2.1 Differential operators on modular forms

For $M=\big (\begin {smallmatrix}a & b \\ c & d\end {smallmatrix}\big )\in \operatorname {SL}_{2}(\mathbb {R})$ and an element z of $\mathcal {H}:=\{z=x+iy\in \mathbb {C}|y>0\}$ , denote $j(M,z):=cz+d$ . 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 $\operatorname {Mp}_{2}(\mathbb {R})$ . We write elements of $\operatorname {Mp}_{2}(\mathbb {R})$ as pairs $(M,\phi )$ , with $M\in \operatorname {SL}_{2}(\mathbb {R})$ and $\phi $ a holomorphic function on $\mathcal {H}$ such that $\phi (z)^{2}=j(M,z)$ .

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

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

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.

For a half-integer $\kappa $ , we define

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

which are the raising operator of weight $\kappa $ , the weight lowering operator, the $\xi $ -operator of weight $\kappa $ from [Reference Bruinier and FunkeBF1], and the Laplacian operator of weight $\kappa $ , respectively. For $n\in \mathbb {N}$ , we write

(2.2) $$ \begin{align} R_{\kappa}^{n}:=R_{\kappa+2n-2} \circ R_{\kappa+2n-4} \circ \dots \circ R_{\kappa+2} \circ R_{\kappa} \end{align} $$

for the iterated raising operator.

These differential operators preserve modularity, in the sense that

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

whereas for $\xi _{\kappa }$ , which involves complex conjugation, we have

(2.3) $$ \begin{equation} \xi_{\kappa}\mathcal{A}_{\kappa}^{!}(\Gamma,\rho)\subseteq\mathcal{A}_{2-\kappa}^{!}(\Gamma,\overline{\rho}),\mathrm{\ where\ }\overline{\rho}\mathrm{\ is\ the\ complex\ conjugate\ representation}. \end{equation} $$

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 $\Delta _{\kappa }$ increasing it by at most 1. For more on these modular forms, including their relations with quasi-modular forms and Shimura’s vector-valued modular forms, see [Reference Martin and RoyerMR, Reference ZemelZe3, Reference ZemelZe7].

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

(2.4) $$ \begin{align} \zeta=A_{w}(z):=\tfrac{z-w}{z-\overline{w}}\in\big\{\zeta\in\mathbb{C}\big||\zeta|<1\big\},\qquad\mathrm{with}\qquad1-A_{w}(z)=\tfrac{2it}{z-\overline{w}} \end{align} $$

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

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

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 [Reference Bruinier, van der Geer, Harder and ZagierBGHZ]Footnote ² as

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

Note that the proof of equation (2.5) makes no use of the modularity of f, so that this expansion is valid for every holomorphic function f. We shall need a formula extending equation (2.5) 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

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

Proof We write $f(z)$ as in equation (1.4), and express each $f_{l}$ via equation (2.5), but with $\kappa $ replaced by $\kappa -l$ . Recalling from Lemma 5.1 of [Reference ZemelZe4] that y equals $\frac {t(1-|A_{w}(z)|^{2})}{|1-A_{w}(z)|^{2}}$ , we get

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

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

$$\begin{align*}\lim_{\epsilon\to0}\int_{\partial B_{\epsilon}(w)}\frac{f(z)}{(1-A_{w}(z))^{\kappa-2}}A_{w}(z)^{\mu}dz=\begin{cases} -\frac{4\pi t^{|\mu|}}{(|\mu|-1)!}R_{\kappa}^{|\mu|-1}f(w), & \mu<0, \\ 0, & \mu\geq0. \end{cases} \end{align*}$$

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 [Reference LiL]).

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

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

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 symmetric space of G is the space of oriented negative definite lines in $V_{\mathbb {R}}:=M_{2}(\mathbb {R})^{0}$ . We can identify $\mathcal {H}$ with the connected component of this symmetric space that contains the line spanned by $\big (\begin {smallmatrix}0 & -1 \\ 1 & 0\end {smallmatrix}\big )$ as a positive generator, via the map taking $z\in \mathcal {H}$ to $\mathbb {R}Z^{\perp }(z)$ with

$$\begin{align*}Z^{\perp}(z):=\frac{1}{\sqrt{N}y}\bigg(\begin{array}{cc}x & -|z|^{2} \\ 1 & -x\end{array}\bigg),\quad\mathrm{and\ we\ set}\quad Z(z)&:=\frac{1}{\sqrt{N}}\bigg(\begin{array}{cc}z & -z^{2} \\ 1 & -z\end{array}\bigg)\\[3pt] & =\frac{1}{\sqrt{N}}\binom{z}{1}(1\ \ -z).\end{align*}$$

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 [Reference ZemelZe4] proves the equalities

(2.6) $$ \begin{align} \begin{aligned} \big(\lambda,Z^{\perp}(z)\big)&=-2\xi\cosh d(z,z_{\lambda})=-2\xi\bigg(\frac{|z-{z_{\lambda}}|^{2}}{2yy_{\lambda}}+1\bigg)=-2\xi \frac{1+|A_{{z_{\lambda}}}(z)|^{2}}{1-|A_{{z_{\lambda}}}(z)|^{2}}, \\[6pt] \big(\lambda,Z(z)\big)&=-2\xi\frac{(z-{z_{\lambda}})(z-\overline{{z_{\lambda}}})}{2y_{\lambda}}=\frac{4\xi y_{\lambda}A_{{z_{\lambda}}}(z)}{\big(1-A_{{z_{\lambda}}}(z)\big)^{2}}, \end{aligned} \end{align} $$

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 kernelFootnote ³ 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

(2.7) $$ \begin{align} L_{m,h}:=\{\lambda \in L+h|Q(\lambda)=m\}. \end{align} $$

Typical examples can be found in [Reference BorcherdsBO], [Reference Alfes-Neumann and SchwagenscheidtAE], [Reference Li and ZemelLZ], or [Reference ZemelZe6]; e.g.,

