(V)}\iota_{\ell}(0,h)\frac{\varepsilon_{\ell}}{\sqrt{N}}\sum_{l=0}^{k-1}l!c_{\ell}(0,l)Q_{l}(0)\frac{(\sqrt{N}\beta_{\ell})^{k-l}\Xi_{k-l}(\omega_{\ell,h})}{(2\pi
v)^{l/2}}.$

But using Equations (3.41), (2.7), (2.15), and (2.9), this becomes just a
multiple of $\delta_{h,0}$ times the sum of the residues of $L_{z}^{k-1}fdz$
at the cusps, with $L_{z}^{k-1}f\in M_{2}^{!}(\Gamma)$, which therefore
vanishes. This proves the corollary. ∎

We can now deduce the theorems mentioned in the Introduction.

###### Proof of Theorem 1.1.

We apply Theorem 4.3 to the scaled lattice $L_{\Delta}:=\Delta L$ for $L$ from
Equation (2.5), with $Q=-\frac{\det}{|\Delta|}$. Then $L_{\Delta}^{*}=L^{*}$
and it is well-known (see [GKZ] or [AE]) that if
$g\in\mathcal{A}_{\frac{3}{2},\rho_{L_{\Delta}}}^{!}$ and
$\Gamma:=\operatorname{PSL}_{2}(\mathbb{Z})$ then

$\Upsilon_{\Delta}(g)(z):=\frac{1}{[\Gamma:\Gamma_{L_{\Delta}}]}\sum_{\delta\in
D_{L_{\Delta}}}\chi_{\Delta}(\delta)g_{\delta}(4z)\in\mathcal{A}^{!}_{\frac{3}{2}}(\tilde{\Gamma}_{0}(4)),$

is in the Kohnen plus space, where $\chi_{\Delta}$ is the character from
Equation (1.1) and $\tilde{\Gamma}_{0}(4)\subset\mathrm{Mp}_{2}(\mathbb{Z})$
is the metaplectic cover of $\Gamma_{0}(4)$. We denote
$\mathcal{I}^{*}_{\Delta}(z,f):=\Upsilon_{\Delta}\big{(}\mathcal{I}^{*}_{1,L_{\Delta}}(z,f)\big{)}$
for $*\in\\{\mathrm{nh},\mathrm{neg},\mathrm{c},\mathrm{cc}\\}$. When
$f=J\cdot E_{2}^{*}$, we have $k=p=1$, and only non-zero coefficients of the
principal part are $c_{\ell}(\Delta,0)=1$,
$c_{\ell}(\Delta,1)=-\frac{3}{\pi}$, and $c_{\ell}(0,0)=-24$ for any
$\ell\in\operatorname{Iso}(V)$. The fact that $c_{\ell}(0,1)=0$ and $Q_{0}=0$
implies that $\mathcal{I}^{\mathrm{cc}}_{\Delta}(z,f)=0$, and we have

(4.16)
$\Gamma\backslash\mathcal{Q}_{\Delta^{2}}=\Gamma\big{\\{}[0,|\Delta|,C]\big{|}C\in\mathbb{Z}/\Delta\mathbb{Z}\big{\\}},~{}\Gamma_{L_{\Delta},\infty}\backslash\ell_{\infty}=\Gamma_{L_{\Delta},\infty}\big{\\{}[0,0,C]\big{|}C\in\mathbb{Z}/\Delta\mathbb{Z}\big{\\}},$

with the value
$\chi_{\Delta}\big{(}[0,|\Delta|,C]\big{)}=\chi_{\Delta}\big{(}[0,0,C]\big{)}=\big{(}\frac{C}{\Delta}\big{)}$,
the real part $r_{[0,|\Delta|,C]}=-\frac{C}{|\Delta|}$, and
$\mathbb{B}_{1}(\omega_{\ell_{\infty},[0,0,C]})=\mathbb{B}_{1}\big{(}\frac{C}{|\Delta|}\big{)}=\frac{C}{|\Delta|}-\frac{1}{2}$
for $0<C<|\Delta|$. Since $L_{z}f=\frac{3}{\pi}J$, $\chi_{\Delta}$ is anti-
symmetric, and the sign from Equation (4.1) is that of $-A$ from $\lambda$ as
in Equation (2.5), we have

