\documentclass[9pt]{article} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsmath} \usepackage{graphicx} \usepackage{color} \definecolor{caca}{rgb}{0.5,0.5,0.5} % color values Red, Green, Blue \usepackage[colorlinks]{hyperref} \usepackage{newlfont} \usepackage{mathrsfs} \usepackage[active]{srcltx} % SRC Specials for DVI Searching \begin{document} \color{caca} \title{\bf{Construction of Vector Valued Modular Forms of positive dimension}} \author{Pepe Gimenez} \date{\today} \maketitle \noindent {\setlength{\baselineskip}% {1.5\baselineskip} \newtheorem{thm}{Theorem} \begin{thm}Let $b_1,\ldots , b_{\mu}$ be a set of column vectors such that $b_j \in \mathbb{C}^p$, $b_{\mu}\neq 0$, $\rho:\Gamma \longrightarrow GL(p,\mathbb{C})$ a $p$-dimensional complex representation , $\varepsilon$ a Multiplier System , and $r> 2\alpha$ (\ref{eq:alpha}), $r,\mu \in \mathbb{Z^+}$ then if \begin{equation} F(\tau)=\left(% \begin{array}{c} \sum_{\nu=1}^\mu b_\nu(1)e^{2\pi i(m_1-\nu)\tau}+ \sum_{m=0}^\infty a_m(1)e^{2 \pi i(m+m_1) \tau} \\ \vdots \\ \sum_{\nu=1}^\mu b_\nu(p)e^{2\pi i(m_p-\nu)\tau}+ \sum_{m=0}^\infty a_m(p)e^{2 \pi i(m+m_p) \tau} \\ \end{array}% \right), \end{equation} where $a_m(\nu ,r,\varepsilon)$ is a column vector in which the $j^{th}$ component is given by \begin{equation} a_m(\nu ,r,\varepsilon)(j)=2 \pi \sum_{c=1}^{\infty}\frac{1}{c} \left[ \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\varepsilon^{-1}(V_{c,d}) \left( \sum_{s=1}^p x_{js}(c,d) e^{2 \pi i m_s \frac{d'-d}{c}}B_{c,\nu,m, \rho, \varepsilon, r}^s\right) e^{-2 \pi i \frac{dm+d'\nu}{c}} \right] \end{equation} where \begin{equation} \rho^{-1}(V_{c,d})=\left(% \begin{array}{ccc} x_{11}(c,d) & \cdots & x_{1p}(c,d) \\ \vdots & \ddots & \vdots \\ x_{p1}(c,d) & \cdots & x_{pp}(c,d) \\ \end{array}% \right) \end{equation} where \begin{equation} V_{c,d}=\left(% \begin{array}{cc} d' & - \frac{dd'+1}{c} \\ c & d \\ \end{array}% \right) \in \Gamma(1) \end{equation} \begin{eqnarray}\label{Eq:defBjknum} \lefteqn{B_{c, \nu,m, \rho, \varepsilon, r}^s}\\ &&=\left( \frac{\nu - m_s}{m+m_s}\right)^{\frac{r+1}{2}}I_{r+1}\left(\frac{4 \pi}{c}(\nu - m_s)^{\frac{1}{2}}(m+m_s)^{\frac{1}{2}}\right)\quad if \quad m+m_s > 0\\ &&= \frac{2 \pi}{(r+1)!} \left(\frac{2 \pi \nu}{c} \right)^{r+1} \quad if \quad m=m_s=0 \end{eqnarray} \begin{equation} a_m=\left(% \begin{array}{c} \sum_{\nu=1}^\mu b_\nu (1) a_m(\nu ,r,\varepsilon)(1) \\ \vdots \\ \sum_{\nu=1}^\mu b_\nu (p) a_m(\nu ,r,\varepsilon)(p) \\ \end{array}% \right)=\left(% \begin{array}{c} a_m (1) \\ \vdots \\ a_m (p) \\ \end{array}% \right) \end{equation} then \begin{enumerate} \item $F(\tau)$ is regular in the complex upper