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

Furthermore, 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 [Reference BorcherdsBo, Reference ScheithauerSch, Reference StrömbergStr, Reference ZemelZe1] 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

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

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

(2.10) $$ \begin{align} L\cap\ell=\mathbb{Z}\beta_{\ell}u_{\ell},\quad\text{and set}\quad\varepsilon_{\ell}:=\tfrac{\alpha_{\ell}}{\beta_{\ell}}. \end{align} $$

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

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

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

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

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

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

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

is $\Gamma $ -invariant, and $Y_{T}:={\Gamma }\backslash \mathcal {H}_{T}$ is a truncated modular curve, with a fundamental domainFootnote ⁴

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

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

$$\begin{align*}\mathcal{F}_{T}:=\big\{z=x+iy\in\mathcal{H}\big||x|\leq\tfrac{1}{2},\ |z|\geq1,\ y \leq T\}. \end{align*}$$

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

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

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.

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,

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

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

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

The canonical image $r_{\lambda }+\alpha _{\ell _{\lambda }}\mathbb {Z}\in \mathbb {Q}/\alpha _{\ell _{\lambda }}\mathbb {Z}$ from equation (2.17), 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

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

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.

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 (2.17). Then we have

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

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

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

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

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

Theorem 4.1 of [Reference BorcherdsBo] implies that for fixed $z\in \mathcal {H}$ , we have $\Theta _{k,L}(\tau ,z)\in \mathcal {A}_{k+\frac {1}{2}}\big (\operatorname {Mp}_{2}(\mathbb {Z}),\rho _{L}\big )$ , whereas for fixed $\tau \in \mathcal {H}$ , it is easy to verify that $\Theta _{k,L,h}(\tau ,z)\in \mathcal {A}_{-2k}(\Gamma )$ for every $h \in D_{L}$ . After collecting terms, we can use equation (2.7) to rewrite

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

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

(2.22) $$ \begin{align} \operatorname{He}_{n}(\xi):=(-1)^{n}e^{\xi^{2}/2}\big(\tfrac{d}{d\xi}\big)^{n}e^{-\xi^{2}/2}=\big(\xi-\tfrac{d}{d\xi}\big)^{n}\cdot1=\sum_{b=0}^{\lfloor n/2 \rfloor}\frac{(-1)^{b}n!}{b!(n-2b)!2^{b}}\xi^{n-2b}. \end{align} $$

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

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

The function $\Theta _{k,\ell }$ from equation (2.23) appears in the asymptotic expansion of the theta kernel $\Theta _{k,L}$ from (2.20), as is given in the following result. It is essentially part (2) of Proposition 4.2 of [Reference Alfes and EhlenANS], which refers to Theorem 5.2 of [Reference BorcherdsBo] for the proof, and can also be proved using properties of appropriate variants of the lattice sums from equation (3.34) below.

Lemma 2.9 Given $\ell \in \operatorname {Iso}(V)$ , there exists a constant $C_{\ell }>0$ such that

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

For $f\in \mathcal {A}_{2k}^{!}(\Gamma )$ , we follow [Reference Alfes and EhlenANS, Reference Bruinier, Funke and ImamoğluBFI] (among others) to define its regularized Shintani lift, using the fundamental domain from equation (2.14), to be the theta integral

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

which is an element of $\mathcal {A}_{k+\frac {1}{2}}\big (\operatorname {Mp}_{2}(\mathbb {Z}),\rho _{L}\big )$ . 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 [Reference Bruinier and FunkeBF2]).

3.1 Familiar functions

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

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

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

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

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

(3.2) $$ \begin{align} \Gamma(\mu,t)= \begin{cases} e^{-t}(\mu-1)!\sum_{a=0}^{\mu-1}\frac{t^{a}}{a!}=e^{-t}t^{\mu}\big(1-\tfrac{d}{dt}\big)^{\mu-1}\tfrac{1}{t}, & \text{ when }0<\mu\in\mathbb{N}\text{ and }t\in\mathbb{R}, \\ \frac{(-1)^{\mu}}{|\mu|!}\bigg(\Gamma(0,t)+e^{-t}\sum_{a=0}^{|\mu|-1}\frac{a!}{(-t)^{a+1}}\bigg), & \text{ when }-\mu\in\mathbb{N},\text{ and }t\neq0, \end{cases} \end{align} $$

$\Gamma (0,t)$ 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) $$ \begin{align} \tfrac{d}{dt}\Gamma(\mu,t)=-e^{-t}t^{\mu-1} \end{align} $$

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

Modifying the antiderivative from equation (3.1), we now define

(3.4) $$ \begin{align} \mathrm{e}(\xi):=-\tfrac{\operatorname{sgn}(\xi)}{\sqrt{2}}\Gamma\big(\tfrac{1}{2},\tfrac{\xi^{2}}{2}\big)=-\operatorname{sgn}(\xi)\int_{|\xi|}^{\infty}e^{-w^{2}/2}dw\quad\mathrm{for}\quad\xi\neq0. \end{align} $$

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

$$\begin{align*}\lim_{\xi\to0^{+}}\mathrm{e}(\xi)-\lim_{\xi\to0^{-}}\mathrm{e}(\xi)=-\sqrt{2}\Gamma\big(\tfrac{1}{2},0\big)=-\sqrt{2\pi}.\end{align*}$$

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

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

where $\delta (\xi )$ is the Dirac delta distribution. Our next goal is to find higher-order antiderivatives 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. They will show up in equation (3.17) and Proposition 3.14, whence their importance. Set

(3.6) $$ \begin{align} \begin{aligned} P_{0}(\xi) &=1\quad\text{and}\quad Q_{0}(\xi)=0, \qquad\text{as well as} \\ P_{\nu}'(\xi) &=P_{\nu-1}(\xi)\quad\text{and}\quad P_{\nu}(\xi)+Q_{\nu}'(\xi)-\xi Q_{\nu}(\xi)=Q_{\nu-1}(\xi)\text{ for }\nu\geq1. \end{aligned} \end{align} $$