$\mathcal{I}^{\mathrm{neg}}_{\Delta}(z,f)=\frac{-12}{\sqrt{2\pi}}\sum_{0<D\in\mathbb{Z}}\Bigg{(}\sum_{0\ll\lambda\in\Gamma\backslash\mathcal{Q}_{\Delta
D}}\frac{\chi_{\Delta}(\lambda)}{|\Gamma_{\lambda}|}J(z_{\lambda})\Bigg{)}\frac{h_{1}\big{(}2\sqrt{2\pi
Dy}\big{)}}{2\sqrt{2\pi Dy}}q^{-D}$

as well as

$\mathcal{I}^{\mathrm{nh}}_{\Delta}(z,f)=48|\Delta|H(-\Delta)+\sum_{0>D\in\mathbb{Z}}\Bigg{(}\sum_{\lambda\in\Gamma\backslash\mathcal{Q}_{\Delta
D}}\chi_{\Delta}(\lambda)\operatorname{Tr}_{\lambda}(f)\Bigg{)}q^{-D},$

where for the latter constant we use Equation (4.16) and the equality

$\sum_{C=1}^{|\Delta|-1}\bigg{(}\frac{C}{\Delta}\bigg{)}\bigg{(}\frac{C}{|\Delta|}-\frac{1}{2}\bigg{)}=\frac{1}{|\Delta|}\sum_{C=1}^{|\Delta|-1}C\bigg{(}\frac{C}{\Delta}\bigg{)}=\frac{\sqrt{|\Delta|}}{\pi}L\Big{(}\big{(}\tfrac{\cdot}{\Delta}\big{)},1\Big{)}=2H(-\Delta).$

Moreover, the congruence from Equation (4.9) gives $a|\Delta$, the
fundamentality of $\Delta$ leaves only $a=|\Delta|$ after $\Upsilon_{\Delta}$,
and then Equation (4.16) and a standard Gauss sum evaluation yield
$\mathcal{I}^{\mathrm{c}}_{\Delta}(z,f)=\frac{12}{\sqrt{2\pi}}\sqrt{|\Delta|}\cdot\frac{J_{1}\big{(}2\sqrt{2\pi|\Delta|y}\big{)}}{2\sqrt{2\pi|\Delta|y}}q^{|\Delta|}$.

Now, let $\tilde{f}_{-\Delta}(z)$ denote the holomorphic part
$\mathcal{I}^{\mathrm{nh}}_{\Delta}(z,f)$, which has the required expansion.
Since Lemma 3.16 and Equation (3.32) imply that $\xi_{\frac{3}{2}}$ takes the
functions $\frac{h_{1}(2\sqrt{2\pi Dy})}{(2\sqrt{2\pi Dy})}q^{-D}$ and
$\frac{J_{1}\big{(}2\sqrt{2\pi|\Delta|y}\big{)}}{2\sqrt{2\pi|\Delta|y}}q^{|\Delta|}$
to $-\frac{q^{D}}{4\sqrt{2\pi D}}$ and
$+\frac{q^{\Delta}}{4\sqrt{2\pi|\Delta|}}$ respectively, the complement
$\mathcal{I}^{\mathrm{neg}}_{\Delta}(z,f)+\mathcal{I}^{\mathrm{c}}_{\Delta}(z,f)$
of $\tilde{f}_{-\Delta}(z)$ is indeed harmonic, with the asserted
$\xi_{\frac{3}{2}}$-image $\frac{3}{2\pi}f_{-\Delta}$. This completes the
proof of the theorem. ∎

