text:
\section{Introduction}\label{Sect_Introduction}
The $cotangent\  Dirichlet\  series$

	\begin{equation}
	\xi ( s, \alpha ) := \sum_{n=1}^{\infty} \frac{\cot \pi n \alpha}{n^s}
	\end{equation}

	for an irrational real algebraic number $\alpha$ over $\mathbb{Q}$
	has been studied by B.C. Berndt \cite{berndt1}.
He showed that $\xi ( s, \alpha )$ is absolutely convergent if $\mathrm{Re}(s)$ is larger than the degree of $\alpha$,
	and evaluated the special values of $\xi(s, \alpha)$ for positive odd integers $s$ and real quadratic numbers $\alpha$.
After him, T. Arakawa improved the bounds of convergence from the degree of $\alpha$ to $1$
	by use of the Thue-Siegel-Roth theorem.
\begin{theorem}\label{1.Berndt_Theorem}
$\mathrm{(B.C. Berndt}$ \cite[Theorem 5.1, 5.2]{berndt1} with  $\mathrm{T. Arakawa}$ \cite[Lemma 1]{arakawa1}$\mathrm{)}$\\
$\mathrm{(i)}$
If $s$ is a real number with $\mathrm{Re}(s) > 1$ and $\alpha$ is an irrational real algebraic number, then
	$\xi (s, \alpha)$ is absolutely convergent.
\\
$\mathrm{(ii)}$
Let $\alpha = (a + b \sqrt{c})/2$  and $\epsilon = \pm 1$,
	where
	$a, b$ and $c$ are integers with $c > 0$ and
	$a^2 -cb^2 = 4 \epsilon$. If $l$ is an integer with $l > 1$, then
	\begin{equation}\label{1.Berndt_formula}
		(1 - \epsilon \alpha^{2l-2})\xi(2l-1, \alpha)
		=
		\frac{(-1)^{l-1}  (2\pi)^{2l-1}}{(2l)!}
			\sum_{k=0}^{l} \binom{2l}{2k}  \alpha^{2k-1} B_{2k}B_{2l-2k}.
	\end{equation}
Here $B_k$ are the k-th Bernoulli numbers defined by
	$\sum\limits_{m=0}^{\infty} (B_m/m!)x^{m} = x / (\mathrm e^x -1)$.
\end{theorem}
For real numbers $s$ and $x$,
	let $\mathrm e(s)$ denote $\mathrm e^{2 \pi i s}$,
	and
	$\langle x \rangle$ (resp. $\{x \}$) the real number which satisfies
	$0 < \langle x \rangle \leq 1$ and $x - \langle x \rangle \in \mathbb{Z}$
	(resp. $0 \leq \{x \} < 1$ and $x - \{ x \} \in \mathbb{Z}$).
Furthermore let $\chi (x)$ be the characteristic function of integers, i.e.,
	\begin{equation*}
		\chi (x) :=
		 	\begin{cases}
				 1 & \mathrm{if} \quad  x \in \mathbb{Z}, \\
				 0 & \mathrm{if} \quad  x \notin \mathbb{Z},
		 	\end{cases}
	\end{equation*}
	and $\zeta(s, x)$ the Hurwitz zeta function defined by
	\begin{equation*}
		\zeta(s, x) := \sum_{n = 0}^{\infty} \frac{1}{(n + x)^s}.
	\end{equation*}
For any pair $\vec{\omega} = (\omega_1, \omega_2)$ of positive real numbers
	and for complex numbers $z, t \in \mathbb{C}$,
	we set
	\begin{equation*}
		G_2 (z, \vec{\omega}; t) :=
			\frac{\exp(-zt)}{(1-\exp(-\omega_1t ))(1-\exp(-\omega_2 t))}.
	\end{equation*}

T. Arakawa \cite{arakawa1} established transformation formulae for the infinite series
	\begin{equation} \label{1.definition_H}
		H (\alpha,s,x,y)
		:=
		\sum_{n=1}^{\infty} \frac{\mathrm e(ny)}{n^{1-s}} \frac{\mathrm e (n \langle x \rangle \alpha)}{1-\mathrm e(n \alpha)}
		+
		\mathrm e(s/2)
		\sum_{n=1}^{\infty} \frac{\mathrm e(-ny)}{n^{1-s}} \frac{\mathrm e(n \langle -x \rangle  \alpha)}{1-\mathrm e(n \alpha)}.
	\end{equation}
\begin{theorem}\label{1.Arakawa_Theorem}
$\mathrm{(T. Arakawa}$ \cite[Theorem 1]{arakawa1}$\mathrm{)}$
	Let $\alpha$ be any irrational real algebraic number,
		and let $V = \begin{pmatrix} a & b \\ c & d \end{pmatrix}  \in \mathrm{SL}_2 (\mathbb{Z})$
		with $c > 0$ and $c \alpha + d > 0$.
Put $\beta = c \alpha + d$,
	and
	set $p' = pa + qc$, $q' = pb + qd$,
		and $\rho = \{ q' \} c - \{p' \} d$ for any pair $(p, q)$ of real numbers.
	If $\mathrm{Re}(s) < 0$, then
		\begin{eqnarray}
			& & \beta^{-s} H(V\alpha, s, p, q) \nonumber \\
			& = &  H(\alpha, s, p', q')
				- \chi(p) (2 \pi)^{-s} \mathrm e(s/4) \beta^{-s} \Gamma(s) \bigl(\zeta(s, \langle q \rangle)
				+\mathrm e(s/2) \zeta(s, \langle - q \rangle) \bigr) \label{1.Arakawa_Equation} \nonumber \\
			& & \quad
				- \chi(p') (2 \pi)^{-s} \mathrm e(-s/4) \beta^{-s} \Gamma(s) \bigl(\zeta(s, \langle -q' \rangle)
				+\mathrm e(s/2) \zeta(s, \langle q' \rangle) \bigr)\nonumber \\
			& & \quad
				+(2 \pi)^{-s}  \mathrm e(-s/4) L(\alpha, s, p', q', c, d).
		\end{eqnarray}
	Here $\Gamma(s)$ is the gamma function and
		\begin{equation*}
			L(\alpha, s, p', q', c, d)
			=
			- \sum_{j=1}^{c}
				\int\limits_{I(\lambda, \infty)}
					t^{s-1} G_2(1-\{ \frac{jd + \rho}{c} \} +  \beta \frac{j -\{ p' \}}{c}, (1, \beta); t) dt
		\end{equation*}
		where $I(\lambda, \infty)$ is the integral path consisting of the oriented half line
		$(+\infty, \lambda)$, a counterclockwise circle of radius $\lambda$ around the origin,
		and the oriented half line $(\lambda, +\infty)$.
\end{theorem}
\begin{remark}
	Let $u, \omega$ be positive real numbers, and $s$ a complex number.
	The Barnes zeta function
	\begin{equation*}
		\zeta_2(s, \omega, u)
		:=
		\sum_{m,n=0}^{\infty}
			\frac{1}{(u + m +n\omega)^s}
	\end{equation*}
	has been intensively studied by Barnes \cite{barnes1}.
	$L(\alpha, s, p', q', c, d)$  is rewritten in terms of Barnes zeta function as follows (see \cite[(1.18)]{arakawa2}) .
	\begin{equation*}
		L(\alpha, s, p', q', c, d)
		=
		- \Gamma(s) (\mathrm e (s) - 1)\sum_{j=1}^{c}
			\zeta_2 (s, \beta, 1-\{ \frac{jd + \rho}{c} \} +  \beta \frac{j -\{ p' \} }{c} ).
	\end{equation*}

\end{remark}

We define the function $\tilde{\xi} (s,\alpha,x,y)$ by
	\begin{equation}\label{1.definition_generalized_H}
		\tilde{\xi} (s,\alpha,x,y) := - H (\alpha,1-s,- y,x)
	\end{equation}
	which is modified the arguments $\alpha, s, x, y$ of $H (\alpha,s,x,y)$.
The purpose of the modification is to express our results easily
	(for example, see (\ref{Sect1.mod}) in which the argument $s$ is adjusted to that of $\xi (s, \alpha)$,
	and
	(\ref{4.theorem1a}) in which transformation formulae on the modular group is effectively described).
The cotangent Dirichlet series $\xi ( s, \alpha )$ is expressed by
	the function $\tilde{\xi} (s,\alpha,x,y)$ with $(x,y)=(0,1)$:
\begin{equation}\label{Sect1.mod}
	\xi (s, \alpha)
	=
	-2i \big{(} \frac{1}{1+\mathrm e(s/2)} \tilde{\xi} (s, \alpha, 0, 1) + \frac{1}{2} \zeta (s) \big{)},
\end{equation}
	where
	$\zeta (s) := \sum\limits_{n=1}^{\infty} 1/n^s$ is the Riemann zeta function.
Therefor we call $\tilde{\xi} (s,\alpha,x,y)$ $generalized$ $cotangent$ $seires$.

The first aim of this paper is to establish elliptic analogue
	 $\tilde{\xi} (s, \alpha, x', x, y', y; \tau)$  to $\tilde{\xi} (s, \alpha,x,y)$
	 which we call
	 \emph{elliptic generalized cotangent Dirichlet series} (see Definition \ref{2.def_H}).
The second is
	to give transformation formulae of the elliptic generalized cotangent Dirichlet series
	at some integer arguments, or at $s \in \mathbb{Z}$ with $s > 2$ (see Theorem \ref{4.Theorem1}).
It should be noted that the argument $s$ is allowed any complex number $s$ with $\mathrm{Re}(s) > 1$
	in Arakawa's transformation formulae of $\tilde{\xi} (s, \alpha,x,y)$.
In particular, if $\alpha$ is a real quadratic number,
	then $\xi (s, \alpha)$ is meromorphic with respect to $s$
	and its transformation formulae hold at all complex numbers $s$.
(We note that
	T. Arakawa \cite{arakawa2} gave certain relations
	between residues at poles of $\xi(s, \alpha)$ and special values of partial zeta-function and Hecke L-functions.)
On the other hand, in our transformation formulae, the argument $s$  is restricted positive integers at least $3$.
The reason of the restriction is that it is difficult to apply Arakawa's way of establishing transformation formulae of  $\tilde{\xi} (s,\alpha,x,y)$ to our case.
For applying, we need  an integral representation of $\tilde{\xi} (s, \alpha, x', x, y', y; \tau)$ and an elliptic analogue of Barnes zeta function.
However,
	we can give Berndt's Theorem \ref{1.Berndt_Theorem} (ii) from our transformation formulae,
	because
	it is derived from Arakawa's transformation formulae in the case that $s$ is an odd integer with $s \geq 3$
	which follow from our transformation formulae of $\tilde{\xi} (s, \alpha, x', x, y', y; \tau)$ by $\tau \rightarrow i \infty$.

The paper is organized as follows:
In Section \ref{Sect_H}, we define the elliptic generalized cotangent Dirichlet seires $\tilde{\xi} (s, \alpha, x', x, y', y; \tau)$
	and show that it is absolutely convergent if $\mathrm{Re}(s)  > 2$.
In Section \ref{Sect_EDRSum},	 we prepare $Elliptic\  Dedekind$-$Rademacher\  sums$
	for next Section \ref{Sect_Transformation}
	in which we establish transformation formulae of $\tilde{\xi} (s, \alpha, x', x, y', y; \tau)$
	with integers $s > 2$.
In Section \ref{Sect_BerndtArakawaResults},
	from our transformation formulae,
	we derive (\ref{1.Arakawa_Equation}) with integers $s < -1$
	and Theorem \ref{1.Berndt_Theorem} (ii).
Section \ref{Sect_Lemma1} and \ref{Sect_Lemma2} devote the proofs of Lemma \ref{4.Lemma1} and \ref{4.Lemma2}
	in Section \ref{Sect_Transformation} respectively.

Throughout the paper, let
	$\alpha$ be an irrational algebraic number over $\mathbb{Q}$,
	$s$ a complex number,
	$y', y, x', x$ real numbers,
	$\tau$ a complex number with positive imaginary part,
	and $\mathrm{SL}_{2} (\mathbb{Z})$ the modular group.
	If $\mathbb{A}$ is a ring,
		$\mathrm{M}_2 (\mathbb{A})$ means the set of two by two matrixies whose entries are in $\mathbb{A}$.
We use the following notions:
	$\mathrm e(x) := \mathrm e ^{2\pi i ix}$,
	$V z := (az + b)/(cz + d)$
	and
	$j(V;z) := cz + d$
	where $z \in \mathbb{C}$ and
	$V = \bigl( \begin{smallmatrix} a & b \\ c & d  \end{smallmatrix} \bigr)
		\in \mathrm{SL}_2 (\mathbb{Z})$.

\section{Elliptic generalized cotangent Dirichlet series} \label{Sect_H}
In this section, we fulfill the first aim of this paper, or define the elliptic generalized cotangent Dirichlet series  $\tilde{\xi} (s, \alpha, x', x, y', y; \tau)$
	as analogue to generalized cotangent Dirichlet series $\tilde{\xi} (s, \alpha, y, x; \tau)$.
For the aim, we introduce the function $\underline{F} (x', x;X;\tau)$ which is analogue to
	$\dfrac{\mathrm e(n\langle x \rangle \alpha)}{1-\mathrm e(n \alpha)}$
	used in $\tilde{\xi} (s,\alpha,x,y) $.

Let $q=\mathrm e(\tau)$.
The function $\underline{F} (x', x;X;\tau)$ is built by Jacobi's theta function
	\begin{align}\label{2.theta}
 		\theta (x ;\tau)
 		&:=
 		\sum_{m \in \mathbb Z} \mathrm e (\frac 12(m+\frac 12)^2\tau +(m+\frac 12)(x+\frac 12))
 		\\
 		& =
 		i q^{1/8} (\mathrm e(\frac{x}{2}) - \mathrm e(-\frac{x}{2}))
 			\prod_{m=1}^{\infty}
		(1-\mathrm e(-x)q^m)(1-\mathrm e(x)q^m)(1-q^m)  \nonumber
	\end{align}
	which is an odd and quasi periodic entire function:
\begin{equation}\label{2.theta_quasi}
\begin{split}
	\theta (-x;\tau) &= -\theta (x;\tau), \\
 	\theta (x+1 ;\tau) &=  -\theta (x ;\tau) , \\
	\theta (x+\tau ;\tau) &=  -\mathrm e(-\frac{\tau}{2} -x)  \theta (x ;\tau).
\end{split}
\end{equation}
Let $\theta '(x ;\tau)$ denote the derivative of $\theta (x ;\tau)$ with respect to $x$.
For any two tuple of real numbers $\vec{x} = (x',x) \in \mathbb{R}^2 \setminus \mathbb{Z}^2$,
	the function $\underline{F} (\vec{x};X;\tau)$ is defined by
	\begin{equation}\label{2.definition_K}
		\underline{F} (\vec{x};X;\tau) :=
		\mathrm e (x X)
		\frac{\theta '(0;\tau) \theta(-x' +x\tau +X;\tau)}{\theta(-x' + x \tau;\tau) \theta(X;\tau)}
	\end{equation}
	which is Kronecker double series introduced in \cite{weil} and essentially a meromorphic Jacobi form studied in \cite[Section 3]{zagier1}.
Some fundamental properties of $\underline{F} (\vec{x};X;\tau)$ are following:
As a function with respect to $X$,
	it is meromorphic with only simple poles on the lattice
 	$\mathbb Z +\tau \mathbb Z$.
By (\ref{2.theta_quasi}) it has properties
	\begin{align}
		\underline{F} (\vec{x};X +1;\tau) &= \mathrm e (x) \underline{F} (\vec{x};X;\tau),&
		\underline{F} (\vec{x};X +\tau;\tau) &= \mathrm e (x') \underline{F} (\vec{x};X;\tau),& \label{2.key_formula1} \\
		\underline{F} (-\vec{x};-X;\tau) &= - \underline{F} (\vec{x};X;\tau),&
		\underline{F} (\vec{x} +\vec{a};X ;\tau) &= \underline{F} (\vec{x};X;\tau)& \label{2.key_formula2}
	\end{align}
	where $\vec{a} = (a', a) \in \mathbb{Z}^2$.