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

Proof Let $\mathrm {g}(\xi ):=e^{-\xi ^{2}/2}$ denote the usual Gaussian. Then, for each $k\geq 1$ , the polynomial

$$\begin{align*}p_{k}(\xi):=\mathrm{g}(\xi)^{-1}\frac{d}{d\xi}\big(\mathrm{g}(\xi)\xi^{k-1}\big)=(k-1)\xi^{k-2}-\xi^{k}\in\mathbb{Q}[\xi]\end{align*}$$

has degree k, parity $(-1)^{k}$ , and leading coefficient $-1$ . It is thus clear that $\{1\}\cup \{p_{k}\}_{k\geq 1}$ is a basis of $\mathbb {Q}[\xi ]$ over $\mathbb {Q}$ , that respects the parity decomposition. Therefore, there exist unique $\tilde {P}\in \mathbb {Q}[\xi ]$ , $c\in \mathbb {Q}$ , and $Q\in \mathbb {Q}[\xi ]$ such that

$$\begin{align*}\tilde{P}'=p,\quad\tilde{P}(0)=0,\quad\mathrm{and}\quad\mathrm{g}(\xi)\big(q(\xi)-\tilde{P}(\xi)+c)&=\frac{d}{d\xi}\big(\mathrm{g}(\xi)Q(\xi)\big)\\ & =\mathrm{g}(\xi)\big(Q'(\xi)-\xi Q(\xi)\big).\end{align*}$$

Setting $P:=\tilde {P}-c$ proves the first assertion. For even q and odd p, $\tilde {P}$ and P are also even, and Q is thus odd. If q is odd and p is even, then $\tilde {P}$ is also odd, and then integrating from $-\infty $ to $\infty $ shows that $c=0$ in this case, completing the proof of the parity assertion. The assertion about the degrees and the leading coefficients when $\deg q\leq \deg p$ are now easily checked from this construction. This proves the lemma.

For convenience, we list the polynomials $P_{\nu }$ and $Q_{\nu }$ for $0\leq \nu \leq 4$ :

$$ \begin{align*} P_{0}(\xi)&=1,~ P_{1}(\xi)=\xi,~ P_{2}(\xi)=\frac{\xi^{2}+1}{2},~ P_{3}(\xi)=\frac{\xi^{3}+3\xi}{6},~ P_{4}(\xi)=\frac{\xi^{4}+6\xi^{2}+3}{24},\\ Q_{0}(\xi)&=0,~ Q_{1}(\xi)=1,~ Q_{2}(\xi)=\frac{\xi}{2},~ Q_{3}(\xi)=\frac{\xi^{2}+2}{6},~ Q_{4}(\xi)=\frac{\xi^{3}+5\xi}{24}. \end{align*} $$

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

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

considered, at the moment, as formal power series in $\xi $ and t, on which derivatives operate in the usual, formal way. They can be characterized in the following way.

Proposition 3.3 allows us to determine the series $\Psi $ and $\Upsilon $ explicitly as the expansions of real-analytic functions, which also shows that their Taylor expansion in (3.7) converges absolutely for all t and $\xi $ .

Regarding Property (i) in that proposition, see Remark 3.5.

Theorem 3.4 The power series $\Psi $ and $\Upsilon $ from equation (3.7) describe the functions

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

These functions also satisfy the differential equations

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

Proof It suffices to show that the asserted series have the properties from Proposition 3.3. Property $(ii)$ is immediate for both (by substituting $t=0$ ), Property $(i)$ for $\Psi $ is easy to check by expanding it in t, and the two differential equations (that from Property $(iii)$ and the one asserted here) are easily established by simple differentiation. Differentiation also shows that

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

from which Property $(iii)$ for $\Upsilon $ quickly follows.

Next, note that Property $(i)$ for $\Upsilon $ is equivalent to $\partial _{t}^{\nu }\Upsilon (\xi ,t)\big |_{t=0}$ being a polynomial in $\xi $ for every $\nu \in \mathbb {N}$ . 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 $\nu $ , the derivative $\partial _{t}^{\nu }\Upsilon (\xi ,t)$ 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 If one considers Property $(iii)$ from Proposition 3.3 as differential equations for functions, then solving them implies that there exist functions $\psi (t)$ and $\phi (t)$ such that as functions, $\Psi (\xi ,t)$ and $\Upsilon (\xi ,t)$ are

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

respectively. The initial conditions from Property $(ii)$ amount to the equalities $\psi (0)=1$ and $\phi (0)=0$ . Showing that the polynomial property of the Taylor expansions from Property $(i)$ in Proposition 3.3 is equivalent to $\psi $ and $\phi $ being 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

$$\begin{align*}\nu P_{\nu}(\xi)-\xi P_{\nu-1}(\xi)=P_{\nu-2}(\xi)\quad\mathrm{and}\quad\nu Q_{\nu}(\xi)-\xi Q_{\nu-1}(\xi)=Q_{\nu-2}(\xi)\quad\mathrm{for\ all}\quad\nu\geq2.\end{align*}$$

For the explicit expressions for the polynomials from equation (3.6), recall the well-known formula for the generating function of the (probabilists’) Hermite polynomials, stating that

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

It can be proved, for example, by comparing equation (2.22) with the Taylor series of $e^{-(\xi -t)^{2}/2}$ at $t=0$ . Theorem 3.4 then implies the following result.