###### Proof of Theorems 1.2 and 1.4.

We apply Theorem 4.3 to the lattice $L$ in Equation (2.5), where
$N=\alpha_{\ell}=\beta_{\ell}=\varepsilon_{\ell}=1$, combined with the
isomorphism from [K] to scalar-valued modular forms. For Theorem 1.2, we let
$f=\big{(}\frac{\pi}{3}E_{2}^{*}\big{)}^{k}$,
$c_{n,l}=(-1)^{l}\binom{k}{l}\big{(}\frac{\pi}{3}\big{)}^{k-l}\delta_{n,0}$
for $n\leq 0$. Then $p=k$,
$L_{z}^{2b}f=\frac{k!}{(k-2b)!}(\frac{\pi}{3}E_{2}^{*})^{k-2b}$, $c(0,k)=1$,
$\mathcal{I}^{\mathrm{c}}_{k,L,0}(4z,f)=\mathcal{I}^{\mathrm{c}}_{k,L,1}(4z,f)=0$,
for $d<0\in\mathbb{Z}$ we have
$\operatorname{Tr}_{m,h}(L_{z}^{k}f)=\operatorname{Tr}_{m,h}(k!)=2k!H(-d)$ for
$m=\frac{d}{4}$ and $h=\frac{d}{2}+\mathbb{Z}$, for $m=\frac{d}{4}>0$ we have
$\iota(m)=1$ if and only if $d=a^{2}$ is a square and $\iota_{\ell}(m,h)=1$
for $h=\mathbb{d}{2}+\mathbb{Z}$ if and only if $a\in\mathbb{N}$,
$P_{k}(0)=\frac{1}{2^{k/2}(k/2)!}$ by Corollary 3.7, and for $l\leq k$ the
expression $Q_{l}(0)\Xi_{k-l}(\omega_{\ell,h})$ is
$-\delta_{h,0}\delta_{l,k-1}\frac{Q_{k-1}(0)}{\sqrt{2\pi}}$. Since
$\operatorname{Tr}_{d}$ for $d>0$ from Equation (1.7) is
$\operatorname{Tr}_{m,h}$ with these $m$ and $h$, and Equation (3.42) reduces
the trace from Equation (4.8) to that from Equation (1.8), we get

$\displaystyle\mathcal{I}^{\mathrm{nh}}_{k,L,0}(4z,f)+\mathcal{I}^{\mathrm{nh}}_{k,L,1}(4z,f)$
$\displaystyle=\sum_{b=0}^{k/2}\sum_{d=0}^{\infty}\frac{\frac{k!}{(k-2b)!}\operatorname{Tr}_{d}\big{(}(\frac{\pi}{3}E_{2}^{*})^{k-2b}\big{)}}{(16\pi
y)^{b}b!}q^{d},$
$\displaystyle\mathcal{I}^{\mathrm{neg}}_{k,L,0}(4z,f)+\mathcal{I}^{\mathrm{neg}}_{k,L,1}(4z,f)$
$\displaystyle=\sum_{0>d\in\mathbb{Z}}\frac{\sqrt{2\pi}|d|^{\frac{k-1}{2}}h_{k}\big{(}2\sqrt{2\pi|d|y}\big{)}2H(-d)k!}{\big{(}2\sqrt{2\pi|d|y}\big{)}^{k}}q^{d},$
$\displaystyle\mathcal{I}^{\mathrm{cc}}_{k,L,0}(4z,f)+\mathcal{I}^{\mathrm{cc}}_{k,L,1}(4z,f)$
$\displaystyle=-2i^{k}k!\sum_{0<a\in\mathbb{N}}a^{k}\frac{I_{k}\big{(}2r\sqrt{2\pi
y}\big{)}-\tilde{\Omega}_{k}\big{(}2a\sqrt{2\pi y}\big{)}}{\big{(}2a\sqrt{2\pi
y}\big{)}^{k}}q^{a^{2}}$ $\displaystyle\quad+\bigg{(}-k!\frac{\log(\sqrt{8\pi
y})+C_{k}}{\big{(}\frac{k}{2}\big{)}!(16\pi
y)^{k/2}}+\frac{k!\pi}{3}\frac{Q_{k-1}(0)}{\sqrt{2\pi}(8\pi
y)^{(k-1)/2}}\bigg{)},$

where only discriminants $d\equiv 0,1(\mathrm{mod\ }4)$ appear. Adding these
together and dividing by $k!$ then gives the expression from Theorem 1.2. Note
that if $d=a^{2}$ for $a\in\mathbb{N}$, then
$\operatorname{Tr}_{d}(L_{z}^{2b}f)$ is $i^{k-2b}$ times the regularized
completed $L$-function of the image of $L_{z}^{2b}f$ under the Hecke operator
$U_{a}$.