If $y \notin \mathbb{Z}$, we find from (\ref{1.definition_H}) and (\ref{1.definition_generalized_H})
	that the infinite series $\tilde{\xi} (s, \alpha, y, x)$ is express as
	\begin{equation} \label{2.infiniteExpression_H}
		\tilde{\xi} (s, \alpha, y, x)
		=
		\sum_{m \in \mathbb{Z} \atop (m \neq 0)}
			\frac{\mathrm e(mx)}{m^s} \frac{\mathrm e(\alpha m \langle -y \rangle)}{\mathrm e(\alpha m) -1}.
	\end{equation}
By considering $\underline{F} (\vec{x};X;\tau)$ to be an elliptic generalization to $\mathrm e(\langle x \rangle X)/(\mathrm e(X) -1)$,
	we define the elliptic analogue $\tilde{\xi} (s, \alpha, x', x, y', y; \tau)$
	of $\tilde{\xi} (s, \alpha, y, x; \tau)$ as follows:
\begin{definition}\label{2.def_H}
	Let $\alpha$ be an irrational real algebraic number
		and $s$ a complex number with $\mathrm{Re}(s) > 2$.
	For four real numbers $x', x, y', y$ with $(y', y) \notin \mathbb{Z}^2$,
		the elliptic generalized cotangent Dirichlet series $\tilde{\xi} (s, \alpha, x', x, y', y; \tau)$
		is defined by
		\begin{equation} \label{3.infinite_series}
			\tilde{\xi} (s, \alpha, x', x, y', y; \tau)
			:=
			\sideset{}{^\prime}\sum_{m', m}
			\frac{\mathrm e (m'x' +mx)}{(\tau m' +m)^{s}}
			\underline{F} (-\vec{y}; \alpha (\tau m' +m); \tau)
		\end{equation}
		where the summation
		ranges over all elements in $\mathbb{Z}^2$ except $(0,0)$.
\end{definition}
We prove the convergency of $\tilde{\xi} (s, \alpha, x', x, y', y; \tau)$.
\begin{lemma}\label{3.Lemma1}
	For any complex number $s$ with $\mathrm{Re}(s) > 2$,
		$\tilde{\xi} (s, \alpha, x', x, y', y; \tau)$
		absolutely converges.
\end{lemma}
\begin{proof}
	Let $LHS$ mean
		the absolute value of $\tilde{\xi} (s, \alpha, x', x, y', y; \tau)$.
	Set $\vec{x} = (x', x)$ and $\vec{y} = (y', y)$.
	Because the function $\underline{F} (x', x' ;X;\tau)$ with respect to $X$
 		is meromorphic with only simple poles on the lattice
 		$\mathbb Z +\tau \mathbb Z$,
 		there is a positive real number $C = C (x', x, \tau)$ depending on $x', x, \tau$
 		which satisfy the following property:
 	If $X= \tau \xi' +\xi$ is a complex number with
 		$-1/2 \leq \xi', \xi \leq 1/2$, then
		\begin{equation*}
 			\lvert X \underline{F} (\vec{x}; X; \tau) \rvert  \leq C .
		\end{equation*}
 	For any real number $r$,
 		let $[[ r ]]$ and
 		$\langle \langle r \rangle \rangle$ mean the integer and the real number satisfying
 		$-1/2 < \langle \langle r \rangle \rangle \leq 1/2$ and $r = [[r]] +  \langle \langle r \rangle \rangle $.
 	By (\ref{2.key_formula1}),
		\begin{equation*}
 			\underline{F} (\vec{x}; \alpha (\tau m' +m); \tau)
			=
			\mathrm e(x' [[\alpha m' ]] +x [[\alpha m]])
			\underline{F} (\vec{x}; \tau \langle \langle \alpha m'\rangle \rangle +\langle \langle \alpha m \rangle \rangle; \tau) ,
		\end{equation*}
		thus we have
		\begin{equation*}
			LHS \leq C \sideset{}{^\prime}\sum\limits_{m', m}
 			\dfrac{1}{{\lvert \tau m' +m \rvert }^{s}}
			\frac{1}{\lvert \tau \langle \langle \alpha m'\rangle \rangle +\langle \langle \alpha m \rangle \rangle  \rvert}.
		\end{equation*}
	If $r'$ and $r$ are real numbers, it holds that
 		\begin{equation}\label{3.lemma1aa}
 			\begin{split}
  				&  \abs{\tau r' +r} \geq  \abs{\mathrm{Im}(\tau)} \times \bigl(  \max \{ \abs{r'},  \abs{r} \}\bigr), \\
  				&  {\lvert \tau r' +r \rvert}^2  \geq 2 (\lvert \tau \rvert - \lvert \mathrm{Re}\ \tau \rvert) \lvert r' r \rvert.
 			\end{split}
 	 	\end{equation}
 	The first inequality is derived from
		\begin{equation*}
			\abs{\tau r' +r}^2
				=
				\abs{\tau}^2 r'^2 + (\tau + \overline{\tau}) r' r + r^2
				=
				{(\mathrm{Re}(\tau) r'  + r)^2 + (\mathrm{Im}(\tau) r' })^2 \geq  (\mathrm{Im}(\tau) r')^2
		\end{equation*}
		and $\abs{\tau r' +r}^2 = \abs{\tau}^2 \abs{ r' + (1/ \tau)r}^2$.
	Since
		\begin{equation*}
			0 \leq (\abs{\tau} r' \pm r)^2 =  \abs{\tau}^2 r'^2 + r^2 \pm 2 \abs{\tau} r'r,
		\end{equation*}
	one has
		\begin{equation*}
			\abs{\tau r' \pm r}^2 - (\tau + \overline{\tau}) r'r
			=
			\abs{\tau}^2 r'^2 + r^2
			=
			\mp 2 \abs{\tau} r'r
			\geq
			2 \abs{\tau} \abs{r' r}.
		\end{equation*}
	Thus the second is derived from
		\begin{equation*}
			\abs{\tau r' \pm r}^2  \geq 2 \abs{\tau} \abs{r' r} + (\tau + \overline{\tau}) r'r \geq (2 \abs{\tau} - \abs{\tau + \overline{\tau}} )\abs{r' r}
			=
			2(\abs{\tau}-\abs{\mathrm{Re}\ \tau})\abs{r' r}.
		\end{equation*}
	Since $\lvert \tau \rvert - \lvert \mathrm{Re}\ \tau \rvert$ is positive because of $\mathrm{Im}(\tau) > 0$,
	there is a positive real number $D = D(\tau)$ depending on $\tau$ such that the following inequality holds:
	\begin{eqnarray}\label{2.proofOfLemma_11}
		& &
		LHS
		\\
		&\leq&
		C D
		\Biggl[
			\sideset{}{^\prime}\sum_{m'} \sideset{}{^\prime}\sum_{m}
				\frac{1}{{\lvert m' \rvert}^{s/2} {\lvert m \rvert}^{s/2}
				{\lvert \langle \langle \alpha m' \rangle \rangle \rvert}^{1/2}
		 		{\lvert \langle \langle \alpha m \rangle \rangle \rvert}^{1/2}}
			\nonumber \\
			& & \hspace{50pt}
			+
			\sideset{}{^\prime}\sum_{m'} \frac{1}{{\lvert m' \rvert}^{s}
				\lvert \langle \langle \alpha m' \rangle \rangle \rvert}
			+
			\sideset{}{^\prime}\sum_{m} \frac{1}{{\lvert m \rvert}^{s}
				\lvert \langle \langle \alpha m \rangle \rangle \rvert}
		\Biggr]
		\nonumber \\
		& \leq &
		C D
		\Biggl[
			\Bigl(
				\sideset{}{^\prime}\sum_{m'}
					\frac{1}
					{{\lvert m' \rvert}^{s/2} {\lvert \langle \langle \alpha m' \rangle \rangle \rvert}^{1/2}}
			\Bigr)
			\Bigl(
				\sideset{}{^\prime}\sum_{m}
					\frac{1}{{\lvert m \rvert}^{s/2} {\lvert \langle \langle \alpha m \rangle \rangle \rvert}^{1/2}}
			\Bigr)
			\nonumber \\
			& & \hspace{50pt}
			+
			\sideset{}{^\prime}\sum_{m'} \frac{1}{{\lvert m' \rvert}^{s}
				\lvert \langle \langle \alpha m' \rangle \rangle \rvert}
			+
			\sideset{}{^\prime}\sum_{m} \frac{1}{{\lvert m \rvert}^{s}
				\lvert \langle \langle \alpha m \rangle \rangle \rvert}
		\Biggr]
		\nonumber,
	\end{eqnarray}
		where the summation $\sum_{m'}'$ and $\sum_{m}'$
		range over all integers except zero respectively.
	T. Arakawa showed that the series
		$\sum\limits_{m=1}^{\infty} \dfrac{1}{m^{1+\epsilon} \lvert \langle \langle \alpha m \rangle \rangle\rvert}$
 		converges if $\epsilon >0$
		in the proof of \cite[Lemma 1]{arakawa1}.
	Because $1/\sqrt{x} < 1/x $ if $0 < x < 1$,
		the series
		$\sum\limits_{m=1}^{\infty} \dfrac{1}{m^{1+\epsilon} {\lvert \langle \langle \alpha m \rangle \rangle\rvert}^{1/2}}$
		does too.
	Therefore the right hand side of (\ref{2.proofOfLemma_11}) converges
		which complete the proof.
\end{proof}

\section{Elliptic Dedekind-Rademacher sums } \label{Sect_EDRSum}
We introduce $Elliptic$ $Dedekind$-$Rademacher$
	$sums\ S_{m,n}(r, x', x, y', y; \tau) $
	for describing transformation formulae of $\tilde{\xi} (s, \alpha, x', x, y', y; \tau)$
	which will be established next section.

These sums are built up by $elliptic\ Bernoulli\ functions\ B_m(x', x;\tau) $ which are defined by
	use of the generating function $\underline{F} (x' ,x ;X;\tau)$:
\begin{equation}\label{3.definition_B}
	\underline{F} (x' ,x ;X;\tau) =
	\sum_{m=0}^{\infty}
		\frac{B_m(x' ,x ;\tau)}{m!} {(2\pi i)}^{m} X^{m-1}.
\end{equation}
Elliptic Bernoulli functions are a part of Kronecker's double series (see \cite{weil}).
They have the following explicit expressions (for example, see \cite{machide})
	\begin{multline} \label{Sect3.FouExp}
	 	B_m(\vec{x};\tau)
	 	=
		m \biggl(
			\sum_{j=1}^{\infty} (x-j)^{m-1} \frac{\mathrm e(-x\tau) q^j}{\mathrm e (-x') -\mathrm e (-x\tau)q^j}
		 	 \\
			-\sum_{j=1}^{\infty} (x+j)^{m-1}\frac{\mathrm e(x\tau)q^j}{\mathrm e (x')-\mathrm e(x\tau)q^j}
			+
			x^{m-1} \frac{\mathrm e (-x'+x\tau)}{\mathrm e (-x'+x\tau) -1}
		\biggr)
		+
		B_m(x),
	\end{multline}
	where $q = \mathrm e(\tau)$ and $B_m(x)$ are Bernoulli polynomials defined by
	$\dfrac{\mathrm e (x X)}{\mathrm e (X)-1} =
		\sum\limits_{m=0}^{\infty}
		\dfrac{B_m(x)}{m!} (2 \pi i X)^{m-1}$.
For $\vec{x} = (x', x) \in \mathbb{R}^2 \setminus \mathbb{Z}^2$ and $\vec{a} = (a', a)  \in \mathbb{Z}^2$,
	they satisfy by (\ref{2.key_formula2}) that
	\begin{equation}\label{2.key_formula3}
		B_m (-\vec{x};\tau) = (-1)^m B_m (\vec{x};\tau),
		\quad
		B_m (\vec{x} +\vec{a} ;\tau) = B_m (\vec{x};\tau).
	\end{equation}
We note that,
	for $ \vec{x} \in \mathbb{R}^2$,
	$B_m(\vec{x};\tau)$
	is discontinuous if $m = 1,2$ and $ \vec{x} \in \mathbb{Z}^2$,
	and continuous otherwise because of (\ref{Sect3.FouExp}).

For defining the elliptic Dedekind-Rademacher sums $S_{m,n}(r, x', x, y', y; \tau)$,
	we prepare the $numerator$ and $denominator$ maps $n$ and $d$ defined by
	\begin{equation} \label{3.Def_ndNumber}
 		\begin{matrix}
			& n: \mathbb{Q} \to \mathbb{Z},  \quad & r \mapsto p,\\
			& d: \mathbb{Q} \to \mathbb{Z}_{>0},  \quad &  r \mapsto q
 		\end{matrix}
	\end{equation}
	where $p, q$ are a unique pare of integers such that
	$\gcd (p, q) =1, q \geq 1$ and $r = p/q$.

\begin{definition}\label{3.definition_ERDSUM}
	For nonnegative integers $m, n$, any rational number $r \neq 0$ and real numbers $x', x, y', y$,
		we define elliptic Dedekind-Rademacher sums by
		\begin{equation}\label{2.elliptic_Dedekind-Rademacher_sum}
			S_{m,n}(r, x', x, y', y; \tau)
			:=
			\frac{1}{d(r)}
			\sum_{j', j (   d(r) )}
			B_m \Bigl(\frac{\vec{j} + \vec{y}}{d(r)}; \tau \Bigr)
			B_n \Bigl(n(r) \frac{\vec{j} + \vec{y}}{d(r)} - \vec{x}; \tau \Bigr),
		\end{equation}
		where the summation ranges over $(j',j) \in {(\mathbb{Z}/d(r)  \mathbb{Z})}^2$
		and  $\vec{j}, \vec{x}, \vec{y}$ denotes the vectors $(j',j), (x',x), (y',y)$ respectively.
	Because of discontinuity of $B_1 (\vec{z}; \tau )$ and $B_2 (\vec{z}; \tau )$ at $ \vec{z} \in \mathbb{Z}^2$,
		we suppose that
		$\vec{y}, n(r) \vec{y} - d(r) \vec{x} \notin \mathbb{Z}^2$
		if $m, n \in \{ 1,2 \}$.
\end{definition}
We note
	the relation between these sums and
	the sums $S_{m,n}^{\tau} \bigl(
	\begin{smallmatrix}
        		(a',a)  &  (b',b)  & (c',c)     \\
        		(x',x)  &  (y',y)  & (z',z)    \\
	\end{smallmatrix} \bigr)$
	defined in \cite{machide}:
	\begin{equation}\label{2.relation}
		S_{m,n}(r, x', x, y', y; \tau)
		=
		S_{m,n}^{\tau}
		\begin{pmatrix}
        			(1,1)  &  (n(r),n(r))  & (d(r),d(r))     \\
        			(0,0)  &  (x',x) & (y', y)     \\
		\end{pmatrix}.
	\end{equation}

\section{Transformation formulae of elliptic generalized cotangent Dirichlet series} \label{Sect_Transformation}
In this section, we give transformation formulae of
	elliptic generalized cotangent Dirichlet series $\tilde{\xi} (s, \alpha, x', x, y', y; \tau)$
	at integers $s > 2$,
	which is  the second aim of this paper.

Firstly we modify and prepare some notations for describing the transformation formulae simply.
The four arguments $(x', x, y', y)$ used in
	$\tilde{\xi} (s, \alpha, x', x, y', y; \tau)$ and $S_{m,n}(r, x', x, y', y; \tau)$
	are rewritten in terms of a matrix element of $\mathrm{M}_{2} (\mathbb{R})$, or
	$M=
	\bigl(
	\begin{smallmatrix}
		x' & x  \\
		y' & y
	\end{smallmatrix}
	\bigr). $
Thus
	\begin{equation}\label{4.equa_Modify}
		\begin{split}
			&
			\tilde{\xi} (s,\alpha, M; \tau)
			:=
			\tilde{\xi} (s, \alpha, x', x, y', y; \tau), \\
			&
			S_{m,n}(r, M; \tau)
			:=
			S_{m,n}(r, x', x, y', y; \tau).
		\end{split}
	\end{equation}
Hereafter we sometimes write the matrix $M$ as
	$\bigl( \begin{smallmatrix} \vec{x}  \\ \vec{y} \end{smallmatrix} \bigr) $
	where $\vec{x} = (x', x)$ and $\vec{y} = (y', y)$.
If $m \in \mathbb{Z}$, $z \in \mathbb{C}$,
	$M =
		\bigl( \begin{smallmatrix} \vec{x}  \\  \vec{y} \end{smallmatrix} \bigr)
		\in \mathrm{M}_{2} (\mathbb{R})$
	and
	$V =
		\bigl(
			\begin{smallmatrix} a & b  \\  c & d \end{smallmatrix}
		\bigr)
		\in \mathrm{SL}_{2} (\mathbb{Z})$
	with
	$\vec{y}, c \vec{x}+d \vec{y} \notin \mathbb{Z}^2$,
	then we define polynomials $R_V (l, z, M ; \tau)$ of $z$ by
	\begin{multline}
		R_V (l, z, M; \tau)\\
		:=
		 \begin{cases} \displaystyle{}
		 	\frac{(2 \pi i)^{l+1}}{(l+1)!}
			\sum_{k=-1}^{l}
			\binom{l+1}{k+1}
			(-j(V;z))^k
				S_{k+1,l-k}
				\biggl( \frac{d}{c},
					\bigl(\begin{smallmatrix} -\vec{x}  \\ \vec{y}\end{smallmatrix} \bigr) ; \tau \biggr)
				&
				( c > 0), \\
			0 & (c =0), \\
			R_{-V} (l, z, M; \tau) & (c < 0)
		\end{cases}
	\end{multline}
	where $j(V;z) := c z + d$ is the automorphic factor on $\mathrm{SL}_{2} (\mathbb{Z})$.
We note that these polynomials derive period polynomials as follows (See \cite{fukuhara1} about period polynomials):
If $V =
	\bigl(
	\begin{smallmatrix}
		0 & -1  \\
		1 & 0
	\end{smallmatrix}
	\bigr),
	$
	then
	\begin{equation*}
		R_V (l, z, M; \tau)
		 =
		 \frac{(2 \pi i)^{l+1}}{(l+1)!}
		\sum_{k=-1}^{l}
		\binom{l+1}{k+1}
		(-z)^k B_{k+1}(\vec{y}; \tau) B_{l-k}(\vec{x}; \tau)
	\end{equation*}
Therefore
	\begin{equation*}
		\lim_{\tau \rightarrow i \infty} R_V (l, z, M; \tau)
 		=
		\frac{(2 \pi i)^{l+1}}{(l+1)!}
		\sum_{k=-1}^{l}
		\binom{l+1}{k+1}
		(-z)^k \tilde{B}_{k+1}(y) \tilde{B}_{l-k}(x)
	\end{equation*}
	where $\tilde{B}_n(x)$ are bernoulli functions.
By tending $x$ and $y$ to $0$, we obtain the period polynomials over $\mathbb{Q}$.
\begin{remark}
	When $\tau$ tends to $i \infty$,
		$R_V (m, z, M; \tau) $ are equal to $f_{l+1} (-d,c; z)$ in \cite[(1.2)]{carlitz1} up to a constant.
	Thus
		(\ref{4.lemma2a}) below can be considered as
		elliptic generalizations of Carlitz's reciprocity relations \cite[(1.11)]{carlitz1}.
\end{remark}

The transformation formulae are as follows:
\begin{theorem}\label{4.Theorem1}
	For $V = \bigl(
		\begin{smallmatrix}
        			a  &  b       \\
        			c  &  d
		\end{smallmatrix} \bigr) \in \mathrm{SL}_2 (\mathbb{Z})$,
		let $\mathrm{M}_2 (V)$ be the subset of $\mathrm{M}_2 (\mathbb{R})$
		defined by
		\begin{equation}\label{4.definition_SM}
			\mathrm{M}_2 (V)
			:=
			\biggl{\{}
			\begin{pmatrix}
				x' & x  \\
				y' & y
			\end{pmatrix}
			=
			\begin{pmatrix}
				\vec{x}  \\
				\vec{y}
			\end{pmatrix}
			\in \mathrm{M}_2 (\mathbb{R})
			\bigg{\vert}
			\vec{y}, c \vec{x}  + d \vec{y}\notin \mathbb{Z}^2
			\biggr{\}}.
		\end{equation}
	Let $\alpha$ be an irrational real algebraic number.
	If $l \in \mathbb{Z}$ with $l \geq 3$,
 		$V \in \mathrm{SL}_2 (\mathbb{Z})$
		and	$M \in \mathrm{M}_2(V)$, it holds that
	\begin{equation}\label{4.theorem1a}
		\tilde{\xi} (l, \alpha, M; \tau)
		-
 		j(V;\alpha)^{l-1}
 		\tilde{\xi} (l, V\alpha, VM; \tau)
		=
		R_V (l, \alpha, M; \tau).
	\end{equation}
\end{theorem}
To prove the theorem,
	we need the following two lemmas
	which will be proved in Section \ref{Sect_Lemma1} and \ref{Sect_Lemma2}.
\begin{lemma}\label{4.Lemma1}
	Let $T$ and $S$ denote the matrices
		$\begin{pmatrix} 1  &  1 \\ 0  &  1 \end{pmatrix}$
		and
		$\begin{pmatrix} 0  &  -1 \\1  &  0 \end{pmatrix}$
		respectively.
	If $V \in \{  T^{\pm}, S \}$, then (\ref{4.theorem1a}) holds.
\end{lemma}
\begin{lemma}\label{4.Lemma2}
	If $l \in \mathbb{Z}_{>0}$,
		$z \in \mathbb{C}$,
		$V, V_1, V_2 \in \mathrm{SL}_2 (\mathbb{Z})$ with $V = V_2 V_1$,
		and $M \in \mathrm{M}_2(V) \bigcap \mathrm{M}_2(V_1)$, then

	\begin{equation}\label{4.lemma2a}
		R_{V_1} (l, z, M; \tau)
			+ j(V_1, z)^{l-1} R_{V_2} (l, V_1 z, V_1 M; \tau)
		=
		R_V (l, z, M; \tau).
	\end{equation}

\end{lemma}
We prove Theorem \ref{4.Theorem1}.
\begin{proof}[Proof of Theorem \ref{4.Theorem1}]
	Put
		$G = \{  T^{\pm}, S \}$.
	For any positive integer $n$,
		let $U_n$ be a subset of $\mathrm{SL}_2 (\mathbb{Z})$ defined by

	\begin{equation*}
		U_n
		:=
		\{ V \in \mathrm{SL}_2 (\mathbb{Z}) \vert
			\text{there are } n \text{ matrices } V_1, \ldots ,V_n \in G \text{ with } V = V_n V_{n-1} \cdots V_1
		\}.
	\end{equation*}

	If $V$ is a matrix in $\mathrm{SL}_2 (\mathbb{Z})$,
 		there is a positive integer $n$
		with  $V \in U_n$ since $G$ generates $\text{SL}_2 (\mathbb{Z}$) (see \cite[Theorem 2.1]{apostol1}).
	We will prove (\ref{4.theorem1a}) by induction on $n$.
	If $n$ equals $1$, the claim has been shown in Lemma \ref{4.Lemma1}.
	Suppose that it is true in case of $n-1$.
	Let $V$ be a matrix in $U_n$.
	Then there are two matrices $V_1$ and $V_2$ such that
		$V_1 \in G, V_2 \in U_{n-1}$ and $V = V_2 V_1$.
	By virtue of $j(V;\alpha) = j(V_1;\alpha)j(V_2;V_1\alpha)$,

	\begin{eqnarray*}
		& &
		\tilde{\xi} (l, \alpha, M; \tau)
		-
 		j(V;\alpha)^{l-1}
 		\tilde{\xi} (l, V\alpha, VM; \tau) \\
 		& = &
		\tilde{\xi} (l, \alpha, M; \tau)
		-
 		j(V_1;\alpha)^{l-1}
		\tilde{\xi} (l, V_1 \alpha, V_1 M; \tau) \\
		& & +
		j(V_1;\alpha)^{l-1}
 		\Bigl(
			\tilde{\xi} (l, V_1 \alpha, V_1 M; \tau)
		-
 		j(V_2;V_1\alpha)^{l-1}
			\tilde{\xi} (l, V_2 V_1 \alpha, V_2  V_1 M; \tau))
		\Bigr) \\
		& \stackrel{\text{(induction)}}{=} &
			R_{V_1} (l, \alpha, M; \tau) +  j(V_1, \alpha)^{l-1} R_{V_2} (l, V_1 \alpha, V_1 M; \tau)
		\\
		& \stackrel{(\text{\ref{4.lemma2a}})}{=}  &
		 R_V (l, \alpha, M; \tau)
	\end{eqnarray*}

		which gives (\ref{4.theorem1a}) when $M \in \mathrm{M}_2(V) \bigcap \mathrm{M}_2(V_1)$.
	The set $\mathrm{M}_2(V) \bigcap \mathrm{M}_2(V_1)$ is dense in $\mathrm{M}_2 (\mathbb{R})$,
		so we obtain (\ref{4.theorem1a})
		with $M \in \mathrm{M}_2(V)$, which completes the proof.
\end{proof}

\section{A part of Arakawa's transformation formulae and Berndt's Theorem \ref{1.Berndt_Theorem} (ii)} \label{Sect_BerndtArakawaResults}
In this section, by use of Theorem \ref{4.Theorem1} we verify
	a part of Arakawa's transformation formulae,
	or (\ref{1.Arakawa_Equation}) with integers $s < -1$.
Berndt's Theorem \ref{1.Berndt_Theorem} (ii) is induced from \cite[Proposition 2.2]{arakawa2}
	which is derived from Arakawa's transformation formulae with integers $s < -1$,
	thus we obtain Berndt's Theorem \ref{1.Berndt_Theorem} (ii) as a corollary.