Corollary 3.7 For every $\nu \in \mathbb {N}$ , we have

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

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 $Q_{\nu +1}(\xi )=\frac {Q_{\nu }'(\xi )+P_{\nu }(\xi )}{\nu +1}$ , and using equation (3.6), we get

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

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 $ , whereas if $\nu $ is odd, then

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

Remark 3.8 The recursion (3.6) extends naturally to $\nu \in \mathbb {Z}$ , in which case $P_{\nu }(\xi )=0$ and

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

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 in the sense that $\operatorname {He}_{\nu }'=\nu \operatorname {He}_{\nu -1}$ (see, e.g., the exponential example in Section 5 of [Reference ZemelZe5] and some of the references therein), and Corollary 3.7 shows that the same applies also to the polynomials $\nu !P_{\nu }$ . This means explicitly that the equalities

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

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., [Reference ZemelZe5], 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) $$ \begin{align} \sum_{\nu=0}^{l}\frac{(-1)^{\nu}}{(l-\nu)!}(\xi+\zeta)^{l-\nu}P_{\nu}(\xi)=P_{l}(\zeta)\in\mathbb{Q}[\xi,\zeta]. \end{align} $$

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

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

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

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

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

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

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 produces

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

(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

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

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

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

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

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 nth harmonic number, then we have

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

Proof Theorem 3.4 and Lemma 3.9 evaluate the generating series

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

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

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

To see this, we differentiate the left-hand side with respect to w and apply the usual trick of completion to the square in order to obtain $\frac {1}{\sqrt {2\pi }}e^{(\zeta -h)w+w^{2}/2}$ times $\int _{-\infty }^{\infty }\mathrm {g}(\xi +h-w)d\xi $ . As the latter integral equals $\sqrt {2\pi }$ , we deduce that both sides have the same derivative with respect to w, and since they both vanish for $w=0$ , they are equal for every w. Substituting equation (3.14) into our generating series yields

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

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

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

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

and deduce the following properties.

Lemma 3.12 The generating series of the polynomials from equation (3.15) is

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

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

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

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

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

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

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

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

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

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

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

3.4 Singular Schwartz functions

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

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

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

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

Remark 3.15 The decay of $\mathrm {g}$ and $\mathrm {e}$ from equations (3.1) and (3.4) imply that

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

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

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

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

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

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

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 reason for introducing them will be clear in Proposition 3.18.

The parity from Proposition 3.14 implies that

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

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

Note that for all $k\in \mathbb {N}$ , the functions from equations (2.19) and (3.18), and the expansions from equations (2.20) and (2.21), are related by

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

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

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

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

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 $\big (\lambda ,Z(z)\big )\neq 0$ .

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

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

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

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

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

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. The results for $\eta <0$ can be obtained from those with $\eta>0$ because we have

$$\begin{align*}\mathbf{g}_{\kappa,l}(\xi;-\eta)=(-1)^{\kappa+l}\mathbf{g}_{\kappa,l}(-\xi;\eta)\hspace{6pt} \mathrm{and\ hence}\hspace{6pt} \widehat{\mathbf{g}_{\kappa,l}}(t;-\eta)=(-1)^{\kappa+l}\widehat{\mathbf{g}_{\kappa,l}}(-t;\eta). \end{align*}$$

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

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

(3.27) $$ \begin{align} \partial_{\eta}I_{\nu,j}(\eta,t)=I_{\nu-1,j}(\eta,t)+\frac{\operatorname{He}_{\nu}(0)}{\nu!j!}\int^{\infty}_{-t}(w+t)^{j}e^{-\eta w}dw=\frac{\operatorname{He}_{\nu}(0)}{\nu!\eta^{j+1}}e^{\eta t}-I_{\nu-1,j}(\eta,t) \end{align} $$

for all $\nu $ and j (note the difference in the summation over $\mu $ in $I_{\nu ,j}$ and in $I_{\nu -1,j}$ ) and thus in particular $I_{\nu }'(\eta )=\frac {\operatorname {He}_{\nu }(0)}{\nu !\eta }-I_{\nu -1}(\eta )$ , while equation (3.1) and simple differentiation give

(3.28) $$ \begin{align} I_{-1}(\eta)=-e^{\eta^{2}/2}\mathrm{e}(\eta)\qquad\mathrm{and}\qquad\partial_{t}I_{\nu,j}(\eta,t)=\begin{cases} I_{\nu,j-1}(\eta,t), & \text{ if }j\geq1, \\ e^{\eta t}\mathrm{g}(t),& \text{ if }j=0\text{ and }\nu=-1. \end{cases} \end{align} $$

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

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

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

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

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

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

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

We can now deduce the following useful property.

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

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

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, using the polynomials from equation (3.30), that

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

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

$$\begin{align*}\widehat{\mathbf{g}_{\kappa,l}}(0;\eta)=\begin{cases} -\frac{\sqrt{2\pi}i^{\kappa+l}}{\kappa \cdot l!}\bigg(\operatorname{He}_{\kappa+l}(\eta)-\eta^{\kappa}\operatorname{He}_{l}(\eta)\bigg),& \text{ if }\kappa\neq0\text{ and }\kappa+l\geq0, \\ \sqrt{2\pi}\cdot(-i)^{l}\big(I_{l}(\eta)-\tilde{\Omega}_{l}(\eta)\big),& \text{ if }\kappa=0, \end{cases}\end{align*}$$

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

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

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

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

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

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

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

(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) $$ \begin{align} J_{\nu}'=-J_{\nu-1},\quad J_{\nu}(-\eta)=(-1)^{\nu-1}J_{\nu}(\eta),\quad\mathrm{and}\quad\eta^{3}\frac{d}{d\eta}\frac{J_{\nu}(\eta)}{\eta^{\nu}}=\frac{J_{\nu-2}(\eta)}{\eta^{\nu-2}}. \end{align} $$

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\begin{align*}\int_{-t}^{t}\!\!e^{-\eta w}\frac{\mathrm{g}(w)}{w}dw&=\int_{0}^{\infty}\!\frac{e^{-\eta w}\!-\!e^{\eta w}}{w}\mathrm{g}(w)dw-\int_{t}^{\infty}\!\frac{e^{-\eta w}-e^{\eta w}}{w}\mathrm{g}(w)dw\\[6pt] & \quad =\sqrt{2\pi}J_{0}(\eta)+o_{\eta,\epsilon}(e^{-(1-\epsilon)t^{2}/2}). \end{align*}$$