To prove the second claim, we can apply Lemmas 3.16 and 3.21 and Equations
(3.10) and (3.43), and recall the value $\frac{(-1)^{k/2}}{2^{k/2}(k/2)!}$ of
$\tilde{\Omega}_{k}(0)$ from Remark 3.13, to see that applying $(-16\pi
L_{z})^{k/2}$ to $(\mathcal{I}_{k,L,0}(4z,f)+\mathcal{I}_{k,L,1}(4z,f))/k!$
gives the holomorphic series

$\sum_{d=0}^{\infty}\operatorname{Tr}_{d}(1)q^{d}+2\sqrt{2\pi}\sum_{0>d\in\mathbb{Z}}H(-d)h_{0}(2\sqrt{2\pi|d|y})\frac{q^{d}}{\sqrt{|d|}}$

from the $\mathcal{I}^{\mathrm{nh}}_{k,L,h}$’s and the
$\mathcal{I}^{\mathrm{neg}}_{k,L,h}$’s, while the
$\mathcal{I}^{\mathrm{nh}}_{k,L,h}$’s are taken to

$-\bigg{(}\log\sqrt{8\pi y}+\frac{\gamma+\log
2}{2}-H_{k}\bigg{)}+\frac{2\pi}{3}\sqrt{y}+\sum_{0<a\in\mathbb{N}}2q^{a^{2}}\Big{(}I_{0}\big{(}2a\sqrt{2\pi
y}\big{)}+H_{k}\Big{)}.$

The Laplacian $\Delta_{\frac{1}{2}}=-\xi_{\frac{3}{2}}\xi_{\frac{1}{2}}$ from
Equation (1.5) now annihilates all the holomorphic terms, while the images of
the logarithmic term and those with $I_{0}$ amount, via Equations (3.27) and
(3.28), to $-\frac{\theta(z)}{4}$. This modular form thus differs from
$2\pi\hat{\mathbf{Z}}_{+}(z)$ by a weight $\frac{1}{2}$ harmonic Maass form
without any singularity and supported on square-indices, which must be a
constant multiple of $\theta(z)$. Recalling that $\operatorname{Tr}_{0}(1)$
from Equation (1.8) equals $\gamma$, our constant term is
$H_{k}+\frac{\gamma-\log(16\pi)}{2}$, and comparing it with the vanishing
constant term of $2\pi\hat{\mathbf{Z}}_{+}(z)$ (see Equation (1.19) of [DIT])
yields the desired result (note that the traces defined in Equation (1.8) of
that reference involve division by $2\pi$ while ours do not, so that this
statement is consistent with comparing the actual formulae). This proves
Theorem 1.2.

For Theorem 1.4 we apply Theorem 4.3 and Corollary 4.4 using the same lattice,
recalling the vanishing of the main terms of $\mathcal{I}^{\mathrm{cc}}_{k,L}$
implied by the vanishing of $c(0,k)$. As Equations (3.10) and (3.41) evaluate
the remaining term from Equation (4.11) to be the asserted one, and the first
term in Equation (4.11) produces the one mentioned in Remark 1.5, this proves
the (extended) theorem. ∎

For Remark 1.3 one combines the proof of Theorem 1.2 with the twist from the
proof of Theorem 1.1.

######  Remark 4.5.

Using [Ze3] and [Ze7], the sum $\sum_{a=0}^{\lfloor
p/2\rfloor}\frac{1}{v^{a}a!}L_{\tau}^{a}\mathcal{I}_{k,L}(\tau,f)$ is a
(vector-valued) quasi-modular form of weight $k+\frac{1}{2}$ and depth
$\big{\lfloor}\frac{p}{2}\big{\rfloor}$, and one checks that the contribution
of $\mathcal{I}^{\mathrm{nh}}_{k,L,h}(\tau,f)$ is just $\sum_{0\leq
m\in\mathbb{Z}+Q(h)}\operatorname{Tr}_{m,h}(f)q_{\tau}^{m}$ for every $h\in
D_{L}$. Moreover, applying this combination to
$\mathcal{I}^{\mathrm{neg}}_{k,L,h}(\tau,f)$,
$\mathcal{I}^{\mathrm{c}}_{k,L,h}(\tau,f)$, and
$\mathcal{I}^{\mathrm{cc}}_{k,L,h}(\tau,f)$ can be shown, using Lemmas 3.16
and 3.21 and Equation (3.32) to replace $h_{l}$ from Equation (4.13), $J_{l}$
in Equation (4.9), and $I_{k}$ appearing in Equation (4.10) by
$\sum_{a}h_{l-2a}{(-2)^{a}a!}$, $\sum_{a}J_{l-2a}{2^{a}a!}$, and
$\sum_{a}I_{k-2a}{2^{a}a!}$ respectively. Moreover, after substituting
Equations (3.17), (3.31), and (3.29), the respective coefficients of $h_{0}$,
$J_{0}$, and $I_{0}$ in these combinations is just the corresponding
denominator $\eta^{l}$ or $\eta^{k}$. It would be interesting to investigate
these functions further. We also note that as long as $2a\leq k$, the term
$L_{\tau}^{a}\mathcal{I}_{k,L}(\tau,f)$ is easily verified to be
$\frac{1}{(-4\pi)^{a}}$ times the Shintani lift
$\mathcal{I}_{k-2a,L}(\tau,L_{z}^{2a}f)$.