Let $\mathrm s(x)$ and $\mathrm c(x)$ be $\sin(2 \pi x)$ and $\cos (2 \pi x)$ respectively.
We firstly introduce the behaviors of the real parts
	of
	$\dfrac{(l+1)!}{(2\pi i)^{l+1}} \tilde{\xi} (s,\alpha, M; \tau)$
	and
	$S_{m,n}(r, x', x, y', y; \tau)$ when $\tau \rightarrow i \infty$.
To do this, we need the functions
	\begin{equation*}
		\begin{split}
			& Cl_l(x)
			:=
			\begin{cases}
				\displaystyle{}
				\sum_{m = 1}^{\infty} \frac{\mathrm s(m x)}{m^l} & ( l \ even),  \\
				\displaystyle{}
				\sum_{m = 1}^{\infty} \frac{\mathrm c (m x)}{m^l} & ( l \ odd),
			\end{cases}
			\\
			& S_{m,n}(r, x, y)
			:=
			\sum_{j (   d(r) )}
			\tilde{B}_m \Bigl(\frac{j + y}{d(r)} \Bigr)
			\tilde{B}_n \Bigl(n(r) \frac{j + y}{d(r)} - x \Bigr), \\
			& \mathfrak{C}(r, x, y)
			:=
			\frac{1}{c} \sideset{}{^\prime}\sum_{j(   c)}
				\cot \Bigl(\pi (\frac{j + y}{d(r)}) \Bigr) \cot \Bigl(\pi  (n(r) \frac{j + y}{d(r)} - x ) \Bigr),
		\end{split}
	\end{equation*}
	where the prime of the last summation means excluding $j$
	such that $a\frac{j+z}{c}-x \in \mathbb Z$ or $b\frac{j+z}{c}-y \in \mathbb Z$.
We note that
	$Cl_l(x)$ are Clausen functions (see \cite{Lewin1}),
	and $S_{m,n}(r, x, y)$ and $\mathfrak{C}(r, x, y)$ respectively are
	the generalized Dedekind-Rademacher sums in \cite{HWZ}
	and the cotangent sum in \cite{dieter1}:
	\begin{equation*}
	\begin{split}
		S_{m,n}(r, x, y) &= S_{m,n} \bigl( \begin{smallmatrix} 1 & n(r) & d(r) \\ 0 & x & y \end{smallmatrix} \bigr),\\
		\mathfrak{C}(r, x, y) &= \mathfrak{c}(1, n(r), d(r) ;0, x, y).
	\end{split}
	\end{equation*}
The behaviors of the real parts are following:
\begin{proposition}\label{DegenerationProperties}
	Let $ \mathrm{Re}\  z $ denote  the real part of a complex number $z$.\\
	{\bf (i)}
	Let $\alpha$ be an irrational real algebraic number,
		$l$ an integer with $l > 2$,
	and $x', x, y', y$ real numbers with $(y', y) \notin \mathbb{Z}^2$.
	Then we have
	\begin{multline} \label{3.proposition1ii}
		\mathrm{Re} \Bigl( \frac{(l+1)!}{(2 \pi i)^{l+1}}  \lim_{\tau \rightarrow i \infty} \tilde{\xi} (l, \alpha, x', x, y', y; \tau) \Bigr) \\
		=
		\begin{cases}
			\displaystyle \frac{(l+1)!}{(2 \pi i)^{l}} \Bigl( \tilde{\xi} (l, \alpha, x, y) - \chi (y) Cl_l(x) \Bigr) &(l\ odd), \\
			\displaystyle \frac{(l+1)!}{(2 \pi i)^{l}} \Bigl( \tilde{\xi} (l, \alpha, x, y) - i \chi (y) Cl_l(x) \Bigr)  & (l\  even).
		\end{cases}
	\end{multline}
	{\bf (ii)}
	Let $r$ be a rational number $r \neq 0$ and
		$x', x, y', y$ real numbers with $\vec{x}, n(r) \vec{x} - d(r) \vec{y} \notin \mathbb{Z}^2$.
	Then we have
	\begin{multline} \label{2.proposition1ii}
		 \mathrm{Re} \lim_{\tau \rightarrow i \infty} S_{m,n}(r, x', x, y', y; \tau)\\
		=
		\begin{cases}
			S_{m,n}(r, x, y) - \frac{1}{4} \mathfrak{C}(r, x, y) & (m= n=1, y, n(r) y - d(r) x \in \mathbb{Z}), \\
			S_{m,n}(r, x, y) & (otherwise).
		\end{cases}
	\end{multline}
\end{proposition}
\begin{proof}
	In order to prove (i),
		we introduce the Fourier expansion for the Jacobi form (see \cite[p. 70]{weil} or \cite[p. 456]{zagier1})
		\begin{equation*}
			F(\xi,X;\tau) := \frac{\theta '(0;\tau) \theta(\xi + X;\tau)}{\theta(\xi ;\tau) \theta(X;\tau)}.
		\end{equation*}
	If $\abs{\mathrm{Im}\ \xi}, \abs{\mathrm{Im}\ X} < \abs{\mathrm{Im}\ \tau}$, then
		\begin{equation*}
			F(\xi,X;\tau)
			=
			\pi \bigl( \cot \pi \xi + \cot \pi X \bigr)
				-2\pi i \sum_{i,j=1}^{\infty} (\mathrm e(i\xi+jX)- \mathrm e(-i\xi-jX)) \mathrm e(ij\tau).
		\end{equation*}
	Let $\xi', \xi, X'$ and $X$ be real numbers with $ \abs{X'} \leq 1/2$ and $ \abs{\xi} < 1$.
	We find from the Fourier expansion that
	\begin{eqnarray}\label{3.proposition1aa}
		& &
		\underline{F} (\xi',\xi; X'\tau +X; \tau)\nonumber \\
		& = &
		\mathrm e(\xi (X'\tau +X) ) F(-\xi' + \xi \tau, X'\tau +X ;\tau) \nonumber \\
		& = &
		\pi  \mathrm e(\xi (X' \tau +X)) \Bigl( \cot \pi(-\xi'+\xi \tau) + \cot \pi(X+ X'\tau ) \Bigr)  \\
		& &
			-2 \pi i \mathrm e(\xi X) \sum_{i,j=1}^{\infty} \Bigl( \mathrm e(-i\xi'+jX) \mathrm e((i + X')(j +\xi)\tau)  \nonumber \\
		& & \qquad \qquad \qquad \qquad \qquad
			- \mathrm e(i\xi'-jX) \mathrm e((i - X')(j - \xi)\tau) \Bigr).\nonumber
	\end{eqnarray}
	If $c = \min \{1 - \abs{\xi}, 1/2 \}$,
		then it follows from $j \pm \xi \geq c j$ and $i \pm X' \geq ci$ $(i, j \in \mathbb{Z})$ that
		\begin{multline} \label{3.proposition1bb}
			\biggl{\lvert}\sum_{i,j=1}^{\infty} \Bigl( \mathrm e(-i\xi'+jX) \mathrm e((i + X')(j +\xi)\tau)\\
				- \mathrm e(i\xi'-jX) \mathrm e((i - X')(j - \xi)\tau) \Bigr) \biggr{\rvert}
			\leq
			2  \sum_{i,j=1}^{\infty} \mathrm e (c^2 ij \tau).
		\end{multline}
	Let the symbols $\iinp{\xi}$ and $\langle \langle \xi \rangle \rangle$ be as in the proof of Lemma \ref{3.Lemma1},
		and $\{ x \}$ as in Section \ref{Sect_Introduction}.
	For any integer $m' \in \mathbb{Z}$,
	let $y_0 = \frp{-y}$ if $m = 0$,
		and let $y_{m'}$ be the real number such that
		$y_{m'} \equiv -y \pmod{1}$,
		$\abs{y_{m'}} <1$ and $y_{m'} \ffrp{\alpha m'} \geq 0$ otherwise.
	It is derived from (\ref{2.key_formula1}), (\ref{2.key_formula2}) and
		$\alpha m' = \iinp{\alpha m'} + \ffrp{\alpha m'}$ that
		\begin{equation*}
			\underline{F} (-\vec{y}; \alpha (\tau m' +m); \tau)
			=
			\mathrm e(-\iinp{\alpha m'} y') \underline{F} (-y',y_{m'};  \ffrp{\alpha m'} \tau + \alpha m; \tau),
		\end{equation*}
		thus we find from (\ref{3.proposition1aa}) and (\ref{3.proposition1bb}) that
		\begin{eqnarray} \label{3.proposition1cc}
			& & \lim_{\tau \rightarrow i \infty} \tilde{\xi} (l, \alpha, x', x, y', y; \tau) \nonumber \\
			& = &
			\lim_{\tau \rightarrow i \infty} \pi
				\sideset{}{^\prime}\sum\limits_{m', m}
				\dfrac{\mathrm e (\vec{m} \cdot \vec{x})}{(\tau m' +m)^{l}} \mathrm e(-\iinp{\alpha m'} y')
				\mathrm e(y_{m'} ( \ffrp{\alpha m'} \tau +\alpha m)) \\
			& & \qquad \qquad \times
				\Bigl(	 \cot \pi(y'+y_{m'} \tau) + \cot \pi(\alpha m+ \ffrp{\alpha m'} \tau )  \Bigr) \nonumber \\
			& \bigl( = &
			\lim_{\tau \rightarrow i \infty} \pi
				\sideset{}{^\prime}\sum\limits_{m', m} S_{m',m} \bigr) \nonumber
		\end{eqnarray}
		where $S_{m',m}$ are the summands in the summation.
	Let $c$ be a positive real number such that
		$\abs{z \cot \pi z} \leq c$
		for any complex number $z$ with $\abs{\mathrm{Re}\ z}, \abs{\mathrm{Im}\  z} \leq 1/2$.
	Since $\cot \pi z = i (\mathrm e(z) + 1)/(\mathrm e(z)-1)$ converges at $\pm i$ when $z$ tends to $\mp i \infty$,
		there is a positive real number $c'$ such that
		\begin{equation*} \abs{ \cot \pi z} \leq \frac{c}{\abs{z}} + c' \end{equation*}
		for $z \in \mathbb{C} \setminus \{ 0 \}$  with $\abs{\mathrm{Re}\ z} \leq 1/2$.
	Set $C = \max \{ c, c'\}$.
	Since $y_{m'} \ffrp{\alpha m'}  \geq 0$, one obtains
		\begin{equation*}
			\sideset{}{^\prime}\sum_{m', m \atop (m' \neq 0)} \abs{S_{m',m}}
			\leq
			C \sideset{}{^\prime}\sum_{m', m \atop (m' \neq 0)} \dfrac{1}{\abs{\tau m' +m}^{l}}
			\Bigl( \frac{1}{\abs{\ffrp{y'}+ y_{m'} \tau}} + \frac{1}{\abs{\ffrp{\alpha m} +\ffrp{\alpha m'} \tau }} + 2 \Bigr)
		\end{equation*}
		which together with (\ref{3.lemma1aa}) and $l > 2$ gives
		\begin{equation}\label{3.proposition1dd}
			\lim\limits_{\tau \rightarrow i \infty} \sideset{}{^\prime}\sum_{m', m \atop (m' \neq 0)} S_{m',m} =0.
		\end{equation}
	Thus we have
		\begin{eqnarray*}
			& &  \lim_{\tau \rightarrow i \infty} \tilde{\xi} (l, \alpha, x', x, y', y; \tau) \\
			& = &
			\pi \sideset{}{^\prime}\sum_m \lim_{\tau \rightarrow i \infty}
				\frac{\mathrm e(mx)}{m^{l}} \mathrm e (m \frp{-y} \alpha)
				\Bigl( \cot \pi(y'+\frp{-y} \tau) + \cot \pi m \alpha \Bigr) \\
			& = &
			\pi \times
			\begin{cases}
				\displaystyle
				\sideset{}{^\prime}\sum_m  \frac{\mathrm e(m x)}{m^{l}} \cot \pi m \alpha
					+
					\cot \pi y' \sideset{}{^\prime}\sum_m \frac{\mathrm e(m x)}{m^{l}} ,
					&
					\quad (y \in \mathbb{Z}), \\
				\displaystyle
				\sideset{}{^\prime}\sum_m  \frac{\mathrm e(m (x+ \{-y\} \alpha))}{m^{l}} \cot \pi m \alpha
					-
					i \sideset{}{^\prime}\sum_m \frac{\mathrm e(x + \{ -y \} \alpha))}{m^{l}},
					&
					\quad (y \notin \mathbb{Z}),
			\end{cases} \\
			& = &
			2 \pi \times
			\begin{cases}
				\displaystyle
				\sum_{m = 1}^{\infty}  \frac{\mathrm c (m x)}{m^{l}} \cot \pi m \alpha
					+
					i \cot \pi y' \sum_{m = 1}^{\infty} \frac{\mathrm s (m x)}{m^{l}},
					&
					(y \in \mathbb{Z}, \ l\ odd), \\
				\displaystyle
				i \sum_{m = 1}^{\infty}  \frac{\mathrm s (m x)}{m^{l}} \cot \pi m \alpha
					+
					\cot \pi y' \sum_{m = 1}^{\infty} \frac{\mathrm c (m x)}{m^{l}},
					&
					(y \in \mathbb{Z}, \ l\ even), \\
				\displaystyle
				\sum_{m = 1}^{\infty}  \frac{\mathrm c (m (x + \{ -y \} \alpha))}{m^{l}} \cot \pi m \alpha
					+
					\sum_{m = 1}^{\infty} \frac{\mathrm s (m (x + \{ -y \} \alpha))}{m^{l}},
					&
					(y \notin \mathbb{Z}, \ l\ odd), \\
				\displaystyle
				i \Bigl( \sum_{m = 1}^{\infty}  \frac{\mathrm s (m (x + \{ -y \} \alpha))}{m^{l}} \cot \pi m \alpha
					-
					\sum_{m = 1}^{\infty} \frac{\mathrm c (m (x + \{ -y \} \alpha))}{m^{l}} \Bigr),
					&
					(y \notin \mathbb{Z}, \ l\ even).
			\end{cases}
		\end{eqnarray*}
	Set $\chi' (x) := 1 - \chi (x)$ where
		$\chi (x)$ is the characteristic function of integers defined in Section \ref{Sect_Introduction},
		that is, $\chi' (x) = 0$ if $x \in \mathbb{Z}$ and $\chi' (x) = 1$ otherwise.
	By the above equation,
		we obtain an expression of the left hand side of (\ref{3.proposition1ii}) up to $(2 \pi i)^{l}/(l+1)!$:
		\begin{eqnarray*}
			& &
			\frac{(2 \pi i)^{l}}{(l+1)!} \times
			\mathrm{Re} \Bigl( \frac{(l+1)!}{(2 \pi i)^{l+1}}
				\lim_{\tau \rightarrow i \infty} \tilde{\xi} (l, \alpha, x', x, y', y; \tau) \Bigr) \\
			& = &
			\begin{cases}
				\displaystyle
				- i \Bigl( \sum_{m = 1}^{\infty}  \frac{\mathrm c (m (x + \{ -y \} \alpha))}{m^{l}} \cot \pi m \alpha
					+
					\chi' (y) \sum_{m = 1}^{\infty} \frac{\mathrm s (m (x + \{ -y \} \alpha))}{m^{l}} \Bigr),
					&
					(l\ odd), \\
				\displaystyle
				\sum_{m = 1}^{\infty}  \frac{\mathrm s (m (x + \{ -y \} \alpha))}{m^{l}} \cot \pi m \alpha
					-
					\chi' (y)  \sum_{m = 1}^{\infty} \frac{\mathrm c (m (x + \{ -y \} \alpha))}{m^{l}},
					&
					(l\ even).
			\end{cases}
		\end{eqnarray*}
	On the other hand, if $y \notin \mathbb{Z}$,
		then it follows from (\ref{2.infiniteExpression_H}) and $\langle -y \rangle = \{ -y \}$ that
		\begin{eqnarray*}
			& &
			\tilde{\xi} (l, \alpha, x, y) \\
			& = &
			\frac{1}{2} \sum_{m \in \mathbb{Z} \atop (m \neq 0)}
				\frac{\mathrm e(m (x + \langle -y \rangle \alpha))}{m^s}
				\Bigl( \frac{\mathrm e(\alpha m) + 1}{\mathrm e(\alpha m) -1} - 1 \Bigr)\\
			& = &
			\frac{1}{2i} \sum_{m \in \mathbb{Z} \atop (m \neq 0)}
				\frac{\mathrm e(m (x + \langle -y \rangle \alpha))}{m^s}
				\cot \pi m \alpha
				-
				\frac{1}{2}
					\sum_{m \in \mathbb{Z} \atop (m \neq 0)}
					\frac{\mathrm e(m (x + \langle -y \rangle \alpha))}{m^s}
				\\
			& = &
			\begin{cases}
				\displaystyle
				- i \Bigl( \sum_{m = 1}^{\infty}  \frac{\mathrm c (m (x + \{ -y \} \alpha))}{m^{l}} \cot \pi m \alpha
					+
					\sum_{m = 1}^{\infty} \frac{\mathrm s (m (x + \{ -y \} \alpha))}{m^{l}} \Bigr),
					&
					(l\ odd), \\
				\displaystyle
				\sum_{m = 1}^{\infty}  \frac{\mathrm s (m (x + \{ -y \} \alpha))}{m^{l}} \cot \pi m \alpha
					-
					\sum_{m = 1}^{\infty} \frac{\mathrm c (m (x + \{ -y \} \alpha))}{m^{l}},
					&
					(l\ even).
			\end{cases}
		\end{eqnarray*}
	If $y \in \mathbb{Z}$, then it does from (\ref{1.definition_H}), (\ref{1.definition_generalized_H}) and $\langle -y \rangle = 1$ that
		\begin{eqnarray*}
			& &
			\tilde{\xi} (l, \alpha, x, y) \\
			& = &
			\sum_{m=1}^{\infty} \frac{\mathrm e(mx)}{m^{l}} \frac{\mathrm e (m \alpha)}{1-\mathrm e(m \alpha)}
				+
				\mathrm e(\frac{1-l}{2})
				\sum_{m=1}^{\infty} \frac{\mathrm e(-mx)}{m^{l}} \frac{\mathrm e (m \alpha)}{1-\mathrm e(m \alpha)}\\
			& = &
			\frac{1}{2} \sum_{m=1}^{\infty} \frac{\mathrm e(mx)}{m^{l}} \Bigl( \frac{\mathrm e(\alpha m) + 1}{\mathrm e(\alpha m) -1} + 1 \Bigr)
				+
				\frac{1}{2} \sum_{m=1}^{\infty} \frac{\mathrm e(-mx)}{(-m)^{l}} \Bigl( \frac{\mathrm e(- \alpha m) + 1}{\mathrm e(- \alpha m) -1} - 1 \Bigr)
				\\
			& = &
			\begin{cases}
				\displaystyle
				- i \sum_{m = 1}^{\infty}  \frac{\mathrm c (m x)}{m^{l}} \cot \pi m \alpha
					+
					Cl_l(x),
					&
					\quad (l\ odd), \\
				\displaystyle
				\sum_{m = 1}^{\infty}  \frac{\mathrm s (m x)}{m^{l}} \cot \pi m \alpha
					+
					i Cl_l(x),
					&
					\quad (l\ even).
			\end{cases}
		\end{eqnarray*}
		Thus we get (i).