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

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

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 applying equation (3.2), the second term becomes $-(-1)^{j}j!\Gamma (-j,-\eta t)+\Gamma (0,-\eta t)$ , 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

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

(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

$$\begin{align*}I_{-1,j}(0,0)=Q_{j}(0)+\sqrt{\pi/2}P_{j}(0)=\lim_{\xi\to0^{-}}h_{j}(\xi)=(-1)^{j-1}\lim_{\xi\to0^{+}}h_{j}(\xi)\quad\mathrm{for}\quad j\in\mathbb{N}. \end{align*}$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(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

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

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

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) $$ \begin{align} \mathbf{G}_{\kappa,l}(\omega;\upsilon):=\mathbf{G}_{\kappa,l}(\omega;\upsilon,0)=\lim_{\eta\to0^{+}}\mathbf{G}_{\kappa,l}(\omega;\upsilon,\eta), \end{align} $$

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 as

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

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 $\mathbb {R}$ (and $\mathbb {B}_{0}$ is the constant function 1), and we have

(3.36) $$ \begin{align} \mathbb{B}_{1}(0)=0=\lim_{\omega\to0^{+}}\mathbb{B}_{1}(\omega)+\tfrac{1}{2}=\lim_{\omega\to0-}\mathbb{B}_{1}(\omega)-\tfrac{1}{2}\quad\mathrm{and}\quad\mathbb{B}_{\mu}(\omega)=-\sum_{0 \neq m\in\mathbb{Z}}\frac{\mu!\mathbf{e}(m\omega)}{(2\pi im)^{\mu}}, \end{align} $$

the latter Fourier expansion being valid for every $\omega \in \mathbb {R}/\mathbb {Z}$ and $0<\mu \in \mathbb {N}$ (this is essentially equations (13)–(15) in Section 1.13 of [Reference Erdélyi, Magnus, Oberhettinger and TricomiEMOT]).

We also recall from Section 1.11 of [Reference Erdélyi, Magnus, Oberhettinger and TricomiEMOT] the function

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

Since we shall use this function only 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 [Reference Erdélyi, Magnus, Oberhettinger and TricomiEMOT] (for $j\geq 1$ ) to give, for all $j\in \mathbb {N}$ , the expansion

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

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

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

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) $$ \begin{align} \Phi_{\kappa}(\omega):=\begin{cases} -\mathbb{B}_{\kappa}(\omega)/\kappa, & \kappa\geq1, \\ -\big[\tilde{\psi}^{(|\kappa|)}(-\omega)+(-1)^{\kappa}\tilde{\psi}^{(|\kappa|)}(\omega)\big]\big/2|\kappa|!, & \kappa\leq0, \end{cases} \end{align} $$

as well as

(3.41) $$ \begin{align} \Xi_{\kappa}(\omega):=\frac{(-2\pi i)^{1-\kappa}}{\sqrt{2\pi}}\begin{cases} F(\mathbf{e}(\omega),\kappa)/|\kappa|!+\delta_{\kappa,0}/2, & \kappa\leq0\text{ and }\omega\not\in\mathbb{Z}, \\ -\mathbb{B}_{1-\kappa}(0)/(1-\kappa)!, & \kappa\leq1\text{ and }\omega\in\mathbb{Z}, \\ 0, & \text{otherwise.} \end{cases} \end{align} $$

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

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

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

(3.43) $$ \begin{align} C_{l}:=\frac{\gamma+\log2-2H_{l}+H_{\lfloor l/2 \rfloor}}{2}=\frac{\gamma+\log2}{2}-\sum_{a=1,\ 2 \nmid a}^{l}\frac{1}{a}. \end{align} $$

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 3.27 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

$$\begin{align*}&-\sqrt{2\pi}\upsilon^{\kappa-1}\big[P_{l}(0)\Phi_{\kappa}(\omega)+Q_{l}(0)\Xi_{\kappa}(\omega)\big]\\ & \quad +\begin{cases} -\frac{\sqrt{2\pi} \cdot i^{\kappa+l}\operatorname{He}_{\kappa+l}(0)}{\upsilon\kappa \cdot l!}+o_{\kappa,l,\epsilon}(e^{-2\pi^{2}(1-\epsilon)/\upsilon^{2}}), & \kappa\geq1, \\ \frac{\sqrt{2\pi}}{\upsilon}\delta_{\kappa,0}P_{l}(0)(\log\upsilon+C_{l})+O_{\kappa,l,\omega}(\upsilon^{\kappa}), & \kappa\leq0, \end{cases} \end{align*}$$

with the O-notation concerning the behavior as $\upsilon \to 0^{+}$ .

Proof Consider first the case where $\kappa \geq 1$ . For $\omega \neq 0$ , we use the Poisson Summation Formula (justified by Remark 3.26), and applying Propositions 3.22 and 3.25 gives us

$$ \begin{align*} &\mathbf{G}_{\kappa,l}(\omega;\upsilon)\\ & \quad =\frac{1}{\upsilon}\bigg(\widehat{\mathbf{g}_{\kappa,l}}(0;0)+\lim_{\eta\to0}\sum_{0 \neq m\in\mathbb{Z}}\mathbf{e}(m\omega)\widehat{\mathbf{g}_{\kappa,l}}\big(\tfrac{m}{\upsilon};\eta\big)\bigg) \\ & \quad =-\frac{\sqrt{2\pi}i^{\kappa+l}}{\upsilon\kappa}\bigg(\frac{\operatorname{He}_{\kappa+l}(0)}{l!}+\frac{\kappa\operatorname{He}_{l}(0)}{l!}\lim_{\eta\to0}\sum_{0 \neq m\in\mathbb{Z}}\frac{\mathbf{e}(m\omega)\upsilon^{\kappa}}{(-2\pi m)^{\kappa}}e^{-2\pi\eta m/\upsilon}\Gamma\big(\kappa,-\tfrac{2\pi\eta m}{\upsilon}\big)\bigg) \\ & \quad =\frac{\sqrt{2\pi}}{\upsilon\kappa}\bigg(\frac{-i^{\kappa+l}\operatorname{He}_{\kappa+l}(0)}{l!}-\kappa!P_{l}(0)\sum_{0 \neq m\in\mathbb{Z}}\frac{\mathbf{e}(m\omega)\upsilon^{\kappa}}{(2\pi im)^{\kappa}}\bigg) \\ & \quad =\frac{\sqrt{2\pi}}{\upsilon\kappa}\bigg(\frac{-i^{\kappa+l}\operatorname{He}_{\kappa+l}(0)}{l!}+P_{l}(0)\upsilon^{\kappa}\mathbb{B}_{\kappa}(\omega)\bigg), \end{align*} $$