###  4.3. Orbital Integrals

In order to prove Theorem 4.3, we need to evaluate Equation (4.15), which we
can do for every $m\in\mathbb{Z}+Q(h)$ separately. Moreover, the integral over
$Y_{T}$ can be replaced by an integral over $\mathcal{F}_{T}$ from Equation
(2.11), and integrals over (nice) regions in $\mathcal{H}$ can be expressed
using the following lemma.

######  Lemma 4.6.

Let $\lambda\in V_{\mathbb{R}}$ and $f\in\widetilde{M}^{!}_{2k}(\Gamma)$ of
some depth $p$ be given, and take a connected domain
$\mathcal{R}\subseteq\mathcal{H}$ with a piecewise smooth positively oriented
boundary $\partial\mathcal{R}$. Then we have

$\int_{\mathcal{R}}f(z)\varphi_{k,-1}\big{(}\sqrt{v}\lambda,z\big{)}d\mu(z)=\sum_{\nu=0}^{p}\oint_{\partial\mathcal{R}}(L_{z}^{\nu}f)(z)\varphi_{k-\nu-1,\nu}\big{(}\sqrt{v}\lambda,z\big{)}dz.$

###### Proof.

We apply Lemma 2.3 repeatedly, where in the $\nu$th step $f$ is replaced by
$L_{z}^{\nu}f$ and
$g(z)=\varphi_{k-\nu,\nu-1}\big{(}\sqrt{v}\lambda,z\big{)}$. Then Proposition
3.18 allows us to take
$G(z)=\varphi_{k-\nu-1,\nu}\big{(}\sqrt{v}\lambda,z\big{)}$, and the sum ends
after $\nu=p$ since $L_{z}^{p+1}f=0$ by assumption. This proves the lemma. ∎

In view of Equation (4.12), we will need the following lemma later.

######  Lemma 4.7.

Let $\lambda\in V$ with $\iota(\lambda)=1$, $v>0$, and
$f\in\widetilde{M}^{!}_{2k}(\Gamma)$ of some depth $p$ be given, and denote
$\eta=2\sqrt{2\pi Q(\lambda)v}$. Then we have the equality

$\sum_{b=0}^{\lfloor
p/2\rfloor}\frac{\operatorname{Sing}_{\lambda}(L_{z}^{2b}f)}{(4\pi
v)^{b}b!}=i^{k}\big{(}2\sqrt{Q(\lambda)}\big{)}^{k-1}\sum_{n\in\mathbb{Z}}\sum_{l=0}^{p}c_{\ell_{\lambda}}(n,l)\mathbf{e}\bigg{(}\frac{nr_{\lambda}}{\alpha_{\ell_{\lambda}}}\bigg{)}\phi_{n}\bigg{(}k-l,T;\frac{2\pi}{\alpha_{\ell_{\lambda}}}\bigg{)}\frac{\operatorname{He}_{l}(\eta)}{\eta^{l}}.$

###### Proof.

As Corollary 3.7 presents the coefficient in front of
$\operatorname{Sing}_{\lambda}(L_{z}^{\mu}f)$ on the left hand side of the
first equality as $\frac{P_{\mu}(0)}{(2\pi v)^{\mu/2}}$, substituting Equation
(4.5), with the summation index $l$ replacing $l-\mu$, into Equation (4.4),
expresses the left hand side of the first equality as

$\displaystyle\sum_{\mu=0}^{p}\frac{P_{\mu}(0)}{(2\pi
v)^{\mu/2}}i^{k-\mu}(2\sqrt{Q(\lambda)})^{k-\mu-1}\sum_{l=\mu}^{p}\sum_{n\in\mathbb{Z}}\frac{l!c_{\ell_{\lambda}}(n,l)}{(l-\mu)!}\mathbf{e}\bigg{(}\frac{nr_{\lambda}}{\alpha_{\ell_{\lambda}}}\bigg{)}\phi_{n}\bigg{(}(k-\mu)-(l-\mu),T;\frac{2\pi}{\alpha_{\ell_{\lambda}}}\bigg{)}$
$\displaystyle=i^{k}(2\sqrt{Q(\lambda)})^{k-1}\sum_{j=0}^{p}\sum_{n\in\mathbb{Z}}c_{\ell_{\lambda}}(n,l)\mathbf{e}\bigg{(}\frac{nr_{\lambda}}{\alpha_{\ell_{\lambda}}}\bigg{)}\phi_{n}\bigg{(}k-l,T;\frac{2\pi}{\alpha_{\ell_{\lambda}}}\bigg{)}\sum_{\mu=0}^{l}\frac{l!P_{\mu}(0)}{(l-\mu)!(2i\sqrt{2\pi
Q(\lambda)v})^{\mu}}.$