	We will verify (ii) next.
	By \cite[PROPOSITION 11]{machide} and (\ref{2.relation}),
		we obtain
		\begin{eqnarray*}
			& & \mathrm{Re} \lim_{\tau \rightarrow i \infty} S_{m,n}(r, x', x, y', y; \tau) \\
			& = &
			\begin{cases}
				S_{m,n}(r, x, y; \tau) - \frac{1}{4} \mathfrak{C}(r, x, y) & (m= n=1, (0, x, y) \in (1, n(r), d(r)) \mathbb{R} + \mathbb{Z}^3), \\
				S_{m,n} \bigl(\begin{smallmatrix} x  \\ y\end{smallmatrix};r \bigr) & (otherwise).
			\end{cases}
		\end{eqnarray*}
	By virtue of \cite[LEMMA 8]{machide},
		the condition $(0, x, y) \in (1, n(r), d(r)) \mathbb{R} + \mathbb{Z}^3$ is equivalent to
		the condition
		$y, n(r) y - d(r) x \in \mathbb{Z}$.
	Thus we obtain (ii) .
\end{proof}

We prove the following proposition, or the remainder of the tasks in this section.
\begin{proposition}
	Theorem \ref{4.Theorem1} induce (\ref{1.Arakawa_Equation}) with integers $s$ less than $-1$
		if $\tau \rightarrow i \infty$.
\end{proposition}
\begin{proof}
	For any integer $l$, let $\psi (l)$ denote $1$ if $l$ is odd, and $i$ if $l$ is even.
	By tending $\tau$ to $i \infty$ in (\ref{4.theorem1a}) up to $(l+1)!/(2 \pi i)^{l+1}$
		and taking its real part,
		we obtain by Proposition \ref{DegenerationProperties} that
		\begin{eqnarray*}
			& &
			\frac{(l+1)!}{(2 \pi i)^{l}}
			\Biggl[
				\Bigl(
					\tilde{\xi} (l, \alpha, x, y) - \psi(l) \chi(y) Cl_l(x)
				\Bigl) \\
			& &
			\quad
				-
	 			j(V;\alpha)^{l-1}
				\Bigl(
	 				\tilde{\xi} (l, V\alpha, ax + by, cx + dy) - \psi(l) \chi(cx + dy) Cl_l(ax + by)
				\Bigr)
			\Biggr]\\
			& = &
			\sum_{k=-1}^{m} \binom{m+1}{k+1}
				(-j(V;\alpha))^k S_{k+1,m-k} ( \frac{d}{c},-x, y).
		\end{eqnarray*}
	On the other hand, we can derive the above equation
		from (\ref{1.Arakawa_Equation})
		with integers $s < -1$ and
			$(x, y) = (q', -p') = (q, -p) \bigl( \begin{smallmatrix} d & -c \\ -b & a\end{smallmatrix} \bigr)$
		in a similar way as the proof of \cite[Proposition 2.2]{arakawa2}.
	We omit the rest of the proof because of similarity.

\end{proof}

\section{Proof of Lemma \ref{4.Lemma1}} \label{Sect_Lemma1}
We give a proof of Lemma \ref{4.Lemma1} in this section.
Let $\sideset{}{}\sum\limits_{m', m}$ (resp. $\sideset{}{^\prime}\sum\limits_{m', m}$).
	means running over all integers $(m', m) \in \mathbb{Z}^2$ (resp. $\mathbb{Z}^2 \setminus \{ (0,0) \})$.
If there is the symbol "$e$" on them,
	we promise that the summation rule follows Cauchy principal value, i.e.,
	\begin{equation}\label{sumRules}
	\begin{split}
		\sideset{}{_e}\sum_{m', m} &= \lim_{M \rightarrow \infty} \sum_{m'=-M}^M \sum_{m = -M}^M, \\
		\sideset{}{_e^\prime}\sum_{m', m} &= \lim_{M \rightarrow \infty} \underset{(m', m) \neq (0, 0)}{\sum_{m'=-M}^M \sum_{m = -M}^M}.
	\end{split}
	\end{equation}
In order to prove Lemma \ref{4.Lemma1},
	we need the equation in the following proposition:
\begin{proposition}\label{5.Proposition1}
	Let $\alpha$ be an irrational real algebraic number,
		and $s$ be a complex number with $\mathrm{Re}(s) >3$.
	If $x', x, y', y \in \mathbb{R} \setminus \mathbb{Z}$, then
		\begin{equation}\label{5.proposition1a}
			\Bigl(
			\sideset{}{_e^\prime}\sum_{m', m} \sideset{}{_e^\prime}\sum_{n', n} -\sideset{}{_e^\prime}\sum_{n', n} \sideset{}{_e^\prime}\sum_{m', m}
			\Bigr)
			\frac{\mathrm e(m'y' + my)}{(\tau m' + m)^s}
				\frac{\mathrm e(n'x' + nx)}{\alpha (\tau m' + m) +\tau n' +n}
			=
			0,
		\end{equation}
		or the order of $(m', m)$ and $(n',n)$ in the sum can be changed.
\end{proposition}
We give the proof of Lemma \ref{4.Lemma1}
before verifying (\ref{5.proposition1a}).
\begin{proof}[Proof of Lemma \ref{4.Lemma1}]
	Let
		$\bigl(
		\begin{smallmatrix}
			x' & x  \\
			y' & y
		\end{smallmatrix} \bigr)
		=
		\bigl(
		\begin{smallmatrix}
			\vec{x}  \\
			\vec{y}
		\end{smallmatrix} \bigr)
 		\in \mathrm{M}_2 (V)$.
	It follows from (\ref{2.key_formula1}) that
		\begin{equation*}
			\tilde{\xi} \bigl(l, T \alpha,
			T
			\bigl(
			\begin{smallmatrix}
				\vec{x}  \\
				\vec{y}
			\end{smallmatrix}
			\bigr)
			;\tau \bigr)
			=
			\tilde{\xi} \bigl(l, \alpha+1,
			\bigl(
			\begin{smallmatrix}
				\vec{x} + \vec{y} \\
				\vec{y}
			\end{smallmatrix}
			\bigr)
			;\tau \bigr)
			=
			\tilde{\xi} \bigl(l, \alpha,
			\bigl(
			\begin{smallmatrix}
				\vec{x}  \\
				\vec{y}
			\end{smallmatrix}
			\bigr)
			;\tau \bigr),
		\end{equation*}
	thus we obtain (\ref{4.theorem1a}) in case of $V = T^{+}$.
Replacing $\alpha$ by $T^{-} \alpha$ on the above equation shows the case $V = T^{-}$.

We will prove (\ref{4.theorem1a}) in case of $V =S$.
Let $l$ be an integer with $l \geq4$,
	and $x', x, y', y \in \mathbb{R} \setminus \mathbb{Z}$.
Since
	$Y^l -X^l = (Y-X) \sum_{k=0}^{l-1} X^k Y^{l-1-k}$,
	\begin{equation*}
		\frac{1}{X^l} \frac{1}{X-Y}
			+  \frac{1}{Y^l} \frac{1}{Y-X}
		=
		- \sum_{k=0}^{l-1} \frac{1}{Y^{k+1}} \frac{1}{X^{l-k}}.
	\end{equation*}
	By replacing $Y$ by $- \frac{1}{\alpha} Y$
	and multiplying $\alpha^{-1}$ both sides,
	we get
	\begin{equation*}
		\frac{1}{X^l} \frac{1}{\alpha X+Y}
		- \alpha^{l-1} \frac{1}{(-Y)^l} \frac{1}{\frac{1}{\alpha}Y+X}
		=
		\sum_{k=0}^{l-1} (-\alpha)^k \frac{1}{Y^{k+1}} \frac{1}{X^{l-k}}.
	\end{equation*}
	By substituting $\tau m' +m$ and $\tau n'+ n$ for $X$ and $Y$ respectively,
		and  ranging $(m', m)$ and $(n', n)$ over $\mathbb{Z}^2 \setminus \{(0,0)\}$
		up to $\mathrm e(m'x'+mx)\mathrm e(n'y'+ny)$ as the  rules (\ref{sumRules}),
		we obtain
		\begin{multline}\label{5.proof_of_lemma4_1aa}
			\sideset{}{_e^\prime}\sum_{n', n} \sideset{}{_e^\prime}\sum_{m', m}
			\Bigl(
			\frac{\mathrm e(m'x'+mx)}{(\tau m' + m)^l}
				\frac{\mathrm e(n'y'+ny)}{\alpha (\tau m' + m) +\tau n' +n}
			\\
			-
			\alpha ^{l-1}
			\frac{\mathrm e(n'y'+ny)}{(-\tau n' - n)^l}
				\frac{\mathrm e(m'x'+mx)}{\frac{1}{\alpha} (\tau n' + n) +\tau m' +m}
			\Bigr)\\
			=
			\sum_{k=0}^{l-1} (-\alpha)^k
			\sideset{}{_e^\prime}\sum_{n', n} \frac{\mathrm e(n'y'+ny)}{(\tau n' + n)^{k+1}}
 			\sideset{}{_e^\prime}\sum_{m', m} \frac{\mathrm e(m'x'+mx)}{(\tau m' + m)^{l-k}} .
		\end{multline}
	On the other hand, L. Kronecker \cite{kronecker1} (see \cite{weil} for a proof) showed that
		\begin{equation*}\label{5.proof_of_lemma4_1bb}
		\begin{split}
			\underline{F} (\vec{x};X;\tau)
			&=
			\sideset{}{_e}\sum_{m',m}
			\frac{\mathrm e(-m'x'-mx)}{X +\tau m' +m}
			=
			- \sideset{}{_e}\sum_{m',m}
			\frac{\mathrm e(m'x'+mx)}{-X +\tau m' +m},
			\\
			B_k(\vec{x};\tau)
			&=
			-\frac{k!}{(2\pi i)^k}
			\sideset{}{_e^\prime}\sum_{m',m}
			\frac{\mathrm e(m'x'+mx)}{(\tau m' +m)^k}.
		\end{split}
		\end{equation*}
	Thus, by (\ref{5.proposition1a}),
		the left hand side of (\ref{5.proof_of_lemma4_1aa}) equals
		\begin{eqnarray*}
			& &
			\sideset{}{_e^\prime}\sum_{m', m}
				\frac{\mathrm e(m'x'+mx)}{(\tau m' + m)^l}
				\Bigl(
					\sideset{}{_e}\sum_{n', n}
					\frac{\mathrm e(n'y'+ny)}{\alpha (\tau m' + m) +\tau n' +n}
					-
					\frac{1}{\alpha (\tau m' + m)}
				\Bigr)
			\\
			& &
			-
			\alpha ^{l-1}
			\sideset{}{_e^\prime}\sum_{n', n}
				\frac{\mathrm e(n'y'+ny)}{(-\tau n' - n)^l}
				\sideset{}{_e}\sum_{m', m}
				\Bigl(
					\frac{\mathrm e(m'x'+mx)}{\frac{1}{\alpha} (\tau n' + n) +\tau m' +m}
					-
					\frac{1}{\frac{1}{\alpha} (\tau n' + n)}
				\Bigr)
			\\
			& = &
 			\sideset{}{_e^\prime}\sum_{m', m}
			\frac{\mathrm e(m'x'+mx)}{(\tau m' + m)^l}
				\underline{F} (- \vec{y}; \alpha (\tau m' + m); \tau)
			+ \frac{1}{\alpha}
			\frac{(2\pi i)^{l+1}}{(l+1)!} B_{l+1}(\vec{x};\tau)
			\\
			& &
			-
			\alpha ^{l-1}  \Bigl(
			\sideset{}{_e^\prime}\sum_{n', n}
			\frac{\mathrm e(-n'y'-ny)}{(\tau n' + n)^l}
				\underline{F}(-\vec{x}; -\frac{1}{\alpha} (\tau n' + n);\tau)
			+ (-1)^l \alpha
			\frac{(2\pi i)^{l+1}}{(l+1)!} B_{l+1}(\vec{y};\tau)
			\Bigr).
		\end{eqnarray*}
	We note that one can omit the symbol "$e$" on $\sideset{}{_e^\prime}\sum_{}$ above
		because the series are absolute converges if $l \geq 4$.
	On the other hand, the right hand side of (\ref{5.proof_of_lemma4_1aa}) equals
		\begin{eqnarray*}
			& &
			\frac{(2\pi i)^{l+1}}{(l+1)!}
				\sum_{k=0}^{l-1} (-\alpha)^k \binom{l+1}{k+1} B_{k+1} (\vec{y};\tau) B_{l-k} (\vec{x};\tau)
		\\
		&=&
			R_S (l, \alpha, M; \tau)
			-
			\frac{(2 \pi i)^{l+1}}{(l+1)!}
			\Bigl(
				-
				\frac{1}{\alpha} B_{l+1} (\vec{x};\tau)
				+
				(-\alpha)^{l} B_{l+1} (\vec{y};\tau)
			\Bigr)
		\end{eqnarray*}
	These yield (\ref{4.theorem1a}) with $l \geq 4$ and $V =S$
		since the set
		$\bigl{\{}
		\bigl(
		\begin{smallmatrix}
			x' & x  \\
			y' & y
		\end{smallmatrix} \bigr)
		\big{\vert}
		x', x, y', y \in \mathbb{R} \setminus \mathbb{Z}
		\bigr{\}}$
		is dense in $\mathrm{M}_2 (V)$.
	In order to get (\ref{4.theorem1a}) with $l=3$ and $V = S$,
		we need differential equations for Kronecker's double series
		and their generating function as follows.
		\begin{eqnarray*}
			& &
			( \tau \frac{\partial}{\partial x '} + \frac{\partial}{\partial x })
			\underline{F} (\vec{x}; X;\tau)
			 =
			2 \pi i X \underline{F} (\vec{x}; X;\tau), \\
			& &
			( \tau \dfrac{\partial}{\partial x '} + \dfrac{\partial}{\partial x })
				 B_m (\vec{x};\tau)
			=
			m B_{m-1} (\vec{x};\tau)
		\end{eqnarray*}
		which are derived from the definitions (\ref{2.definition_K}) and (\ref{3.definition_B}) immediately.
	If $ \mathrm{Re}\  s >3$ and $(z', z) \in \{ (x', x), (y', y) \}$,
		we can easily see from the proof of Lemma \ref{3.Lemma1} that the series
		\begin{equation*}
			\sideset{}{^\prime}\sum\limits_{m', m}
			(\tau \frac{\partial}{\partial z'} + \frac{\partial}{\partial z})
			\Bigl(
			\frac{\mathrm e (m'y'+my)}{(\tau m' +m)^{s}}
			\underline{F} (\vec{x}; \alpha (\tau m' +m); \tau)
			\Bigr)
		\end{equation*}
		absolutely converges,
		thus the term wise differentiation of the above series is possible.
	Therefore applying the differential operator $\tau \dfrac{\partial}{\partial x '} + \dfrac{\partial}{\partial x }$
		to (\ref{4.theorem1a}) with $l = 4$
		yields (\ref{4.theorem1a}) with $l=3$.
\end{proof}
To give a proof of Proposition \ref{5.Proposition1},
	we need the following lemma.
Its proof is based on Siegel's way \cite[pp. 31--32]{siegel1}.
We note that it was used in \cite{katayama1} for proving Rmanujan's formulas with respect to $L$-functions.
\begin{lemma}\label{5.Lemma1}
	Let $ y', y, \lambda$ be real numbers,
		and $N_1', N_1, N_2', N_2, m', m$ integers.
	If $y', y \notin \mathbb{Z}, \lambda >0,
		N_1' \leq N_2', N_1 \leq N_2 $ and $ (m', m) \neq	 (0, 0)$,
		then there is a positive real number $D = D(y', y, \lambda, \alpha, \tau)$
			dependent to $y', y, \lambda, \alpha, \tau$
			and
			independent to $N_1', N_1, N_2', N_2, m', m$
			such that
			\begin{equation} \label{5.lemma1a}
				\Abs{
					\sum_{n' = N_1'}^{N_2'} \sum_{n = N_1}^{N_2}
					\frac{\mathrm e(n'y' + ny)}{\alpha (\tau m' +m) + \tau n' +n}
				}
				\leq
				D \abs{\tau m' + m}^{\lambda +1}.
			\end{equation}
\end{lemma}
\begin{proof}
	We consider the four cases as follows:
		\begin{equation*}
			\begin{split}
				 \mathrm{(i)}  & -\alpha m' \notin [N_1', N_2'] ,\quad -\alpha m \notin [N_1, N_2], \\
				 \mathrm{(ii)}  & -\alpha m' \in [N_1', N_2'], \quad -\alpha m \notin [N_1, N_2], \\
				 \mathrm{(iii)}  & -\alpha m' \notin [N_1', N_2'], \quad -\alpha m \in [N_1, N_2], \\
				 \mathrm{(iv)}  & -\alpha m' \in [N_1', N_2'], \quad -\alpha m \in [N_1, N_2].
			\end{split}
		\end{equation*}
	Firstly, we verify (\ref{5.lemma1a}) in case of (i).
	Set the real numbers $z, a_{n'}' , a_{n}, f_{n', n}$ as
		\begin{equation*}
		\begin{split}
				& a_{n'}' = \frac{\mathrm e (n'y')}{\mathrm e(y')-1},\quad a_{n} = \frac{\mathrm e (ny)}{\mathrm e(y)-1},\\
				& z = \alpha (\tau m' +m),\quad f_{n', n} = z + \tau n' +n.
		\end{split}
		\end{equation*}
	Note that $\mathrm e(y')-1 \neq 0$ and $\mathrm e(y)-1 \neq 0$ because of $y', y \notin \mathbb{Z}$.
	Since $\mathrm e (n'y') = a_{n'+1}' - a_{n'}'$ and
		$\mathrm e (ny) = a_{n+1} - a_{n}$,
		we find that
		\begin{eqnarray*}
		& &
			\sum_{n' = N_1'}^{N_2'} \sum_{n = N_1}^{N_2} \frac{\mathrm e(n'y' + ny)}{z + \tau n' +n}
		\\
		& = &
			\sum_{n' = N_1'}^{N_2'} \sum_{n = N_1}^{N_2} \frac{a'_{n'+1}a_{n+1}-a'_{n'+1}a_{n}-a'_{n'}a_{n+1}+a'_{n'}a_{n}}{f_{n',n}}
		\\
		& = &
			\sum_{n' = N_1' +1}^{N_2' +1} \sum_{n = N_1 +1}^{N_2+1} \frac{a'_{n'}a_{n}}{f_{n'-1,n-1}}
			-
			\sum_{n' = N_1'+1}^{N_2'+1} \sum_{n = N_1}^{N_2} \frac{a'_{n'}a_{n}}{f_{n'-1,n}}