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) $$ \begin{align} \mathbf{G}_{\kappa,l}(0;\upsilon)=\lim_{\omega\to0^{+}}\big(\mathbf{G}_{\kappa,l}(\omega+\mathbb{Z};\upsilon)-\mathbf{g}_{\kappa,l}(\upsilon\omega;0)\big). \end{align} $$

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

$$\begin{align*}-\lim_{\xi\to0^{+}}\mathbf{g}_{1,l}(\xi;0)=\lim_{\xi\to0^{+}}(-1)^{l+1}h_{l}(\xi)=I_{-1,l}(0,0)=P_{l}(0)\sqrt{\pi/2}+Q_{l}(0) \end{align*}$$

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

$$\begin{align*}\mathbf{g}_{\kappa,1}(\xi;0)=\xi^{\kappa}h_{0}(\xi)-\xi^{\kappa-1}h_{1}(\xi)=-\xi^{\kappa-1}\mathrm{g}(\xi);\quad\!\!\mathrm{hence}\!\!\quad \mathbf{G}_{\kappa,1}(\omega;\upsilon)=-\lim_{\eta\to0^{+}}\sum_{0 \neq \xi\in\mathbb{Z}+\omega}\mathrm{g_{\kappa}}(\upsilon\xi;\eta). \end{align*}$$

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

(3.45) $$ \begin{align} &-\frac{\sqrt{2\pi}i^{\kappa-1}}{\upsilon}\bigg(\lim_{\eta\to0^{+}}I_{-1,-\kappa}(\eta,0)+\lim_{\eta\to0^{+}}\sum_{0 \neq m\in\mathbb{Z}}e^{-2\pi\eta m/\upsilon}I_{-1,-\kappa}\big(\eta,\tfrac{2\pi m}{\upsilon}\big)\mathbf{e}(m\omega)\bigg)\nonumber \\ &=-\frac{\sqrt{2\pi}i^{\kappa-1}}{\upsilon}\bigg(I_{-1,-\kappa}(0,0)+\sum_{m=1}^{\infty}\Big(P_{|\kappa|}\big(\tfrac{2\pi m}{\upsilon}\big)+o_{\kappa,\epsilon}(e^{-2\pi^{2}(1-\epsilon)m^{2}/\upsilon^{2}})\Big)\sqrt{2\pi}\mathbf{e}(m\omega)\bigg) \nonumber\\ &=-\frac{\sqrt{2\pi}i^{\kappa-1}}{\upsilon}\bigg(I_{-1,-\kappa}(0,0)+ \sqrt{2\pi}\sum_{j=0}^{|\kappa|}\big(\tfrac{2\pi}{\upsilon}\big)^{j}\frac{P_{|\kappa|-j}(0)}{j!}F(\mathbf{e}(\omega),-j)\bigg) +o_{\kappa,\epsilon}(e^{-2\pi^{2}(1-\epsilon)/\upsilon^{2}}).\nonumber\\ \end{align} $$

Now, the summands with $j<|\kappa |$ give $O_{\kappa ,\omega }(\upsilon ^{\kappa })$ , and the same applies to 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 (3.45) 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

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

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 (3.24), (3.34), and (3.35) and the trick from the proof of Lemma 8.5 of [Reference Bruinier, Funke and ImamoğluBFI] evaluate $\mathbf {G}_{\kappa ,0}(\omega ;\upsilon )$ as

(3.46) $$ \begin{align} \sum_{0 \neq \xi\in\mathbb{Z}+\omega}\frac{\mathrm{e}(\upsilon\xi)}{(\upsilon\xi)^{|\kappa|+1}}=-\!\!\sum_{0 \neq \xi\in\mathbb{Z}+\omega}\frac{\operatorname{sgn}(\xi)}{(\upsilon\xi)^{|\kappa|+1}}\operatorname{CT}_{s=0}(\upsilon|\xi|)^{-s}\bigg(\kern-1.2pt\int_{0}^{\infty}e^{-w^{2}/2}w^{s}dw-\int_{0}^{\upsilon|\xi|}e^{-w^{2}/2}w^{s}dw\kern-1.2pt\bigg). \end{align} $$

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

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

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 [Reference Erdélyi, Magnus, Oberhettinger and TricomiEMOT] 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

$$\begin{align*}-\sqrt{2\pi}\upsilon^{\kappa-1}\bigg(-\frac{\tilde{\psi}^{(|\kappa|)}(-\omega)+(-1)^{\kappa} \tilde{\psi}^{(|\kappa|)}(\omega)}{2|\kappa|!}-\delta_{\kappa,0}\frac{\gamma+\log2+2\log\upsilon}{2}\bigg), \end{align*}$$

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,

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

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

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

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 zeroth 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.