Equation (3.11) and Corollary 3.7 now express the sum over $\mu$ as
$\frac{l!P_{l}(i\eta)}{(i\eta)^{l}}=\frac{\operatorname{He}_{l}(\eta)}{\eta^{l}}$,
as desired. This proves the lemma. ∎

Now, the coefficient of $q^{m}$ in Equation (4.15) is evaluated for $m<0$,
$m>0$ with $\iota(m)=0$, $m>0$ with $\iota(m)=1$, and $m=0$ respectively in
the following four propositions.

######  Proposition 4.8.

For every $h\in D_{L}$ and $0>m\in\mathbb{Z}+Q(h)$ we have the equality

$\lim_{T\to\infty}\\!v^{\frac{1-k}{2}}\\!\int_{Y_{T}}\\!\\!f(z)\\!\\!\sum_{\lambda\in
L_{m,h}}\\!\\!\varphi_{k,-1}\big{(}\sqrt{v}\lambda,z\big{)}d\mu(z)\\!=\\!\sum_{l=k}^{p}\\!\frac{4^{k}\sqrt{\pi}|m|^{\frac{k-1}{2}}h_{l}(2\sqrt{2\pi|m|v})\operatorname{Tr}_{m,h}^{(k)}(R_{2k-2l}^{l-k}L_{z}^{l}f)}{\sqrt{2}(4\sqrt{2\pi|m|v})^{l}(l-k)!}.$

###### Proof.

The proof is similar to that of Proposition 3.9 in [BFIL]. Remark 3.15 gives
us the strong decay of $\varphi_{\kappa,\nu}\big{(}\sqrt{v}\lambda,z\big{)}$,
so that we can take the integral over $z\in Y$. Unfolding, cutting out a small
neighborhood $B_{\epsilon}(z_{\lambda})$ for each $\lambda\in\Gamma\backslash
L_{m,h}$, and applying Lemma 4.6 yields

$\int_{Y}f(z)\sum_{\lambda\in
L_{m,h}}\varphi_{k,-1}\big{(}\sqrt{v}\lambda,z\big{)}d\mu(z)=\lim_{\epsilon\to
0}\sum_{\lambda\in\Gamma\backslash
L_{m,h}}\\!\frac{-1}{|\Gamma_{\lambda}|}\sum_{l=0}^{p}\oint_{\partial
B_{\epsilon}(z_{\lambda})}(L_{z}^{l}f)(z)\varphi_{k-1-l,l}\big{(}\sqrt{v}\lambda,z\big{)}dz.$

Substituting Equations (3.18) and (2.3) (with $\zeta=\sigma\sqrt{|m|}$ for
$\sigma:=-\operatorname{sgn}\big{(}\lambda,Z^{\perp}(z_{\lambda})\big{)}$),
and multiplying by $v^{\frac{1-k}{2}}$, shows that the desired left hand side
is the sum over $\lambda$ and $l$ of

$\sum_{\lambda\in\Gamma\backslash
L_{m,h}}\sum_{l=0}^{p}\frac{-\big{(}4y_{\lambda}\sigma\sqrt{|m|}\big{)}^{k-1-l}}{(2\pi)^{(l+1)/2}v^{l/2}|\Gamma_{\lambda}|}\lim_{\epsilon\to
0}\oint_{\partial
B_{\epsilon}(z_{\lambda})}\frac{(L_{z}^{l}f)(z)A_{z_{\lambda}}(z)^{k-1-l}dz}{\big{(}1-A_{z_{\lambda}}(z)\big{)}^{2k-2l-2}}h_{l}\big{(}2\sigma\sqrt{2\pi
mv}\tfrac{1+\epsilon^{2}}{1-\epsilon^{2}}\big{)}.$