			\\
			& &
			\hspace{0pt}

			-
			\sum_{n' = N_1'}^{N_2'} \sum_{n = N_1+1}^{N_2+1} \frac{a'_{n'}a_{n}}{f_{n',n-1}}
			+
			\sum_{n' = N_1'}^{N_2'} \sum_{n = N_1}^{N_2} \frac{a'_{n'}a_{n}}{f_{n',n}}
		\\
		& = &
			\sum_{n' = N_1'+1}^{N_2'} \sum_{n = N_1+1}^{N_2} a'_{n'}a_{n}
				\Bigl( \frac{1}{f_{n'-1,n-1}}-\frac{1}{f_{n'-1,n}}-\frac{1}{f_{n',n-1}}+\frac{1}{f_{n',n}} \Bigr)

			\\
			& &
			\hspace{0pt}

			-
			a_{N_1} \sum_{n' = N_1'+1}^{N_2'} a'_{n'} \Bigl(\frac{1}{f_{n'-1,N_1}}-\frac{1}{f_{n',N_1}} \Bigr)
			+
			a_{N_2+1} \sum_{n' = N_1'+1}^{N_2'} a'_{n'} \Bigl(\frac{1}{f_{n'-1,N_2}}-\frac{1}{f_{n',N_2}} \Bigr)

			\\
			& &
			\hspace{0pt}

			-
			a_{N_1'} \sum_{n= N_1+1}^{N_2} a_{n} \Bigl(\frac{1}{f_{N_1',n-1}}-\frac{1}{f_{N_1',n}} \Bigr)
			+
			a_{N_2'+1} \sum_{n= N_1+1}^{N_2} a_{n} \Bigl(\frac{1}{f_{N_2',n-1}}-\frac{1}{f_{N_2',n}} \Bigr)

			\\
			& &
			\hspace{0pt}

			+
			\frac{a'_{N_2'+1}a_{N_2+1}}{f_{N_2',N_2}}
			-
			\frac{a'_{N_2'+1}a_{N_1}}{f_{N_2',N_1}}
			-
			\frac{a'_{N_1'}a_{N_2+1}}{f_{N_1',N_2}}
			+
			\frac{a'_{N_1'}a_{N_1}}{f_{N_1',N_1}}.
		\end{eqnarray*}
	If $\xi', \xi, \xi_1', \xi_1, \xi_2', \xi_2$ are real numbers such that
		\begin{equation*}
			\xi_1' \leq \xi_2', \quad
				\xi_1 \leq \xi_2, \quad
 				-\alpha m' \notin [\xi_1', \xi_2'], \quad
				-\alpha m \notin [\xi_1, \xi_2],
		\end{equation*}
 		then elementary integral theory tells us that
 		\begin{equation*}
 			\begin{split}
 				& \frac{1}{f_{\xi_1', \xi}} -  \frac{1}{f_{\xi_2', \xi}} = \tau \int_{\xi_1'}^{\xi_2'} \frac{d \mu'}{(z+ \tau \mu' +\xi)^2}, \\
				& \frac{1}{f_{\xi', \xi_1}} - \frac{1}{f_{\xi', \xi_2}} = \int_{\xi_1}^{\xi_2} \frac{d \mu}{(z+ \tau \xi' +\mu)^2}, \\
 				&  \frac{1}{f_{\xi_1', \xi_1}} -  \frac{1}{f_{\xi_1', \xi_2}} -  \frac{1}{f_{\xi_2', \xi_1}} + \frac{1}{f_{\xi_2', \xi_2}}
				 	= 2\tau \int_{\xi_1'}^{\xi_2'} \int_{\xi_1}^{\xi_2} \frac{d \mu'  d \mu }{(z+ \tau \mu' +\mu)^3}.
 			\end{split}
 		\end{equation*}
	Let $A = A(y', y, \tau)$ be $2(1+\abs{\tau})/\abs{\mathrm e(y')-1} \abs{\mathrm e(y)-1}$
		and $LHS$ be the left hand side of (\ref{5.lemma1a}).
	Since $2(1+\abs{\tau})\abs{a_{n'}'}\abs{a_n}  \leq A$
		if $N_1' \leq n' \leq N_2' + 1$ and $N_1 \leq n \leq N_2 + 1$,
		it follows that
		\begin{eqnarray}\label{5.lemma1cc}
			LHS
		&\leq&
			A \Bigl(
				\int_{N_1'}^{N_2'} \int_{N_1}^{N_2}
 				\frac{d \mu'  d \mu }{\abs{z+ \tau \mu' +\mu}^3}

			\\
			& &
			\hspace{20pt}

			+
			\int_{N_1'}^{N_2'} \frac{d \mu'}{\abs{z+ \tau \mu' +N_1}^2}
			+
			\int_{N_1'}^{N_2'} \frac{d \mu'}{\abs{z+ \tau \mu' +N_2}^2}

			\nonumber
			\\
			& &
			\hspace{20pt}

			+
			\int_{N_1}^{N_2} \frac{d \mu}{\abs{z+ \tau N_1' +\mu}^2}
			+
			\int_{N_1}^{N_2} \frac{d \mu}{\abs{z+ \tau N_2' +\mu}^2}

			\nonumber
			\\
			& &
			\hspace{20pt}

			+
			\frac{1}{\abs{f_{N_1', N_1}}}
			+
			\frac{1}{\abs{f_{N_1', N_2}}}
			+
			\frac{1}{\abs{f_{N_2', N_1}}}
			+
			\frac{1}{\abs{f_{N_2', N_2}}}
			\Bigr).

			\nonumber

		\end{eqnarray}
	Put $t' = \alpha m'$ and $t = \alpha m$.
	It follows from (\ref{3.lemma1aa}) that
		\begin{eqnarray*}
		& &
			\int_{N_1'}^{N_2'} \int_{N_1}^{N_2}
 				\frac{d \mu'  d \mu }{\abs{z+ \tau \mu' +\mu}^3}
		%
		\\
		&\leq&
		%
			\frac{1}{\bigl(2 (\lvert \tau \rvert - \lvert \mathrm{Re}\ \tau \rvert) \bigr)^{3/2}}
				\int_{N_1'}^{N_2'} \int_{N_1}^{N_2}
 				\frac{d \mu'  d \mu }{\abs{t' + \mu'}^{3/2} \abs{t + \mu}^{3/2}}.
		\end{eqnarray*}
	Thus, because $-t' \notin [N_1', N_2']$ and $-t \notin [N_1, N_2]$ by the condition (i),
		there are a real number  $b = b(\tau)$ depending on $\tau$ such that
		\begin{eqnarray*}
		& &
			\int_{N_1'}^{N_2'} \int_{N_1}^{N_2}
 			\frac{d \mu'  d \mu }{\abs{z+ \tau \mu' +\mu}^3}
		%
		\\
		&\leq&
		%
			b \Bigl(
			 \frac{1}{\abs{t' + N_2'}^{1/2} \abs{t + N_2}^{1/2}}
			 	+  \frac{1}{\abs{t' + N_2'}^{1/2} \abs{t + N_1}^{1/2}}

			\\
			& & \hspace{20pt}

				+  \frac{1}{\abs{t' + N_1'}^{1/2} \abs{t + N_2}^{1/2}}
				+  \frac{1}{\abs{t' + N_1'}^{1/2} \abs{t + N_1}^{1/2}}
			\Bigr ).
		\end{eqnarray*}
	Since
		\begin{equation*}
			\int \frac{1}{y^2 + x^2} dx
			=
			\frac{1}{y} \arctan \frac{x}{y},
		\end{equation*}
		we have
		\begin{eqnarray*}
		& &
			\int_{N_1}^{N_2} \frac{d \mu}{\abs{z+ \tau N' +\mu}^2}
		%
		\\
		&=&

			\int_{N_1}^{N_2} \frac{d \mu}{\bigl(t+\mu+(\mathrm{Re}\ \tau)(t'+N') \bigr)^2 +(\mathrm{Im}\ \tau)^2(t'+N')^2}
		%
		\\
		&=&
		%
			\frac{1}{ \abs{\mathrm{Im}\ \tau} \abs{t'+N'}}
				\biggl[
					\arctan
						\frac{t+\mu+ (\mathrm{Re}\ \tau) (t'+N')} {\abs{\mathrm{Im}\ \tau} \abs{t'+N'}}
				\biggr]_{\mu= N_1}^{\mu = N_2}
		%
		\\
		&\leq&
		%
			\frac{\pi}{ \abs{\mathrm{Im}\ \tau} \abs{t'+N'}}.
		\end{eqnarray*}
	In a similar way, we get
		\begin{equation*}
			\int_{N_1'}^{N_2'} \frac{d \mu'}{\abs{z+ \tau \mu' +N}^2}
		%
		\leq
		%
			\frac{\pi}{\abs{\tau}^2 \abs{\mathrm{Im}\ \frac{1}{\tau}} \abs{t+N}}.
		\end{equation*}
	Furthermore we easily see from (\ref{3.lemma1aa}) that
		\begin{equation*}
			\frac{1}{\abs{f_{N_{j'}', N_j}}}
			=
			\frac{1}{\abs{\tau t'+t+\tau N_{j'}' + N_{j}}}
			\leq
			\Bigl( \frac{1}{2(\lvert \tau \rvert - \lvert \mathrm{Re}\ \tau \rvert)\abs{t'+N_{j'}'}\abs{t+N_{j}}} \Bigr)^{1/2}
		\end{equation*}
		if $j', j = 1,2$.
	Therefore we conclude from (\ref{5.lemma1cc}) that
		there is a positive real number $B=B(y', y,\tau)$ depending on $y', y,\tau$ such that
		\begin{multline*}
			LHS
			\leq
			B \Bigl(
				\bigl(\frac{1}{\abs{\alpha m' + N_1'}^{1/2}}+ \frac{1}{\abs{\alpha m' + N_2'}^{1/2}} \bigr)
				\bigl(\frac{1}{\abs{\alpha m + N_1}^{1/2}}+ \frac{1}{\abs{\alpha m + N_2}^{1/2}} \bigr) \\
				+
				\frac{1}{\abs{\alpha m' + N_1'}} + \frac{1}{\abs{\alpha m' + N_2'}}
				+
				\frac{1}{\abs{\alpha m + N_1}} + \frac{1}{\abs{\alpha m + N_2}}
			\Bigr).
		\end{multline*}
	On the other hand, by the theorem of Thue-Siegel-Roth in
		the diophantine approximation theory,
		there is a positive real number $C = C (\alpha, \lambda) > 1$ depending on $\alpha, \lambda$ such that
		\begin{equation*}
			\Abs{\alpha - \frac{k}{l}} > \frac{1}{C l^{\lambda +2}} \quad ( k, l \in \mathbb{Z}, \ l  > 0).
		\end{equation*}
	We see from this that
		\begin{equation*}
			\frac{1}{\abs{\alpha l + k}}
			<
			C \abs{l}^{\lambda +1}
		\end{equation*}
		for $k, l \in \mathbb{Z}$ with $(k,l) \neq (0,0)$,
		which deduces
		from $C \abs{l}^{\lambda +1}< C ( \abs{l}^{\lambda +1} +1) \leq C ( \abs{l}^{(\lambda +1)/2} +1)^2$
		that
		\begin{equation*}
			\frac{1}{\abs{\alpha l + k}^{1/2}}
			<
			C^{1/2} ( \abs{l}^{(\lambda +1)/2} +1).
		\end{equation*}
	Thus, by (\ref{3.lemma1aa}), we have
		\begin{eqnarray*}
			& &
			LHS
			\\
			& \leq &
			B
			\Bigl(
				4C(\abs{m'}^{(\lambda +1)/2} + 1)(\abs{m}^{(\lambda +1)/2} + 1)
				+
				2C(\abs{m'}^{\lambda +1} + \abs{m}^{\lambda +1} + 2)
			\Bigr)
			\\
			&\leq&
			8BC \Bigl(
				\abs{m'}^{(\lambda +1)/2} \abs{m}^{(\lambda +1)/2}
				+
				\abs{m'}^{\lambda +1} + \abs{m}^{\lambda +1} +1
				\Bigr)
			\\
			& \leq &
			8BC \Bigl(
				\frac{1}{\{2(\lvert \tau \rvert - \lvert \mathrm{Re}\ \tau \rvert)\}^{(\lambda+1)/2}}
				+
				\frac{2}{\mathrm{Im}(\tau)^{\lambda + 1}}
				+
				C'
				\Bigr)
				\abs{\tau m' +m}^{\lambda + 1}
		\end{eqnarray*}
		where $C'$ is the maximum of $\{ 1/(\tau k' + k)  \vert (k', k) \in \mathbb{Z}^2 \setminus (0,0) \}$.
 	Because $B, C$ and $C'$ are dependent to only $y', y, \lambda, \alpha, \tau$,
		we obtain (\ref{5.lemma1a}) in case of (i).

	We consider the case (ii).
	We can easily obtain (\ref{5.lemma1a}) if $(m', N_1', N_2) = (0, 0, 0)$,
		so we assume that  $(m', N_1', N_2) \neq (0, 0, 0)$.
	Let $[x]$ denote the integer part of a real number $x$, i.e., $0 \leq x- [x] < 1$.
	Suppose that  $m' = 0$.
	Then $ -\alpha m' \notin [N_1', -1], - \alpha m' \notin [1, N_2']$, $- \alpha m' \in [0,0]$.
	Set $g_{n',n}:= \mathrm e(n'y' + ny)/ (\alpha (\tau m' +m) + \tau n' +n)$.
	We have
		\begin{eqnarray*}
			\Abs{
				\sum_{n' = N_1'}^{N_2'} \sum_{n = N_1}^{N_2} g_{n',n}
			}
		\leq
			 \Abs{
			 	\sum_{n' = N_1'}^{-1} \sum_{n = N_1}^{N_2} g_{n',n}
			 }
			 +
			 \Abs{
			 	\sum_{n' = 1}^{N_2'} \sum_{n = N_1}^{N_2} g_{n',n}
			 }
			 +
			 \Abs{
			 	\sum_{n = N_1}^{N_2} g_{0,n}
			 }
		\end{eqnarray*}
		where empty sums mean $0$.
	We see from the proof of (i) that the first and second terms in the last formula
		have the upper bound $\lvert \tau m'+m \rvert^{\lambda+1}$ up to constants depending only $y',y,\lambda, \alpha, \tau$,
		and from Siegel's way \cite[pp. 31--32]{siegel1} (which is also used in \cite[proof of Lemma 2]{katayama1}) that
		the third term
		$\lvert \sum_{n = N_1}^{N_2} g_{0,n} \rvert = \lvert \sum_{n = N_1}^{N_2} \mathrm e(ny)/(m+n) \rvert $
		have the upper bound $\lvert m \rvert^{\lambda+1}$ up to constant depending only $y, \lambda$.
	Therefore we verify the case (ii) with $m'=0$.
	If $m' \neq 0$, then
		$-\alpha m' \notin [N_1', [-\alpha m']]$ and $-\alpha m' \notin [ [-\alpha m'] +1, N_2']$,
		thus
		we can prove this case as same as the above case that $m'=0$.
	Therefore  we completes the proof of (\ref{5.lemma1a}) in case of (ii).

	We can prove similarly the other cases (iii) and (iv) as the case (ii), thus we omit the proofs.
\end{proof}

In the end of the section, We prove Proposition \ref{5.Proposition1}.
\begin{proof}[Proof of Proposition \ref{5.Proposition1}]
	Since $\mathrm{Re}\ s > 3$, there are positive real numbers $\lambda$ and $s_0$
		with $\mathrm{Re}\ s - \lambda -1 > s_0>2$.
	Let $\epsilon$ be an arbitrary positive real number.
	Since the series $\displaystyle{} \sideset{}{^\prime}\sum\limits_{m', m} \dfrac{1}{\abs{\tau m' +m}^{s_0}}$
		converges, there is a positive integer $M$ satisfying
		\begin{equation*}
			\sum_{m' \geq M \atop or\ m \geq M}   \dfrac{1}{\abs{\tau m' +m}^{s_0}}
			 <
			 \epsilon
		\end{equation*}
	where the summation runs over $(m', m) \in \mathbb{Z}$ such that $m' \geq M$ or $m \geq M$.
	Moreover since the series $\displaystyle{} \sideset{}{_e^\prime}\sum\limits_{n', n} \dfrac{\mathrm e (n'y' + ny)}{\alpha (\tau m' + m) +\tau n' +n}$ converges
		for every $(m', m) \in \mathbb{Z}^2 \setminus (0, 0)$,
		there is a positive integer $N_0$ such that
		\begin{equation*}
			 \quad
			\Abs{\sum_{\abs{n'}  \geq N \atop or\  \abs{n} \geq N}
				\frac{\mathrm e(n'y' + ny)}{\alpha (\tau m' + m) +\tau n' +n}} < \epsilon
		\end{equation*}
		if $N > N_0$ and $(m', m) \in \{ \mathbb{Z}^2 \setminus (0,0) \  \vert \ \abs{m'}, \abs{m} \leq M \}$.
	Thus we find from (\ref{5.lemma1a}) that
		there is a positive real number $D$ not depending on $N$ and $\epsilon$ such that
 		\begin{eqnarray*}\label{5.proposition1aa}
			& &
			\Abs{
				\Bigl(
				\sideset{}{_e^\prime}\sum_{m', m} \sideset{}{_e^\prime}\sum_{n', n} -\sideset{}{^\prime}
					\sum_{\abs{n'}< N \atop and\  \abs{ n} <N} \sideset{}{_e^\prime}\sum_{m', m}
				\Bigr)
				\frac{\mathrm e(m' x' + mx)}{(\tau m' + m)^s}
					\frac{\mathrm e(n'y' + ny)}{\alpha (\tau m' + m) +\tau n' +n}
			}\\
			&= &
			\Abs{
				\Bigl(
					\sum_{\abs{m'}  \geq M \atop or\  \abs{m} \geq M}
					+
					\sideset{}{^\prime}\sum_{\abs{m'}< M \atop and\  \abs{ m} <M}
				\Bigr)
				\sum_{\abs{n'}  \geq N \atop or\  \abs{n} \geq N}
					\frac{\mathrm e(m' x' + mx)}{(\tau m' + m)^s}
					\frac{\mathrm e(n'y' + ny)}{\alpha (\tau m' + m) +\tau n' +n}
			}\\
			& \leq &
			D \sum_{\abs{m'}  \geq M \atop or\  \abs{m} \geq M}
				\frac{1}{\abs{\tau m' + m}^s}
					\abs{\tau m' + m}^{\lambda +1}
				+
				\epsilon
					\sideset{}{^\prime}\sum_{\abs{m'}< M \atop and\  \abs{ m} <M}
					\frac{1}{\abs{\tau m' + m}^s} \\
			& \leq &
			\epsilon
				\Bigl(
					D + \sideset{}{^\prime}\sum_{m' , m } \frac{1}{\abs{\tau m' + m}^s}
				\Bigr)
		\end{eqnarray*}
		if $N > N_0$.
		By virtue of the arbitrariness of $\epsilon$, we obtain (\ref{5.proposition1a}).
\end{proof}

\section{Proof of Lemma \ref{4.Lemma2}} \label{Sect_Lemma2}
In order to prove Lemma \ref{4.Lemma2},
	we give the following formulae
	for the elliptic Dedekind-Rademacher sums $S_{1,l} (r,M; \tau)$.

\begin{theorem}\label{6.Theorem1}
	Let $r $ be a rational number, and $l$ a positive integer.
	The maps $n: \mathbb{Q} \to \mathbb{Z}$ and $d: \mathbb{Q} \to \mathbb{Z}_{>0}$ are
		as (\ref{3.Def_ndNumber}).
	Let $\mathrm{M}_2 (r)$ be the subset of $\mathrm{M}_2 (\mathbb{R})$
		defined by
		\begin{equation}\label{6.definition_SM}
			\mathrm{M}_2 (r)
			:=
			\biggl{\{}
			\begin{pmatrix}
				x' & x  \\
				y' & y
			\end{pmatrix}
			=
			\begin{pmatrix}
				\vec{x}  \\
				\vec{y}
			\end{pmatrix}
			\in \mathrm{M}_2 (\mathbb{R})
			\bigg{\vert}
			\vec{x}, d(r) \vec{x} - n(r) \vec{y} \notin \mathbb{Z}^2
		\biggr{\}}.
		\end{equation}
	The relation of $\mathrm{M}_2 (V)$ defined in (\ref{4.definition_SM})
		is that $\mathrm{M}_2 (-d/c) = \mathrm{M}_2 (V)$ if $c \neq 0$.