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

If $Q(\lambda )>0$ , then we recall the geodesic $c_{\lambda }\subseteq \mathcal {H}$ and its image $c(\lambda ) \subseteq Y$ from equation (2.15), and that when $\lambda $ is not split-hyperbolic, i.e., when $\iota (\lambda )=0$ in the notation of equation (2.16), 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) $$ \begin{align} \operatorname{Tr}_{\lambda}(g):=\oint_{c(\lambda)}g(z)\big(\lambda,Z(z)\big)^{k-1}dz. \end{align} $$

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 toward 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., for $g\in \widetilde {M}_{2k}^{!}(\Gamma )$ . Then its Fourier expansion near the cusp associated with some $\ell \in \operatorname {Iso}(V)$ is given, in the coordinates from equation (2.12), by

(4.3) $$ \begin{align} g_{\ell}(z_{\ell}):=(g\mid_{2k}\sigma_{\ell})(z_{\ell})=\sum_{l=0}^{p}\frac{g_{\ell,l}(z_{\ell})}{y_{\ell}^{l}}=\sum_{l=0}^{p}\sum_{n\in\mathbb{Z}}\frac{c_{\ell}(n,l)q_{\ell}^{n}}{y_{\ell}^{l}}= \sum_{l=0}^{p}\sum_{n\leq0}\frac{c_{\ell}(n,l)q_{\ell}^{n}}{y_{\ell}^{l}}+g_{\ell}^{0}(z_{\ell}), \end{align} $$

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 $g_{\ell }^{0}$ . Recall our extension of the incomplete Gamma function in (3.2), and the truncated modular curve $Y_{T}$ from equation (2.13) for any $T>1$ . We also set for n and $\kappa $ in $\mathbb {Z}$ , 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) $$ \begin{align} \begin{aligned} \phi_{n}(\kappa,T;r)&:=\begin{cases} \Gamma(\kappa,rnT)/(rn)^{\kappa}, & n\neq0, \\ -T^{\kappa}/\kappa, & n=0\mathrm{\ and\ }\kappa\neq0, \\ -\log T, & n=0{\ and\ }\kappa=0 \end{cases}\qquad\mathrm{and} \\ \operatorname{Sing}_{\lambda}(g,T)&:=i^{k}(2\sqrt{Q(\lambda)})^{k-1}\sum_{l=0}^{p}\sum_{n\in\mathbb{Z}}c_{\ell_{\lambda}}(n,l)\mathbf{e}\bigg(\frac{nr_{\lambda}}{\alpha_{\ell_{\lambda}}}\bigg) \phi_{n}\bigg(k-l,T;\frac{2\pi}{\alpha_{\ell_{\lambda}}}\bigg). \end{aligned} \end{align} $$

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

We can now define the trace to be

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

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

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

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

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 (2.12), then it becomes

$$\begin{align*}(\pm1)^{k}\int_{r_{\pm\lambda}+iR}^{r_{\pm\lambda}+iT}g_{\ell}(z_{\ell})\big(\sigma_{\ell}^{-1}(\pm\lambda),Z(z_{\ell})\big)^{k-1}dz_{\ell}&=(\pm i)^{k}(2\sqrt{Q(\lambda)})^{k-1}\\ & \quad \times \int_{R}^{T}g_{\ell}(r_{\pm\lambda}+iy_{\ell})y_{\ell}^{k-1}dy_{\ell} \end{align*}$$

via equation (2.17). This expression is a differentiable function of T, and equations (3.3) and (4.3) 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. This regularized integral can also be viewed as the special value at $s=k$ of an appropriately regularized L-function of g, as in, e.g., [Reference Bringmann, Fricke and KentBFK].

Following [Reference Alfes and EhlenANS, Reference Bruinier, Funke and ImamoğluBFI], we now give an equivalent expression for the regularized theta lift $\mathcal {I}_{k,L}(\tau ,f)$ of f from equation (2.24).

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 (2.24) can be written as

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

where $\phi _{0}$ is defined in equation ( 4.4 ).Footnote ⁵

Proof We argue as in Proposition 5.2 of [Reference Alfes and EhlenANS], focusing on the modifications required for our statement, and leaving many details out as they are identical. For fixed T, the constant term in the integral over $Y_{T}$ is the substitution $s=0$ , and near a cusp $\ell $ , Lemma 2.9 expresses $\Theta _{k,L}(\tau ,z)$ as a multiple of $\Theta _{k,\ell }(\tau )$ plus decreasing terms, whose integral vanishes as $T\to \infty $ . Here, instead of the first equation on page 2319 of [Reference Alfes and EhlenANS], the integral of our nearly holomorphic modular form yields

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

The same identifications of these constant terms with our functions $\phi _{0}(k-l,T)$ yield the desired result. This proves the proposition.

For k, $f\in \mathcal {A}_{0}^{!}(\Gamma )$ , and $g\in \mathcal {A}_{2k}^{!}(\Gamma )$ as above and an index $m\neq 0$ , we can define the combinations

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

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 $g\in \widetilde {M}^{!}_{2k}(\Gamma )$ has the usual expansion and $Q(\lambda )=m=0$ , we will define the trace to be

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

where $\iota _{\ell }(m,h)$ , $\omega _{\ell ,h}$ , and $\Phi _{\kappa }$ are defined in equations (2.18), (2.11), and (3.40), respectively.

The main term of the Shintani lift from Theorem 4.3 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) $$ \begin{align} \begin{aligned} &\operatorname{Tr}^{\mathrm{c}}_{m,h}(f,v):=-(-2i\sqrt{m})^{k}\cdot\sqrt{2\pi}\sum_{\ell\in\Gamma\backslash\operatorname{Iso}(V)} \big(\iota_{\ell}(m,h)+(-1)^{k}\iota_{\ell}(m,-h)\big)\frac{\varepsilon_{\ell}}{\sqrt{N}} \\ &\times\sum_{\substack{0>n\in\mathbb{Z} \\ n\equiv0\bmod 2\varepsilon_{\ell}\sqrt{m/N}}}\mathbf{e}\bigg(\frac{nr_{\lambda}}{\alpha_{\ell}}\bigg)\sum_{l=k}^{p}\bigg(\frac{2\pi n}{\alpha_{\ell}}\bigg)^{l-k}\frac{l!c_{\ell}(n,l)}{(l-k)!}\cdot\frac{J_{l}\big(2\sqrt{2\pi mv}\big)}{\big(2\sqrt{2\pi mv}\big)^{l}}, \end{aligned} \end{align} $$

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

where $I_{k}$ and $\tilde {\Omega }_{k}$ are defined in equation (3.26) and Remark 3.13, respectively. The name represents the fact that only the constant terms $c_{\ell }(0,k)$ contribute to it.