Evaluating this limit via Corollary 2.2, using the parity from Proposition
3.14, and applying Equations (4.1) and (4.7) yields the desired right hand
side. This proves the proposition. ∎

######  Proposition 4.9.

If $h\in D_{L}$ and $0<m\in\mathbb{Z}+Q(h)$ with $\iota(m)=0$ then we have

$v^{\frac{1-k}{2}}\lim_{T\to\infty}\int_{Y_{T}}f(z)\sum_{\lambda\in
L_{m,h}}\varphi_{k,-1}\big{(}\sqrt{v}\lambda,z\big{)}d\mu(z)=\sum_{b=0}^{\lfloor
p/2\rfloor}\frac{\operatorname{Tr}_{m,h}(L_{z}^{2b}f)}{(4\pi v)^{b}b!}.$

###### Proof.

An element $\lambda\in L_{m,h}$ is not perpendicular to $\ell$ for any
$\ell\in\operatorname{Iso}(V)$, and its stabilizer $\Gamma_{\lambda}$ is
infinite cyclic by Lemma 2.5. Therefore the functions
$\varphi_{\kappa,\nu}\big{(}\sqrt{v}\lambda,z\big{)}$ again decay strongly,
via Remark 3.15, towards any cusp $\ell$. We combine this with the usual
unfolding argument to express our left hand side as

$v^{\frac{1-k}{2}}\lim_{T\to\infty}\sum_{\lambda\in\Gamma\backslash
L_{m,h}}\int_{\mathcal{F}_{\lambda,T}}f(z)\varphi_{k,-1}\big{(}\sqrt{v}\lambda,z\big{)}d\mu(z)=v^{\frac{1-k}{2}}\sum_{\lambda\in\Gamma\backslash
L_{m,h}}\int_{\mathcal{F}_{\lambda}}f(z)\varphi_{k,-1}\big{(}\sqrt{v}\lambda,z\big{)}d\mu(z),$

where $\mathcal{F}_{\lambda}$ is a fundamental domain for the action of
$\Gamma_{\lambda}$ on $\mathcal{H}$ and
$\mathcal{F}_{\lambda,T}:=\mathcal{F}_{\lambda}\cap\mathcal{H}_{T}$ (this is
well-defined modulo $\Gamma$ by Equation (3.20) and the modularity of $f$). We
now remove an $\epsilon$-neighborhood of the geodesic $c_{\lambda}$ from
$\mathcal{F}_{\lambda}$ for applying Lemma 4.6, substitute the value of each
$\varphi_{k-\nu-1,\nu}$ from Equation (3.18), and gather powers of $v$ to
write our expression as

$-\sum_{\lambda\in\Gamma\backslash
L_{m,h}}\sum_{\nu=0}^{p}\frac{\lim_{\epsilon\to
0^{+}}\big{(}h_{\nu}(\epsilon)-h_{\nu}(-\epsilon)\big{)}}{(2\pi)^{(\nu+1)/2}v^{\nu/2}}\int_{c_{\lambda}\cap\mathcal{F}_{\lambda}}(L_{z}^{\nu}f)(z)(\lambda,Z(z))^{k-\nu-1}dz.$

But the integral is $\operatorname{Tr}_{\lambda}(L_{z}^{\nu}f)$ from Equation
(4.2), summing over $\lambda$ replaces it by
$\operatorname{Tr}_{m,h}(L_{z}^{\nu}f)$ from Equation (4.7), and we have
$\lim_{\epsilon\to
0^{+}}\big{(}h_{\nu}(\epsilon)-h_{\nu}(-\epsilon)\big{)}=-\sqrt{2\pi}P_{\nu}(0)$
by Proposition 3.14. The desired formula now follows from Corollary 3.7. This
proves the proposition. ∎

When $\iota(m)=1$ the coefficient $a(\Theta_{k,\ell},m,h,v)$ from (2.19), as
well as the traces from Equations (4.9) and (4.10), may be non-zero. Recalling
the functions $\phi_{n}$ from Equation (4.4), the limit of the corresponding
coefficient from Equation (4.15) is evaluated as follows.

######  Proposition 4.10.

Let $h\in D_{L}$ and $0<m\in\mathbb{Z}+Q(h)$ with $\iota(m)=1$ be given. Then
for large $T>1$ we have

$\begin{split}v^{\frac{1-k}{2}}&\int_{Y_{T}}f(z)\sum_{\lambda\in