	If $V = \bigl(
		\begin{smallmatrix}
		        a  &  b       \\
		        c  &  d
		\end{smallmatrix} \bigr) \in \mathrm{SL}_2 (\mathbb{Z})$ with
		 $j(V; r) \neq 0$ and
		 $M  \in \mathrm{M}_2( V) \bigcap \mathrm{M}_2 (r)$,
		 then
		\begin{multline}\label{6.theorem1a}
			S_{1,l} (r, M;  \tau)
				- j(V; r)^{l-1}
				S_{1,l} (V r, VM; \tau)\\
				=
				\frac{(-1)^{l} l!}{(2\pi i)^{l+1}}
				R_V(l, r, M; \tau)
					-
					\frac{(-1)^{l} l}{l+1}
					\frac{c}{j (V;r)}
					\frac{B_{l+1}(d(r) \vec{x} - n(r) \vec{y};\tau)}{d(r)^{l+1}}.
		\end{multline}
\end{theorem}
\begin{remark}
Theorem \ref{6.Theorem1} reproduces a part of Halbritter's result \cite[Theorem 2]{halbritter} when $\tau \rightarrow i \infty$.
\end{remark}
For any
	$V = \bigl(
	\begin{smallmatrix}
	          a & b      \\
	          c & d
	\end{smallmatrix} \bigr)
	\in \mathrm{SL}_2 (\mathbb{Z})$,
	set $\tilde{j} (V; r) := \dfrac{c}{j(V;r)} = \dfrac{d}{dr} j(V;r)$.
Before proving the theorem,
 	we give the proof Lemma \ref{4.Lemma2} by use of (\ref{6.theorem1a}).
\begin{proof}[Proof of Lemma \ref{4.Lemma2}]
	Because $\det V = ad - bc = 1$ and $Vr = (ar + b)/(cr + d) = (a n(r) + b d(r))/(c n(r) + d d(r) )$,
		we have
		\begin{equation*}
			\gcd (a n(r) + b d(r), c n(r) + d d(r) ) =1.
		\end{equation*}
	Since $j(V;r)=cr+d=(cn(r)+dd(r))/d(r)$,
		we obtain
		\begin{equation}\label{6.proof_Lemma4.2aa}
			\begin{pmatrix} n(Vr)  \\ d(Vr) \end{pmatrix}
				=
				\mathrm{sgn}(j(V; r)) V \begin{pmatrix} n(r)  \\ d(r) \end{pmatrix}
				\end{equation}
		where $\mathrm{sgn}(j(V; r))$ equals $1$ if $j(V; r) >0$, and $-1$ if $j(V; r) < 0$.
	We also get
		\begin{equation}\label{6.proof_Lemma4.2bb}
			n(r) \eta - d(r) \xi
			=
			\mathrm{sgn}(j(V; r)) \bigl( n(Vr) (c\xi +d\eta) - d(Vr) (a\xi +b\eta) \bigr)
		\end{equation}
		for complex numbers $\xi, \eta$ by virtue of
		\begin{equation*}
			\det
				\begin{pmatrix} n(V r)  &  a\xi +b\eta \\ d(V r)  &  c\xi +d\eta    \end{pmatrix}
			=
			\mathrm{sgn}(j(V; r))  \det
				 V \begin{pmatrix} n(r)  &  \xi \\ d(r)  &  \eta     \end{pmatrix}.
		\end{equation*}
	Since
	\begin{eqnarray*}
		0
		&=&
		\Bigl{\{}
				S_{1,l} (r, M;  \tau) -  j(V; r)^{l-1} S_{1,l} (V r, VM; \tau)
			\Bigr{\}}
		\\
		& &
			-
			 \Bigl{\{}
				S_{1,l} (r, M;  \tau) -  j(V_1; r)^{l-1} S_{1,l} (V_1 r, V_1M; \tau)
		\\
		& &\qquad
				+
				j(V_1; r)^{l-1}
					\Bigl(
						S_{1,l} (V_1 r, V_1M; \tau)  - j(V_2; V_1 r)^{l-1} S_{1,l} (V_2 V_1 r, V_2 V_1 M; \tau)
					\Bigr)
			\Bigr{\}},
	\end{eqnarray*}
	we find from (\ref{6.theorem1a}), (\ref{6.proof_Lemma4.2aa}) and (\ref{6.proof_Lemma4.2bb}) that
		\begin{eqnarray*}\label{6.proof_Lemma4.2cc}
			& &
			 \frac{(l+1)!}{(2\pi i)^{l+1} l}
			 \Bigl( R_V (l, r,  M; \tau)- R_{V_1} (l, r, M; \tau)
				- j(V_1, r)^{l-1} R_{V_2} (l, V_1 r, V_1 M ; \tau) \Bigr) \\
			& = &
			\tilde{j} (V; r)  \frac{ B_{l+1}(d(r) \vec{x} - n(r) \vec{y} ;\tau)}{d(r)^{l+1}}
			-
			\tilde{j} (V_1; r) \frac{ B_{l+1}(d(r) \vec{x} - n(r) \vec{y} ;\tau)}{d(r)^{l+1}} \\
			&  &
				\qquad - j(V_1;r)^{l-1} \tilde{j} (V_2; V_1r) \frac{ B_{l+1}(d(V_1 r) (a_1 \vec{x} + b_1 \vec{y}) - n(V_1 r) (c_1 \vec{x} + d_1 \vec{y}) ;\tau)}{d(V_1r)^{l+1}} \\
			& = &
			 \frac{ B_{l+1}(d(r) \vec{x} - n(r) \vec{y} ;\tau)}{d(r)^{l+1}}
				\Bigl{(}
					\tilde{j} (V; r) - \tilde{j} (V_1; r)
						- \frac{1}{j(V_1, r)^{2} } \tilde{j} (V_2; V_1r)
				\Bigr{)}
		\end{eqnarray*}
		where $V_1 = \bigl( \begin{smallmatrix} a_1 & b_1 \\ c_1 & d_1\end{smallmatrix} \bigr)$.
	By differentiating $\log j(V;r) = \log j(V_1;r) j(V_2;V_1r)$ on $r$,
		we obtain
		\begin{equation*}
			\tilde{j} (V; r) - \tilde{j} (V_1; r)  - \frac{1}{j(V_1, r)^{2} } \tilde{j} (V_2; V_1r) = 0.
		\end{equation*}
	Thus (\ref{4.lemma2a}) follows if $z=r$ is any rational number.
	Since $R_V (l, z, M; \tau)$ are polynomials in $z$,
		(\ref{4.lemma2a}) holds for any complex number $z \in \mathbb{C}$.
\end{proof}
In order to verify (\ref{6.theorem1a}),
	we establish transformation formulae of the functions $\widehat{S}_{1,l}(r, M;   X  ,Y ; \tau)$ defined below
	instead of $S_{1,l} (r,M; \tau)$.
The coefficients of $X^0, Y^0$ on the functions $\widehat{S}_{1,l}(r, M; X, Y ; \tau)$
	nearly equal $S_{1,l} (r,M; \tau)$ (see Lemma \ref{6.Lemma3} below),
	thus the transformation formulae of $\widehat{S}_{1,l}(r, M;  X, Y ; \tau)$ give rise to those of  $S_{1,l} (r,M; \tau)$.

Let us define $\widehat{S}_{1,l}(r, M;  X , Y ; \tau)$.
Assume that
	\begin{equation*}
		l \in \mathbb{Z}_{>0}, \quad
		r = n(r)/ d(r) \in \mathbb{Q}, \quad
		M = \bigl(\begin{smallmatrix} x' & x \\ y' & y \end{smallmatrix} \bigr)
			= \bigl(\begin{smallmatrix} \vec{x}  \\ \vec{y} \end{smallmatrix} \bigr)
				\in \mathrm{M}_2 (\mathbb{R})
	\end{equation*}
	and $X, Y$ are complex variables.
If $M \in \mathrm{M}_2 (r)$,
	then $\widehat{S}_{1,l}(r, M; X, Y ; \tau)$ are defined by
	\begin{multline}\label{8.def_S}
		\widehat{S}_{1,l}(r, M;  X  , Y ; r)
		\\
		:=
		\frac{1}{d(r)}
			\sum_{j', j (  d(r))}
			\underline{F} \Bigl(\frac{\vec{j} + \vec{y}}{d(r)}; n(r) Y -d(r) X;  \tau \Bigr)
				\underline{F}^{(l-1)} \Bigl(n(r) \frac{\vec{j} + \vec{y}}{d(r)} -\vec{x};-Y; \tau \Bigr)
	\end{multline}
	where
	$\underline{F} ^{(m)} (\vec{x};X;\tau)$ denotes
	$\dfrac{1}{(2\pi i)^m} \Bigl ( \dfrac{\partial}{\partial X} \Bigr )^m
		\underline{F} (\vec{x};X;\tau)$.
	We see from (\ref{2.key_formula1}) that $\widehat{S}_{1,l}(r; M,  X , Y ; r)$
		have the following periodicities with respect to $X$ and $Y$:
		\begin{equation}\label{6.periodicity_S}
			\begin{split}
			\widehat{S}_{1,l}(r, M;   X+1  , Y ; \tau)
			& =
			\mathrm e(-y) \widehat{S}_{1,l}(r, M;   X  , Y ; \tau), \\
			\widehat{S}_{1,l}(r, M;  X  + \tau , Y ; \tau)
			& =
			\mathrm e(-y') \widehat{S}_{1,l}(r, M;   X  , Y ; \tau), \\
			\widehat{S}_{1,l}(r, M;  X,  Y+1 ; \tau)
			& =
			\mathrm e(x) \widehat{S}_{1,l}(r, M;  X  , Y ; \tau),  \\
			\widehat{S}_{1,l}(r, M;  X  , Y+\tau ; \tau)
			& =
			\mathrm e(x') \widehat{S}_{1,l}(r, M;   X  , Y ; \tau).
			\end{split}
		\end{equation}

For the purpose of establishing
	 transformation formulae of
	$\widehat{S}_{1,l}(r, M;  X , Y ; \tau)$,
	we also need functions
	$\widehat{R}_V (l, r, M;  X  , Y  ; \tau)$
	whose coefficients of $X^0,Y^0$ nearly equal $R_V (l, r, M; \tau)$ (see Lemma \ref{6.Lemma3}):
If $V = \bigl(
	\begin{smallmatrix}
	        a  &  b       \\
	        c  &  d
	\end{smallmatrix} \bigr) \in \mathrm{SL}_2 (\mathbb{Z})$
	and $M
		= \bigl( \begin{smallmatrix} \vec{x}  \\ \vec{y}\end{smallmatrix} \bigr)  \in \mathrm{M}_2( V)$,
	then the functions $\widehat{R}_V (l, r, M; X  , Y ; \tau)$ are defined by
	\begin{multline} \label{6.definition_R}
		\widehat{R}_V (l, r, M; X  , Y ; \tau)\\
		:=
		\begin{cases} \displaystyle{}
			(-1)^l \sum_{k=0}^{l-1}
			\binom{l-1}{k}
			\bigl(
				- j (V, r)\bigr)^k
				\widehat{S}_{k+1,l-k} \biggl(\frac{d}{c}, \bigl(\begin{smallmatrix} -\vec{x}  \\ \vec{y} \end{smallmatrix} \bigr); X  , Y ; \tau \biggr)
			& ( c \neq 0), \\
			0 & (c =0),
		\end{cases}
	\end{multline}
	where $\widehat{S}_{k+1,l-k} \biggl(\dfrac{d}{c},\bigl(\begin{smallmatrix} -\vec{x}  \\ \vec{y} \end{smallmatrix} \bigr); X,  Y ; \tau \biggr) $
	denotes
	\begin{equation}
		\frac{1}{c} \sum_{j', j (  \abs{c})}
			\underline{F}^{(k)} (\frac{\vec{j} +\vec{y}}{c}; -cX -dY ;\tau)
			\underline{F}^{(l-1-k)} (d\frac{\vec{j} +\vec{y}}{c} + \vec{x}; Y ;\tau)
	\end{equation}
	and the summation takes over all elements in $(\mathbb{Z} / \abs{c} \mathbb{Z})^2$.
We see from (\ref{2.key_formula2}) that
	\begin{equation}
		\widehat{R}_{-V} (l, r, M; X  , Y ; \tau)
		=
		\widehat{R}_V (l, r, M; X  , Y ; \tau).
	\end{equation}
For a proof of the transformation formulae of $\widehat{S}_{1,l}(r, M; X , Y ; \tau)$,
we prepare two lemmas.
\begin{lemma}\label{6.Lemma1}
	Assume that $r = n(r)/ d(r) \in \mathbb{Q}$,
		$\bigl(\begin{smallmatrix} a  &  b  \\ c  &  d    \end{smallmatrix} \bigr) \in \mathrm{SL}_2(\mathbb{Z})$
		and
		$\bigl(\begin{smallmatrix} m'  &  n' \\m  &  n     \end{smallmatrix} \bigr)$,
		$\bigl(\begin{smallmatrix} m_0'  &  n_0' \\ m_0  &  n_0  \end{smallmatrix} \bigr) \in \mathrm{M}_2(\mathbb{Z})$.
	If $\bigl( \begin{smallmatrix} m'  &  n' \\ m  &  n    \end{smallmatrix} \bigr)
		=  \bigl( \begin{smallmatrix} m_0'  &  n_0' \\ m_0 &  n_0  \end{smallmatrix} \bigr)
			\bigl( \begin{smallmatrix} a  &  b \\ c  &  d    \end{smallmatrix} \bigr)$,
		then
		\begin{equation}\label{6.lemma1a}
			r (\tau m' + m) + \tau n' + n
			=
			(ar +b) (\tau m_0' +m_0) + (cr +d) (\tau n_0' +n_0).
		\end{equation}
	In particular, we obtain the equalities of lattice
		\begin{equation}\label{6.lemma1b}
			\begin{split}
			\frac{1}{d(r)} (\tau \mathbb{Z} +\mathbb{Z})
			& =
			r (\tau \mathbb{Z} +\mathbb{Z}) + \tau \mathbb{Z} +\mathbb{Z} \\
			& =
			(ar +b)  (\tau \mathbb{Z} +\mathbb{Z})
				+
				(c r +d)  (\tau \mathbb{Z} +\mathbb{Z}).
			\end{split}
		\end{equation}
\end{lemma}
\begin{proof}
	A direct calculation verifies (\ref{6.lemma1a}).
	The first equality in (\ref{6.lemma1b}) follows from
		the fact that $\gcd (n(r), d(r)) =1$.
	The second from (\ref{6.lemma1a}) and
		$\mathrm{det} \bigl(\begin{smallmatrix} a  &  b \\ c  &  d  \end{smallmatrix} \bigr) = 1$.
\end{proof}
\begin{lemma}\label{6.Lemma2}
	Let $\vec{g} \cdot \vec{h} := g'h' + gh$ be the inner product
		for two vectors $\vec{g} = (g', g)$ and $\vec{h} = (h', h) \in \mathbb{R}^2$.
	Assume that $c \in \mathbb{Z} \setminus \{0\}$, $l \in \mathbb{Z}_{\geq 0}$,
		and $\vec{i} = (i', i) \in \mathbb{Z}^2$.
	Then we have
		\begin{eqnarray}
			& &
			\underline{F}^{(l)} (\vec{x}; X + \frac{\tau i' +i}{c};\tau)
			=
			c^{l-1} \sum_{j', j ( \abs{c})}
				\mathrm e(\vec{i}\cdot  \frac{\vec{j}+\vec{x}}{c})
				\underline{F}^{(l)}(\frac{\vec{j}+\vec{x}}{c}; cX; \tau) , \label{6.lemma2a}
			\qquad
			\\
			& &
			B_m (\vec{x}; \tau)
			=
			c^{m-2} \sum_{j', j ( \abs{c})}
				B_m (\frac{\vec{j}+\vec{x}}{c}; \tau), \label{6.lemma2b}
		\end{eqnarray}
		where the summations take over all elements in $(\mathbb{Z} / \abs{c} \mathbb{Z})^2$.
\end{lemma}
\begin{proof}
	Suppose that $\vec{j} = (j',j) \in \mathbb{Z}^2$ and
 		$X \notin \dfrac{\tau}{c} \mathbb Z + \dfrac{1}{c}\mathbb Z$.
	Set
		\begin{equation*}
			f(z) = \underline{F} (\vec{x}; -z +X; \tau) \underline{F}(\frac{\vec{j}+\vec{x}}{c};cz;\tau).
		\end{equation*}
	We find from (\ref{2.key_formula1}) that
		the function $f(z)$ is a doubly periodic function on $\tau \mathbb{Z} +\mathbb{Z}$,
		and from the positions of the poles of $\underline{F} (\vec{x}; X; \tau)$
		that
		$f(z)$ has the only simple poles on the lattices
 		$\dfrac{\tau}{c} \mathbb Z + \dfrac{1}{c}\mathbb Z$
 		and $X + \tau \mathbb Z + \mathbb Z$.
	Since the sum of the residues of $f(z)$ at its poles in
 		any period parallelogram equals zero,
 		we have
		\begin{equation*}
			\frac{1}{c} \sum_{k', k (   \abs{c})}
			\underline{F}(\vec{x}; X +\frac{\tau k' +k}{c};\tau) \mathrm e(-k'\frac{j'+x'}{c}-k\frac{j+x}{c})
			=
			\underline{F} (\frac{\vec{j}+\vec{x}}{c}; cX;\tau).
		\end{equation*}
	By adding each side of the above equation for $j', j =0, 1, \ldots, \abs{c}-1$,
		we obtain
		\begin{equation*}
			c \underline{F} (\vec{x}; X ;\tau)
			=
 			\sum_{j', j (   \abs{c})} \underline{F}(\frac{\vec{j}+\vec{x}}{c}; cX; \tau)
		\end{equation*}
		because $\sum\limits_{j = 0}^{\abs{c}-1} \mathrm e ( -k \dfrac{j}{\abs{c}} )$ equals $\abs{c}$ if $k$ divides $\abs{c}$,
			or equals $0$ otherwise.
	Replacing $X$ by $X + \dfrac{\tau i' +i}{c}$ and (\ref{2.key_formula1})
		imply (\ref{6.lemma2a}).
	The second equation (\ref{6.lemma2b}) follows from (\ref{6.lemma2a}) with $i' = i =0$
		and (\ref{3.definition_B}).
\end{proof}

For describing the transformation formulae of
	$\widehat{S}_{1,l}(r, M;  X , Y ; \tau)$,
	we use the following notations
	which are only different in the description of variables $X$ and $Y$:
	\begin{equation}
		\begin{split}
			&
			\widehat{S}_{1,l}(r, M;  \begin{smallmatrix} X \\ Y \end{smallmatrix} ; \tau)
			:=
			\widehat{S}_{1,l}(r, M;  X , Y ; \tau), \\
			&
			\widehat{R}_V (l, r, M; \begin{smallmatrix} X  \\ Y \end{smallmatrix} ; \tau)
			:=
			\widehat{R}_V (l, r, M; X  , Y ; \tau)
		\end{split}
	\end{equation}