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

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

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

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

which resembles the non-holomorphic part of a harmonic weak Maass form with cuspidal $\xi $ -image (see also the proof of Proposition 4.5). We also gather the traces from equations 4.9–4.11, and set

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

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 $f\in \widetilde {M}_{2k}^{!}(\Gamma )$ , which is defined in equation (2.24) and lies in $\mathcal {A}_{k+\frac {1}{2}}\big (\operatorname {Mp}_{2}(\mathbb {Z}),\rho _{L}\big )$ , 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

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

where the terms are defined in equations (4.12)–(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 (2.23) respectively, which gives

(4.15) $$ \begin{align} \\[-3pc] &\mathcal{I}_{k,L,h}(\tau,f)\nonumber \\ &\quad =\lim_{T\to\infty}\left(\!\!\!\begin{array}{c}\displaystyle v^{\frac{1-k}{2}}\int_{Y_{T}}f(z)\sum_{m\in\mathbb{Z}+Q(h)}q_{\tau}^{m}\sum_{\lambda \in L_{m,h}}\varphi_{k,-1}\big(\sqrt{v}\lambda,z\big)d\mu(z)+ \\ \displaystyle\sum_{\substack{0 \leq m\in\mathbb{Z}+Q(h) \\ \iota(m)=1}}q_{\tau}^{m}\sum_{\ell\in\Gamma\backslash\operatorname{Iso}(V)}i^{k}\frac{\varepsilon_{\ell}}{\sqrt{N}}a(\Theta_{k,\ell},m,h,v)\sum_{l=0}^{p}c_{\ell}(0,l)\phi_{0}(k-l,T)\end{array}\!\!\!\!\right).\nonumber \end{align} $$

We may interchange the order of integration and summation as both are absolutely convergent for fixed T. Propositions 4.10–4.13 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 Assume that $f\in \widetilde {M}_{2k}^{!,\leq p}(\Gamma )$ , and that $p<k$ . Then

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

For $p=0$ , we recover the part of Theorem 6.1 of [Reference Alfes and EhlenANS] with weakly holomorphic input.

Proof If $p<k$ , then $\mathcal {I}^{\mathrm {neg}}_{k,L,h}$ and $\mathcal {I}^{\mathrm {c}}_{k,L,h}$ vanish identically (because both involve sums over $k \leq l \leq p$ ), and equations (4.10) and (4.11) imply that $\mathcal {I}^{\mathrm {cc}}_{k,L,h}$ consists only of the constant term

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

(because $c_{\ell }(0,k)=0$ when $p<k$ ). Since $k-l>0$ , the only term $\Xi _{k-l}(\omega _{\ell ,h})$ from equation (3.41) that may not vanish is when $l=k-1$ and $\omega _{\ell ,h}=0$ , where it equals the constant $-\frac {1}{\sqrt {2\pi }}$ . However, equation (2.11) shows that the latter equality only holds when $h=0$ , and thus equation (2.18) yields $\iota _{\ell }(0,h)=1$ for all $\ell $ , and the terms involving $\varepsilon _{\ell }$ , $\beta _{\ell }$ , and $\sqrt {N}$ reduce to $\alpha _{\ell }$ when $l=k-1$ by equation (2.10). This reduces our expression for $\mathcal {I}^{\mathrm {cc}}_{k,L,h}$ to $-\frac {Q_{k-1}(0)}{(2\pi )^{k/2}v^{(k-1)/2}}$ times $\sum _{\ell }\alpha _{\ell }c_{\ell }(0,k-1)$ . However, the $c_{\ell }(0,k-1)$ ’s are, up to constant multiple that is independent of $\ell $ , the constant terms of the weakly holomorphic modular form $L_{z}^{k-1}f \in M_{2}^{!}(\Gamma )$ , and by equation (2.12), multiplying each by $\alpha _{\ell }$ yields the residue of the corresponding meromorphic differential $L_{z}^{k-1}fdz$ at the cusp $\ell $ (again, up to a constant multiple that is independent of $\ell $ ). Thus, $\mathcal {I}^{\mathrm {cc}}_{k,L,h}$ is a multiple of the sum of residues of a meromorphic differential on the compact curve associated with $\Gamma $ at all of its poles, which therefore vanishes (this also explains Remark 1.4). This proves the corollary.

When $p \geq k$ , Corollary 4.4 no longer holds, but one can obtain a nearly holomorphic modular form after applying an appropriate weight-changing operator.

Proposition 4.5 For any $f\in \widetilde {M}_{2k}^{!,\leq p}(\Gamma )$ for which the constant terms $c_{\ell }(0,k)$ vanish for every $\ell $ , we have

$$\begin{align*}\xi_{k-2\lfloor p/2 \rfloor+\frac{1}{2}}L_{\tau}^{\lfloor p/2 \rfloor}\mathcal{I}_{k,L}(\tau,f)\in\widetilde{M}_{2\lfloor p/2 \rfloor-k+\frac{3}{2}}^{\leq p-k}\big(\operatorname{Mp}_{2}(\mathbb{Z}),\overline{\rho}_{L}\big). \end{align*}$$

Note that, for $p<k$ , the depth is negative, so that the space $\widetilde {M}_{2\lfloor p/2 \rfloor -k+\frac {3}{2}}^{\leq p-k}(\overline {\rho }_{L})$ is trivial, and indeed the modular forms that are annihilated by the operator in question are those lying in $\widetilde {M}_{k+\frac {1}{2}}^{\leq \lfloor p/2 \rfloor }\big (\operatorname {Mp}_{2}(\mathbb {Z}),\rho _{L}\big )$ .