The transformation formulae are the following.
\begin{proposition}\label{6.Proposition1}
	Let
		$ l \in \mathbb{Z}_{>0}$,
		$r = n(r)/ d(r) \in \mathbb{Q}$,
		$V \in \mathrm{SL}_2 (\mathbb{Z})$,
		$M \in\mathrm{M}_2( V) \bigcap \mathrm{M}_2 (r)$,
		and $X, Y \in \mathbb{C}$.
	If $j(V; r) > 0$,
		then we have
		\begin{equation}\label{6.proposition1a}
			\widehat{S}_{1,l}(r, M;  \begin{smallmatrix} X \\ Y \end{smallmatrix} ; \tau)
				- j(V; r)^{l-1} \widehat{S}_{1,l}(V r, V M;  V \bigl( \begin{smallmatrix} X \\ Y \end{smallmatrix} \bigr) ; \tau)
			=
			\widehat{R}_V (l, r, M; \begin{smallmatrix} X  \\ Y \end{smallmatrix} ; \tau).
		\end{equation}
	Here the maps $n, r$ and the sets $\mathrm{M}_2( V)$, $\mathrm{M}_2 (r)$
		are as (\ref{3.Def_ndNumber}), (\ref{4.definition_SM}) and (\ref{6.definition_SM}).
\end{proposition}
\begin{proof}
	Set $V = \bigl( \begin{smallmatrix} a  &  b \\ c  &  d \end{smallmatrix} \bigr)$
		and
		$M = \bigl( \begin{smallmatrix} \vec{x}  \\ \vec{y} \end{smallmatrix} \bigr)
		=
		\bigl( \begin{smallmatrix} x' & x  \\ y' & y \end{smallmatrix} \bigr)$.
	Let $M$ and $Y$ be fixed.
	We define the function $L(X), R(X)$ and $LR(X)$ by
		\begin{equation*}
			\begin{split}
				L(X)  & := \text{(the left hand side of (\ref{6.proposition1a}))}, \\
				R(X) & := \text{(the right hand side of (\ref{6.proposition1a}))}, \\
				LR(X) & := L(X)- R(X).
			\end{split}
		\end{equation*}
	We find from (\ref{2.key_formula1}) and (\ref{6.periodicity_S}) that
		$L(X)$ and $R(X)$, or $LR(X)$, have the quasi periodicity depending on $y', y$ on $\tau \mathbb{Z} +\mathbb{Z}$:
		\begin{equation}\label{6.proposition1aa}
			LR(X+1) =\mathrm e(-y) LR(X),\quad
			LR(X+\tau) = \mathrm e(-y') LR(X).
		\end{equation}

	The goal of the proof is $LR(X) = 0$.
	For the goal, it is enough to prove that $LR(X)$ is an entire function when
		\begin{equation*}
			Y \notin \dfrac{1}{d(V r)}  (\tau \mathbb{Z} +\mathbb{Z}), \quad
			y', y \notin \mathbb{Z}.
		\end{equation*}
	In fact, if it is true,
		then $LR(X)$ is a bounded function by the periodicity,
		thus $LR(X)$ is a constant function
		for above $Y, y', y$ because of Liouville's theorem.
	Since $y', y \notin \mathbb{Z}$ and (\ref{6.proposition1aa}), the constant should be equal to $0$.
	By the continuity of $LR(X)$ as a function of $Y, y', y$,
		we may omit the restrictions of $Y, y', y$.

	We will attain the goal.
	We define
		$M_0
			= \bigl( \begin{smallmatrix} \vec{x_0}  \\ \vec{y_0} \end{smallmatrix} \bigr)
			= \bigl( \begin{smallmatrix} x_0' & x_0  \\ y_0' & y_0 \end{smallmatrix} \bigr) $
		and
		$\bigl( \begin{smallmatrix} X_0  \\ Y_0 \end{smallmatrix} \bigr) $
		by
	\begin{equation}\label{7.eqn_MXY}
	\begin{split}
		M_0
		=
		\begin{pmatrix} \vec{x_0}  \\  \vec{y_0} \end{pmatrix}
		&: =
		V M
		=
		\begin{pmatrix}  a\vec{x}+b\vec{y} \\   c\vec{x} +d\vec{y} \end{pmatrix}
		=
		\begin{pmatrix} a x' + by' &  ax+by \\ cx' + dy'  &  cx +dy \end{pmatrix},
		\\
		\begin{pmatrix} X_0  \\ Y_0 \end{pmatrix}
		&:=
		V \begin{pmatrix} X  \\ Y \end{pmatrix}
		=
		\begin{pmatrix} a X + b Y \\ c X + d Y \end{pmatrix}.
	\end{split}
	\end{equation}
	Since $j(V;r) > 0$,
		we see from (\ref{6.proof_Lemma4.2aa}) and (\ref{6.proof_Lemma4.2bb}) that
		\begin{equation}\label{6.proposition1cc}
			\begin{split}
			&
			\widehat{S}_{1,l}(V r, V M;  V \bigl( \begin{smallmatrix} X \\ Y \end{smallmatrix} \bigr) ; \tau)
			\\
			=&
			\widehat{S}_{1,l}(\frac{n(Vr)}{d(Vr)},
					\begin{pmatrix} \vec{x_0} \\ \vec{y_0} \end{pmatrix};  \begin{pmatrix} X_0 \\ Y_0 \end{pmatrix} ; \tau)
			\\
			=&
			\frac{1}{d(V r)} \sum_{j', j (  d(Vr))}
				\underline{F} \Bigl(\frac{\vec{j} + \vec{y_0}}{d(Vr)}; n(r) Y -d(r) X;  \tau \Bigr)
			\\
			& \qquad \qquad \qquad \qquad \times
				\underline{F}^{(l-1)} \Bigl(n(V r) \frac{\vec{j} + \vec{y_0}}{d(V r)} -\vec{x_0};-cX-dY; \tau \Bigr).
			\end{split}
		\end{equation}
	Therefore it follows from (\ref{8.def_S}), (\ref{6.definition_R}) and (\ref{6.proposition1cc})
		that all possible poles of the function $L(X)$ are on the lattices
		\begin{equation*}
			\begin{cases}
	 			r Y + \dfrac{1}{d(r)} (\tau \mathbb{Z} + \mathbb{Z}), \quad
	 				- \dfrac{d}{c} Y + \dfrac{1}{c} (\tau \mathbb{Z} + \mathbb{Z}) & ( c \neq 0),\\
	  			r Y + \dfrac{1}{d(r)} (\tau \mathbb{Z} + \mathbb{Z})  & (c =0),
			\end{cases}
		\end{equation*}
		and that those of the function $R(X)$ are on
		\begin{equation*}
  			- \dfrac{d}{c} Y + \dfrac{1}{c} (\tau \mathbb{Z} + \mathbb{Z})
		\end{equation*}
		if $c \neq 0$.
	Because $Y \notin \dfrac{1}{d(V r)}  (\tau \mathbb{Z} +\mathbb{Z})$,
		the two lattices
		do not intersect, i.e.,
		\begin{equation*}
			\bigl(r Y + \frac{1}{d(r)} (\tau \mathbb{Z} + \mathbb{Z}) \bigr)
			\cap
			\bigl(- \frac{d}{c} Y + \frac{1}{c} (\tau \mathbb{Z} + \mathbb{Z}) \bigr)
			=
			\phi.
		\end{equation*}
	Therefore, we only have to prove the following two claims for the entireness of $LR(X) = L(X) - R(X)$:\\
	$\mathrm{(i)}$
		$L(X)$ is holomorphic at  $X  \in r Y + \dfrac{1}{d(r)} (\tau \mathbb{Z} + \mathbb{Z})$.\\
	$\mathrm{(ii)}$
		$L(X) - R(X)$ is holomorphic at $X \in - \dfrac{d}{c} Y + \dfrac{1}{c} (\tau \mathbb{Z} + \mathbb{Z})$ if $c > 0$.\\

	Firstly we prove $\mathrm{(i)}$.
	Let $z \in \dfrac{1}{d(r)} (\tau \mathbb{Z} + \mathbb{Z})$.
	By virtue of Lemma \ref{6.Lemma1},
		there are
		$\bigl(\begin{smallmatrix} m'  &  n' \\m  &  n     \end{smallmatrix} \bigr),
			\bigl(\begin{smallmatrix} m_0'  &  n_0' \\ m_0  &  n_0   \end{smallmatrix} \bigr) \in \mathrm{M}_2(\mathbb{Z})$
		such that
		$\bigl( \begin{smallmatrix} m'  &  n' \\ m  &  n    \end{smallmatrix} \bigr)
		=  \bigl( \begin{smallmatrix} m_0'  &  n_0' \\ m_0 &  n_0  \end{smallmatrix} \bigr)V$
		and
		\begin{equation*}
			\begin{split}
			d(r) z
			& = n(r) (\tau m' + m) + d(r) (\tau n' + n) \\
			& = n(Vr) (\tau m_0' + m_0) + d(Vr) (\tau n_0' +n_0).
			\end{split}
		\end{equation*}
	Let $\vec{m}, \vec{n}, \vec{m_0}$ and $\vec{n_0}$ mean
		the vectors $(m',m), (n',n), (m_0', m_0)$ and $(n_0', n_0)$ respectively,
		that is,
		\begin{equation}\label{7.LemProof11}
			\mbox{}^{t} \begin{pmatrix} \vec{m_0} \\ \vec{n_0} \end{pmatrix} V
			=
			\mbox{}^{t} \begin{pmatrix} \vec{m} \\ \vec{n} \end{pmatrix}
		\end{equation}
		where the symbol $\mbox{}^{t} \gamma$ means the transpose of any matrix $\gamma$.
	Then we set
		\begin{equation}\label{6.prop.aa}
			w
			 : =
			-\vec{m}\cdot \vec{x} -\vec{n}\cdot \vec{y} = -\vec{m_0}\cdot \vec{x_0} -\vec{n_0}\cdot \vec{y_0}
				\in \mathbb{C}
		\end{equation}
		where the second equation follows from (\ref{7.eqn_MXY}) and (\ref{7.LemProof11}).
	Let $O$ stand for Landau notation.
	We see that
		\begin{eqnarray*}
			&  & \widehat{S}_{1,l}(r, M;  \begin{smallmatrix} X+r Y + z \\ Y \end{smallmatrix} ; \tau)
			\\
			& \stackrel{\text{(\ref{8.def_S})}}{=}  &
			\frac{1}{d(r)}
				\sum_{j', j (  d(r))}
					\underline{F} \Bigl(\frac{\vec{j} + \vec{y}}{d(r)}; -d(r) (X + z);  \tau \Bigr)
					\underline{F}^{(l-1)} \Bigl( n(r) \frac{\vec{j}  + \vec{y}}{d(r)}- \vec{x} ;-Y; \tau \Bigr)
			\\
			& \stackrel{\text{(\ref{2.key_formula2})}}{=}  &
			\frac{(-1)^{l}}{d(r)}
				\sum_{j', j (  d(r))}
					\underline{F} \Bigl(\frac{\vec{j} + \vec{y}}{d(r)}; -d(r) X - n(r) (\tau m' + m) -  d(r) (\tau n'  + n);  \tau \Bigr)
			\\
			& & \qquad \qquad \qquad \qquad \times
					\underline{F}^{(l-1)} \Bigl(  - n(r) \frac{\vec{j}  + \vec{y}}{d(r)} + \vec{x} ;Y; \tau \Bigr) \\
			& \stackrel{\text{(\ref{2.key_formula1})}}{=}  &
			\frac{(-1)^{l}}{d(r)}
				\sum_{j', j (   d(r))} \mathrm e \bigl( - \frac{\vec{j} + \vec{y}}{d(r)} \cdot (n(r) \vec{m} + d(r) \vec{n}) \bigr)
			\\
			& & \qquad \qquad \qquad \qquad \times
					\underline{F} \Bigl(\frac{\vec{j} + \vec{y}}{d(r)}; -d(r) X ;  \tau \Bigr)
					\underline{F}^{(l-1)} \Bigl(  - n(r) \frac{\vec{j}  + \vec{y}}{d(r)} + \vec{x};Y; \tau \Bigr)
			\\
			& \stackrel{\text{(\ref{3.definition_B})}}{=}  &
			\frac{(-1)^{l-1}\mathrm e(w)}{d(r)^2 X}
				\sum_{j', j (  d(r))}
				\mathrm e(\vec{m}\cdot  \frac{-n(r) \vec{j} + d(r) \vec{x} - n(r) \vec{y} }{d(r)} )
			\\
			& & \qquad \qquad \qquad \qquad \times
				\underline{F}^{(l-1)} \Bigl( - n(r) \frac{\vec{j}  + \vec{y}}{d(r)} + \vec{x};Y; \tau \Bigr)
				+  O (1).
		\end{eqnarray*}
	Because of $\gcd (n(r), d(r)) = 1$ and (\ref{6.lemma2a}),
			\begin{eqnarray*}
			&  & 	\widehat{S}_{1,l}(r, M;  \begin{smallmatrix} X+r Y + z \\ Y \end{smallmatrix} ; \tau)
			\\
			& = &
			\frac{(-1)^{l-1}\mathrm e(w)}{d(r)^2 X}
				\sum_{j', j (  d(r))}
				\mathrm e(\vec{m}\cdot  \frac{ \vec{j} + d(r) \vec{x} - n(r) \vec{y} }{d(r)} )
			\\
			& & \qquad \qquad \qquad \qquad \times
				\underline{F}^{(l-1)} \Bigl( \frac{\vec{j}  + d(r) \vec{x} - n(r) \vec{y} }{d(r)};Y; \tau \Bigr)
				+  O (1)
			\\
			& = &
			\frac{(-1)^{l-1}\mathrm e(w)}{d(r)^l X}
				\underline{F}^{(l-1)} \Bigl(  d(r) \vec{x} - n(r) \vec{y} ; \frac{1}{d(r)} Y+ \frac{\tau m' + m}{d(r)}; \tau \Bigr) +  O (1).
		\end{eqnarray*}
	In a similar way,
 		we find that
		\begin{eqnarray*}
			& & \widehat{S}_{1,l}(V r, V M;  V \bigl( \begin{smallmatrix} X  + r Y + z\ \\ Y \end{smallmatrix} \bigr) ; \tau)
			\\
			&=&
			\frac{(-1)^{l}}{d(V r)} \sum_{j', j (  d(Vr))}
				\underline{F} \Bigl(\frac{\vec{j} + \vec{y_0}}{d(Vr)}; -d(r) X - n(Vr) (\tau m_0' + m_0) - d(Vr) (\tau n_0' +n_0);  \tau \Bigr)
			\\
			& & \qquad \qquad \qquad \qquad  \times
				\underline{F}^{(l-1)} \Bigl(- n(V r) \frac{\vec{j} + \vec{y_0}}{d(V r)} + \vec{x_0}; c(X+z) + \frac{d(Vr)}{d(r)} Y ; \tau \Bigr)
			\\
			& = &
			\frac{(-1)^{l-1}\mathrm e(w)}{d(r)d(Vr) X}
				\sum_{j', j (  d(r))}
				\mathrm e(\vec{m_0}\cdot  \frac{-n(Vr) \vec{j} + d(Vr) \vec{x_0} - n(Vr) \vec{y_0} }{d(Vr)} )
			\\
			& & \qquad \qquad \qquad \qquad \times
				\underline{F}^{(l-1)} \Bigl( - n(Vr) \frac{\vec{j}  + \vec{y_0}}{d(Vr)} + \vec{x_0};cz + \frac{d(Vr)}{d(r)} Y; \tau \Bigr)
				+  O (1).
			\\
			&=&
			\frac{(-1)^{l-1}\mathrm e(w)}{d(r) d(V r)^{l-1} X}
			\\
			& & \quad \times
				\underline{F}^{(l-1)} \Bigl(  d(V r) \vec{x_0} - n(Vr) \vec{y_0} ;  \frac{1}{d(r)} Y + \frac{cz + \tau m_0' + m_0}{d(Vr)}; \tau \Bigr) +  O (1).
		\end{eqnarray*}
	By (\ref{6.proof_Lemma4.2bb}) and (\ref{7.eqn_MXY}) we have $d(V r) \vec{x_0} - n(Vr) \vec{y_0} = d(r) \vec{x} - n(r) \vec{y}$,
	and by some calculations we get
		\begin{equation*}
			j(V;r)^{l-1}
			=
			\frac{d(V r)^{l-1}}{d(r)^{l-1}},
			\qquad
			\frac{\tau m' +m}{d(r)}
			=
			\frac{cz + \tau m_0' +m_0}{d(Vr)}.
		\end{equation*}
	Therefore
		it follows that
		$L(X  + r Y + z) = O(1)$,
		which prove the claim $\mathrm{(i)}$.

	We prove the claim $\mathrm{(ii)}$ next.
	Let $z = \dfrac{1}{c}(\tau m_0' +m_0)  \in \dfrac{1}{c} (\tau \mathbb{Z} + \mathbb{Z})$.
	By (\ref{3.definition_B}), one has
		\begin{equation}\label{6.proposition1ss}
			\begin{split}
			\underline{F}^{(n)} (\vec{x};X;\tau)
				& =
				\frac{(-1)^n n!}{(2\pi i)^n X^{n+1}} +
				\sum_{m=0}^{\infty}
				\frac{B_{m+n+1}(\vec{x};\tau)}{(m+n+1)m!} (2\pi i)^{m+1} X^m \\
				& =
				\frac{(-1)^n n!}{(2\pi i)^n X^{n+1}}  + O(1),
			\end{split}
		\end{equation}
		thus we find that
		\begin{eqnarray*}
			& & \widehat{S}_{1,l}(V r, V M;  V \bigl( \begin{smallmatrix} X  - (d/c) Y + z\ \\ Y \end{smallmatrix} \bigr) ; \tau)\\
			&\stackrel{\text{(\ref{2.key_formula2})}}{\stackrel{\text{(\ref{6.proposition1cc})}}{=}} &
			\frac{(-1)^{l-1}}{d(V r)} \sum_{j', j (  d(Vr))}
					\underline{F} \Bigl(- \frac{\vec{j} + \vec{y_0}}{d(Vr)}; d(r) (X + z) - \frac{d(Vr) Y}{c} ;  \tau \Bigr) \\
			& & \qquad \qquad \qquad \qquad  \times
					\underline{F}^{(l-1)} \Bigl(- n(V r) \frac{\vec{j} + \vec{y_0}}{d(V r)} + \vec{x_0}; cX + \tau m_0' + m_0 ; \tau \Bigr)	\\
			&\stackrel{\text{(\ref{2.key_formula1})}}{=} &
			\frac{(-1)^{l-1}}{d(V r)} \sum_{j', j (  d(Vr))}
					\mathrm e \bigl(-   \vec{m_0} \cdot \{ n(V r) \frac{\vec{j} + \vec{y_0}}{d(V r)} - \vec{x_0}  \} \bigr)
			\\
			& &    \times
					\underline{F} \Bigl(- \frac{\vec{j} + \vec{y_0}}{d(Vr)}; d(r) (X + z) - \frac{d(Vr) Y}{c} ;  \tau \Bigr)
					\underline{F}^{(l-1)} \Bigl(- n(V r) \frac{\vec{j} + \vec{y_0}}{d(V r)} + \vec{x_0}; cX; \tau \Bigr)	\\
			&\stackrel{\text{(\ref{6.proposition1ss})}}{=} &
			\frac{(l-1)!}{(2 \pi i)^{l-1}c^l}
				\mathrm e( \vec{m_0} \cdot \vec{x_0})
				\sum_{k=0}^{l-1} \frac{( 2 \pi i d(r))^k}{k ! X^{l-k}} \frac{1}{d(V r) }
			\\
			& &    \times
					 \sum_{j', j (  d(Vr))}
						\mathrm e (-  n(V r)  \vec{m_0} \cdot \frac{\vec{j} + \vec{y_0}}{d(V r)} )
						\underline{F}^{(k)} \Bigl(- \frac{\vec{j} + \vec{y_0}}{d(Vr)};  d(r) z - \frac{d(V r) Y}{c};  \tau \Bigr) + O(1)\\
			&\stackrel{\text{(\ref{6.lemma2a})}}{=}&
			\frac{(l-1)!}{(2 \pi i)^{l-1}c^l}
				\mathrm e( \vec{m_0} \cdot \vec{x_0})
			\\
			& &  \times
				\sum_{k=0}^{l-1} \frac{( 2 \pi i d(r))^k}{k ! X^{l-k} d(V r)^{k}}
						\underline{F}^{(k)} \Bigl(- \vec{y_0};  - \frac{Y}{c} +   \frac{d(r) + c n(Vr)}{d(Vr)}  z ;  \tau \Bigr) + O(1).
		\end{eqnarray*}
	Because of $d(r)^k/d(Vr)^k = 1/j(V;r)^k$ and $(d(r) + c n(Vr))/d(Vr) = a$,  we conclude that
		\begin{multline}\label{6.proposition1ee}
			L( X  - \frac{d}{c} Y + z)\\
			=
			- \frac{(l-1)!}{(2 \pi i)^{l-1}c^l}
				\mathrm e( \vec{m_0} \cdot \vec{x_0})
				\sum_{k=0}^{l-1} \frac{( 2 \pi i)^k j(V; r)^{l-1-k}}{k ! X^{l-k}}
						\underline{F}^{(k)} \Bigl(- \vec{y_0};  - \frac{Y}{c} + a z ;  \tau \Bigr) + O(1).
		\end{multline}
	On the other hand,
			since
			$\bigl( \begin{smallmatrix} \vec{x} \\ \vec{y}\end{smallmatrix} \bigr)
				=
				V^{-1} \bigl( \begin{smallmatrix} \vec{x_0} \\ \vec{y_0}\end{smallmatrix} \bigr)$,
			it follows that $\vec{y} = -c\vec{x_0} + a \vec{y_0}$ or
		\begin{equation}\label{6.proposition1fff}
			\vec{m_0} \cdot \vec{y} = a \vec{m_0} \cdot \vec{y_0} -c \vec{m_0}  \cdot \vec{x_0}.
		\end{equation}
	Therefore we can find similarly above way that
		\begin{eqnarray*}
			& &
			R( X  - \frac{d}{c} Y + z)
			\\
			& = &
			\widehat{R}_V (l, r, M; \begin{smallmatrix} X  - (d/c) Y + z  \\ Y \end{smallmatrix} ; \tau)
			\\
			& \stackrel{\text{(\ref{6.definition_R})}}{=} &
			(-1)^l \sum_{k=0}^{l-1}\binom{l-1}{k}
				\bigl( - j (V, r)\bigr)^k
					\frac{1}{c}
			\\
			& & \qquad \qquad \times
			\sum_{j', j (  \abs{c})}
						\underline{F}^{(k)} (\frac{\vec{j} +\vec{y}}{c};	 -c(X+z) ;\tau)
						\underline{F}^{(l-1-k)} (d\frac{\vec{j} +\vec{y}}{c} + \vec{x}; Y ;\tau)
			\\
			& \stackrel{\text{(\ref{8.def_S})}}{\stackrel{\text{(\ref{6.proposition1ss})}}{=}}  &
			- \sum_{k=0}^{l-1}
				\binom{l-1}{k} j (V, r)^k  \frac{k!}{(2 \pi i)^k (cX)^{k+1}}
				\frac{1}{c}
			\\
			& & \qquad \qquad \times
			\sum_{j', j (  \abs{c})}
					\mathrm e(- \vec{m_0} \cdot \frac{\vec{j} +\vec{y}}{c})
						\underline{F}^{(l-1-k)} (\frac{- d \vec{j} - \vec{y_0} }{c} ; -Y ;\tau) + O(1)
			\\
			& \stackrel{\text{(\ref{6.proposition1fff})}}{=} &
			- \sum_{k=0}^{l-1}
				\binom{l-1}{k} j (V, r)^k  \frac{k!
					\mathrm e (\vec{m_0}  \cdot \vec{x_0})}{(2 \pi i)^k (cX)^{k+1}}
				\frac{1}{c}
			\\
			& & \qquad \qquad \times
			\sum_{j', j (  \abs{c})}
					\mathrm e( - a \vec{m_0} \cdot \{  \frac{ d \vec{j} +  \vec{y_0} }{c} \} )
						\underline{F}^{(l-1-k)} (\frac{- d \vec{j} - \vec{y_0} }{c} ; -Y ;\tau) + O(1)\\
			& \stackrel{\text{(\ref{6.lemma2a})}}{=} &
			- \frac{\mathrm e (\vec{m_0}  \cdot \vec{x_0})}{c^{l}}\sum_{k=0}^{l-1}
				\binom{l-1}{k} j (V, r)^k  \frac{k! }{(2 \pi i)^k X^{k+1}}
			\\
			& & \qquad \qquad \qquad \qquad \times
						\underline{F}^{(l-1-k)} (-\vec{y_0}; -\frac{Y}{c} +  a z;\tau) + O(1).
		\end{eqnarray*}
	In the last equation, we use the fact that $ad \equiv 1 \  (\mathrm{mod\ } \abs{c})$
		which follows from $\det V =1$.
		Replacing $k$ by $l-1-k$, the last formula equals the right hand side of (\ref{6.proposition1ee}),
			which proves the claim $\mathrm{(ii)}$.
\end{proof}

The remain of the tasks in this section is to derive THEOREM \ref{6.Theorem1} from PROPOSITION \ref{6.Proposition1}.
In order to do it, we need the following lemma:
\begin{lemma}\label{6.Lemma3}
	Let $M(X)$ means the set of meromorphic functions with the variable $X$,
		and $MF(Y, Z)$ that with the variables $X, Y$.
	For $f(X) \in MF(X)$ and $g(X,Y) \in MF(X,Y)$,
		let $C_{X^m} (f(X))$, $C_{X^m} (g(X,Y))$ and $C_{Y^m} (g(X,Y))$ mean
		the coefficients of $X^m$ in $f(X)$ and $g(X,Y)$, and that of $Y^m$ in $g(X,Y)$  respectively.
	We note that
		$C_{X^m} (f(X)) \in \mathbb{C}$,
		$C_{X^m} (g(X,Y)) \in MF(Y)$
		and
		$C_{Y^m} (g(X,Y)) \in MF(X)$.

	Let $l, r, V, M$ be as in PROPOSITION \ref{6.Proposition1} with extra condition $c \neq 0$.
	Then we have the three equations
		\begin{equation}\label{6.lemma3a}
			C_{X^0} \circ C_{Y^0} (\widehat{S}_{1,l}(r, M;  \begin{smallmatrix} X \\ Y \end{smallmatrix} ; \tau))
			=
			(2 \pi i)^2
			\Bigl[
				\frac{1}{l} S_{1,l}(r, M ; \tau) - \frac{r^l}{l(l+1)} B_{l+1} ( \vec{y};  \tau )
			\Bigr],
		\end{equation}
		\begin{multline}\label{6.lemma3b}
			C_{X^0} \circ C_{Y^0} (\widehat{S}_{1,l}(V r, V M;  V \bigl( \begin{smallmatrix} X \\ Y \end{smallmatrix} \bigr) ; \tau))
			\\
			=
			(2 \pi i)^2
			\Bigl[
				\frac{1}{l} S_{1,l}(V r, V M; \tau)
				+
				\frac{(-1)^{l-1}}{l (l+1) c^l j(V;r)^l} B_{l+1} (c\vec{x}+d\vec{y}; \tau)
				\\
				+
				\frac{c}{(l+1) d(r) d(Vr)^l} B_{l+1}(n(r) \vec{y} - d(r) \vec{x};\tau)
			\Bigr],
		\end{multline}
		\begin{multline}\label{6.lemma3c}
			C_{X^0} \circ C_{Y^0} (\widehat{R}_V (l, r, M; \begin{smallmatrix} X  \\ Y \end{smallmatrix} ; \tau))
			\\
			=
			\frac{(2 \pi i)^2}{l(l+1)}
			\Bigl[
				\frac{(-1)^l (l+1)!}{(2\pi i)^{l+1}}
				R_V (l, r, M; \tau)
				+
				\frac{(-1)^l}{c^l j(V;r)} B_{l+1} (c\vec{x}+d\vec{y};\tau)
				-
				r^lB_{l+1} (\vec{y}; \tau) \
			\Bigr].
		\end{multline}
\end{lemma}

\begin{proof}
	We find from (\ref{8.def_S}), (\ref{6.lemma2a}) and (\ref{6.proposition1ss}) that
		\begin{eqnarray*}
			& & C_{Y^0}(\widehat{S}_{1,l}(r; M,  \begin{smallmatrix} X \\ Y \end{smallmatrix} ; \tau) )
			\\
			&\stackrel{\text{(\ref{8.def_S})}}{\stackrel{\text{(\ref{6.proposition1ss})}}{=}} &
			\frac{2 \pi i}{l d(r)}
				\sum_{j', j (  d(r))}
					\Bigl{\{}
						\underline{F} \Bigl(\frac{\vec{j} + \vec{y}}{d(r)} ;  -d(r) X;  \tau \Bigr) B_l \Bigl(n(r) \frac{\vec{j} + \vec{y}}{d(r)} -\vec{x}; \tau \Bigr)
			\\
			& & \qquad \qquad \qquad \qquad
						-
						(n(r))^l \underline{F}^{(l)} \Bigl(\frac{\vec{j} + \vec{y}}{d(r)} ;  -d(r) X;  \tau \Bigr)
					\Bigr{\}}
			\\
			& \stackrel{\text{(\ref{6.lemma2a})}}{=} &
			\frac{2 \pi i}{l}
				\biggl[ \frac{1}{d(r)}
					\sum_{j', j (  d(r))} \underline{F} \Bigl(\frac{\vec{j} + \vec{y}}{d(r)} ;  -d(r) X;  \tau \Bigr)
						B_l \Bigl(n(r) \frac{\vec{j} + \vec{y}}{d(r)} -\vec{x}; \tau \Bigr)
			\\
			& & \qquad \qquad \qquad \qquad
						- r^l \underline{F}^{(l)} ( \vec{y} ;  - X;  \tau )
				\biggr].
		\end{eqnarray*}
	By using (\ref{6.proposition1ss}) again, we get (\ref{6.lemma3a}).

	We prove (\ref{6.lemma3b}) next.
	By (\ref{6.proposition1cc}), we have
		\begin{multline*}
			C_{Y^0} (\widehat{S}_{1,l}(V r, V M;  V \bigl( \begin{smallmatrix} X \\ Y \end{smallmatrix} \bigr) ; \tau))
			\\
			=
			\frac{1}{d(V r)} \sum_{j', j (  d(Vr))}
				\underline{F} \Bigl(\frac{\vec{j} + \vec{y_0}}{d(Vr)}; -d(r) X;  \tau \Bigr)
				\underline{F}^{(l-1)} \Bigl(n(V r) \frac{\vec{j} + \vec{y_0}}{d(V r)} -\vec{x_0};-cX; \tau \Bigr).
		\end{multline*}
	By (\ref{6.proposition1ss}), we get (\ref{6.lemma3b}).

	Finally we verify (\ref{6.lemma3c}).
	By (\ref{6.definition_R}), (\ref{6.lemma2a}) and (\ref{6.proposition1ss}) we have
		\begin{eqnarray*}
			& &
			C_{Y^0}(\widehat{R}_V (l, r, M; \begin{smallmatrix} X  \\ Y \end{smallmatrix} ; \tau))
			\\
			&\stackrel{\text{(\ref{6.definition_R})}}{\stackrel{\text{(\ref{6.proposition1ss})}}{=}} &
			2 \pi i (-1)^l \sum_{k=0}^{l-1}
				\binom{l-1}{k} \bigl(- j (V, r)\bigr)^k  \frac{1}{l-k} \\
			& & \qquad \qquad \times
				 \frac{1}{c} \sum_{j', j (  \abs{c})}
					\Bigl{\{}
						\underline{F}^{(k)} \Bigl(  \frac{\vec{j} + \vec{y}}{c} ;  -c X;  \tau \Bigr)
							B_{l-k} \Bigl( d \frac{\vec{j} + \vec{y}}{c} + \vec{x}; \tau \Bigr)
			\\
			& & \qquad \qquad \qquad \qquad
						-
						d^{l-k}
							\underline{F}^{(l)} \Bigl(  \frac{\vec{j} + \vec{y}}{c} ;  -c X;  \tau \Bigr)
					\Bigr{\}}
			\\
			& \stackrel{\text{(\ref{6.lemma2a})}}{=} &
			2 \pi i (-1)^l \sum_{k=0}^{l-1}
				\binom{l-1}{k} \bigl(- j (V, r)\bigr)^k \frac{1}{l-k}\\
			& & \qquad \qquad \times
				\Bigl{\{}
					 \frac{1}{c} \sum_{j', j (  \abs{c})}
						\underline{F}^{(k)} \Bigl(  \frac{\vec{j} + \vec{y}}{c} ;  -c X;  \tau \Bigr)
							B_{l-k} \Bigl( d \frac{\vec{j} + \vec{y}}{c} + \vec{x}; \tau \Bigr)
			\\
			& & \qquad \qquad \qquad \qquad
						-
						\frac{d^{l-k}}{c^l}
							\underline{F}^{(l)} \Bigl( \vec{y} ;  - X;  \tau \Bigr)
				\Bigr{\}}
			\\
			& =&
			2 \pi i (-1)^l
				\Bigl[
					\sum_{k=0}^{l-1} \binom{l-1}{k} \frac{\bigl(- j (V, r)\bigr)^k}{l-k}
						 \frac{1}{c} \sum_{j', j (  \abs{c})}
							\underline{F}^{(k)} \Bigl(  \frac{\vec{j} + \vec{y}}{c} ;  -c X;  \tau \Bigr)
								B_{l-k} \Bigl( d \frac{\vec{j} + \vec{y}}{c} + \vec{x}; \tau \Bigr) \\
			& & \qquad \qquad \qquad \qquad
					-
					\frac{1}{c^l} \underline{F}^{(l)} \Bigl( \vec{y} ;  - X;  \tau \Bigr)
						\sum_{k=0}^{l-1} \binom{l-1}{k} \frac{\bigl(- j (V, r)\bigr)^k d^{l-k}}{l-k}
				\Bigr]	.
		\end{eqnarray*}
	Since
		\begin{eqnarray*}
			\sum_{k=0}^{l-1} \binom{l-1}{k} \frac{\bigl(- j (V, r)\bigr)^k d^{l-k}}{l-k}
			& = &
			\frac{1}{l} \sum_{k=0}^{l-1} \binom{l}{k} \bigl(- j (V, r)\bigr)^k d^{l-k}
			\\
			& = &
			\frac{1}{l} \Bigl(
						(d - j(V, r))^{l}
						-
						(- j(V, r))^{l}
					\Bigr)
			\\
			& = &
			\frac{(-1)^l}{l}
				\Bigl( (cr)^l - j(V;r)^l \Bigr),
		\end{eqnarray*}
		we get
		\begin{eqnarray*}
			& &
			C_{Y^0}(\widehat{R}_V (l, r, M; \begin{smallmatrix} X  \\ Y \end{smallmatrix} ; \tau))
			\\
			& = &
			\frac{2 \pi i}{l}
				\Bigl[
					\frac{j(V;r)^l - (cr)^l}{c^l} \underline{F}^{(l)} \Bigl( \vec{y} ;  - X;  \tau \Bigr)
			\\
			& & \quad
					+
					(-1)^l \sum_{k=0}^{l-1} \binom{l}{k} \bigl(- j (V, r)\bigr)^k
						\frac{1}{c} \sum_{j', j (  \abs{c})}
							\underline{F}^{(k)} \Bigl(  \frac{\vec{j} + \vec{y}}{c} ;  -c X;  \tau \Bigr)
								B_{l-k} \Bigl( d \frac{\vec{j} + \vec{y}}{c} + \vec{x}; \tau \Bigr)
				\Bigl].
		\end{eqnarray*}
	Therefor it follows from (\ref{6.proposition1ss}) that
		\begin{eqnarray*}
			& &
			C_{X^0} \circ C_{Y^0} (\widehat{R}_V (l, r, M; \begin{smallmatrix} X  \\ Y \end{smallmatrix} ; \tau))
			\\
			& =&
			\frac{(2 \pi i)^2}{l (l+1)}
					\Bigl[
						\frac{j(V;r)^l - (cr)^l}{c^l} B_{l+1} ( \vec{y} ;  \tau )
				\\
			& & \quad
						+
						(-1)^l \sum_{k=0}^{l-1} \binom{l+1}{k+1} \bigl(- j (V, r)\bigr)^k
							\frac{1}{c} \sum_{j', j (  \abs{c})}
								B_{k+1} \Bigl(  \frac{\vec{j} + \vec{y}}{c} ;  \tau \Bigr)
									B_{l-k} \Bigl( d \frac{\vec{j} + \vec{y}}{c} + \vec{x}; \tau \Bigr)
					\Bigl]
			\\
			& =&
			\frac{(2 \pi i)^2}{l (l+1)}
					\Bigl[
						\frac{j(V;r)^l - (cr)^l}{c^l} B_{l+1} ( \vec{y} ;  \tau )
						+
						\frac{(-1)^l (l+1)!}{(2\pi i)^{l+1}}R_V (l, r, M; \tau)
				\\
			& & \quad
						+
						\frac{(-1)^l}{j(V;r) c} \sum_{j', j (  \abs{c})} B_{l+1} \Bigl( d \frac{\vec{j} + \vec{y}}{c} + \vec{x}; \tau \Bigr)
						-
						\frac{j(V;r)^l}{c}  \sum_{j', j (  \abs{c})} B_{l+1} \Bigl(  \frac{\vec{j} + \vec{y}}{c} ;  \tau \Bigr)
					\Bigl]
		%
		\\
		& =&
			\frac{(2 \pi i)^2}{l (l+1)}
			\Bigl[
				\frac{j(V;r)^l - (cr)^l}{c^l} B_{l+1} ( \vec{y} ;  \tau )
				+
				\frac{(-1)^l (l+1)!}{(2\pi i)^{l+1}}R_V (l, r, M; \tau)
				\\

				& & \quad

				+
				\frac{(-1)^l}{j(V;r) c^l} B_{l+1} (c\vec{x}+d\vec{y}; \tau )
				-
				\frac{j(V;r)^l}{c^l} B_{l+1} ( \vec{y};  \tau )
			\Bigl],
		%
		\end{eqnarray*}
	which with (\ref{6.lemma2b}) gives (\ref{6.lemma3c}).

\end{proof}
We prove Theorem \ref{6.Theorem1} as the end of the section.
\begin{proof}[Proof of Theorem \ref{6.Theorem1}]
	The case that $c =0$ is trivial, so we suppose that $c \neq 0$.
	If $j(V;r) > 0$, (\ref{6.theorem1a}) follows from Proposition \ref{6.Proposition1} and Lemma \ref{6.Lemma3}.
	Since $j(-V; r) = -j(V; r)$, $S_{1,l}((-V) r, (-V) M; \tau) = (-1)^{l-1} S_{1,l}(V r, V M; \tau)$
		and $R_{-V} (l, r, M; \tau)= R_V (l, r, M; \tau)$ by definition,
		the case that $j(V;r) <0$ is reduced to the case that $j(V;r) >0$.
\end{proof}

redpajama_set_name: RedPajamaArXiv