half-plane $ \mathcal{H}$ \item $F(\tau)$ satisfies \begin{equation} F(\tau)- \varepsilon^{-1}(M)(-i(c \tau +d))^r \rho^{-1} (M) F(M\tau)= p_M(\tau) \end{equation}, for all \begin{equation} M=\left(% \begin{array}{cc} a & b \\ c & d \\ \end{array}% \right)\in \Gamma \end{equation} \end{enumerate} where $p_M(\tau)$ is a column vector of polynomials on $\tau$ of degree at most $r$. \end{thm} \newtheorem{lem}[thm]{Lemma} \begin{lem} For $r>2 \alpha$ as $m\rightarrow \infty$, we have \begin{equation} |a_m(\nu ,r,\varepsilon)|=O \left( (m+\kappa_m)^{-\frac{3}{4}- \frac{r}{2}} e^{4 \pi(\nu -\kappa_m)^{\frac{1}{2}}(m+\kappa_M)^{\frac{1}{2}}} \right) \end{equation} where \begin{equation} \kappa_m= \min{m_1,\ldots,m_p} \quad and \quad \kappa_M= \max{m_1,\ldots,m_p} \end{equation} \end{lem} \begin{proof}In \cite {[12]} Marvin Knopp proves that given $0 \leq m_s <1$ and $q \geq 0$ \begin{equation} c^{-q} A_{c,\nu,m_s}(m)=O \left( c^{\frac{2}{3}+ \varepsilon }\right) \end{equation} where \begin{equation} A_{c,\nu,m_s}(m)=\sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}} \varepsilon^{-1}(V_{c,d}) e^{2 \pi i m_s \frac{d'-d}{c}}e^{-2 \pi i \frac{dm+d'\nu}{c}} \end{equation} the strategy is the same as the one in the proof of Knopp first we can fix $s$ and show that \begin{equation} \begin{split}2 \pi \sum_{c=2}^{\infty}\frac{1}{c} \left[ \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\varepsilon^{-1}(V_{c,d}) \left( x_{sj}(c,d) e^{2 \pi i m_s \frac{d'-d}{c}} B_{c,\nu,m, r}^s \right) e^{-2 \pi i \frac{dm+d'\nu}{c}} \right] \\ \leq C(m+m_s)^{-\frac{1}{2}}e^{2 \pi (m+m_s)^{\frac{1}{2}}(\nu-m_s)^{\frac{1}{2}}}\end{split} \end{equation} and show that the summations behaves as the first term as $m\rightarrow \infty$. Here $B_{c,\nu,m, r}^s $ can be understood as $B_{c,\nu,m, \rho, \varepsilon, r}^s$, with a MS $\varepsilon$ and a representation $\rho$ such that \begin{equation} \varepsilon \left(% \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array}% \right) \rho \left(% \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array}% \right)=\left(% \begin{array}{ccccc} e^{2 \pi i m_1} & & s^{th}cloumn & & \\ & \ddots & & & \\ s^{th}row & & e^{2 \pi i m_s} & & \\ & & & \ddots & \\ & & & & e^{2 \pi i m_p} \\ \end{array}% \right) \end{equation} Also in the previous paper I showed that $|\rho(V_{c,d})|=O(c^{2 \alpha})$, which means that $|x_{ij}|=O(c^{2 \alpha})$. And since $r> 2 \alpha$, we can write $r=q+\alpha$ where $q>0$ In the same fashion as the mentioned proof we can see that \begin{eqnarray} \lefteqn{2 \pi \sum_{c=2}^{\infty}\frac{1}{c} \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\left|\varepsilon^{-1}(V_{c,d}) x_{sj}(c,d) e^{2 \pi i m_s \frac{d'-d}{c}}B_{c,\nu,m, r}^s e^{-2 \pi i \frac{dm+d'\nu}{c}} \right| } \\ &&\leq C_1 \frac{(\nu-m_s)^{r+\frac{1}{2}}}{(m+m_s)^{\frac{1}{2}}}\sinh \left( 2 \pi (m+m_s)^{\frac{1}{2}}(\nu-m_s)^{\frac{1}{2}}\right) \sum_{c=2}^\infty x_{sj}(c,d)c^{-r-2}A_{c,\nu,m_s}(m)\\ &&\leq C_2(m+m_s)^{-\frac{1}{2}}e^{2 \pi (m+m_s)^{\frac{1}{2}}(\nu-m_s)^{\frac{1}{2}}}\sum_{c=2}^\infty c^{-\frac{4}{3}+\varepsilon}\\ &&\leq C_3(m+m_s)^{-\frac{1}{2}}e^{2 \pi (m+m_s)^{\frac{1}{2}}(\nu-m_s)^{\frac{1}{2}}} \end{eqnarray} the first term is \begin{equation} \begin{split}2 \pi \varepsilon^{-1}(V_{1,0})x_{1,s}(1,0) e^{2 \pi i m_s (d'-d)}e^{-2 \pi i (dm+d'\nu)}\left( \frac{\nu - m_s}{m+m_s}\right)^{\frac{r+1}{2}}I_{r+1}\left(4 \pi(\nu - m_s)^{\frac{1}{2}}(m+m_s)^{\frac{1}{2}}\right)\\ =O\left((m+m_s)^{-\frac{r}{2}- \frac{3}{4}}e^{4 \pi (m+m_s)^{\frac{1}{2}}(\nu-m_s)^{\frac{1}{2}}}\right)\end{split} \end{equation} therefore we can see that the series \begin{equation} 2 \pi \sum_{c=2}^{\infty}\frac{1}{c} A_{c,\nu,m_s}(m) B_{c,\nu,m, r}^s\ \end{equation} converges uniformly on compacts and absolutely. So can conclude that \begin{eqnarray} a_m(\nu ,r,\varepsilon)(j)&=&2 \pi \sum_{c=1}^{\infty}\frac{1}{c} \left[ \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\varepsilon^{-1}(V_{c,d}) \left( \sum_{s=1}^p x_{sj}(c,d) e^{2 \pi i m_s \frac{d'-d}{c}}B_{c,\nu,m, \rho, \varepsilon, r}^s \right) e^{-2 \pi i \frac{dm+d'\nu}{c}} \right] \\ &&\leq 2 \pi \sum_{s=1}^p\sum_{c=1}^{\infty}\frac{1}{c} \left[ \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\left|\varepsilon^{-1}(V_{c,d}) \left( x_{sj}(c,d) e^{2 \pi i m_s \frac{d'-d}{c}} B_{c,\nu,m, \rho, \varepsilon, r}^s\right) e^{-2 \pi i \frac{dm+d'\nu}{c}}\right| \right] \\ &&=O \left( (m+\kappa_m)^{-\frac{3}{4}- \frac{r}{2}} e^{4 \pi(\nu -\kappa_m)^{\frac{1}{2}}(m+\kappa_M)^{\frac{1}{2}}} \right) \end{eqnarray} \end{proof} \newtheorem{cor}[thm]{Corollary} \begin{cor}The following series converges uniformly on $I_w=\{\tau : \mathcal{I}(\tau) > w > 0 \}$ \begin{equation} \sum_{m=0}^\infty \left|a_m(\nu ,r,\varepsilon)(j)e^{2 \pi i(m+m_j) \tau}\right| \end{equation} \end{cor} \begin{proof} \begin{eqnarray} \lefteqn{\left|a_m(\nu ,r,\varepsilon)(j)e^{2 \pi i(m+m_j) \tau} \right|}\\ && \leq C_1 \left|(m+\kappa_m)^{-\frac{3}{4}- \frac{r}{2}} e^{4 \pi(\nu -\kappa_m)^{\frac{1}{2}}(m+\kappa_M)^{\frac{1}{2}}}e^{2 \pi i(m+m_j)\tau}\right|\\ && \leq C_2m^{-\frac{3}{4}- \frac{r}{2}}e^{-2 \pi m w+4 \pi \mu^{\frac{1}{2}} (m+1)^{\frac{1}{2}}}\\ &&\leq C_3m^{-\frac{3}{4}} \quad \mbox{for $m$ large enough} \end{eqnarray} \end{proof} \begin{proof}[Proof of the Theorem] Let $F_\nu(\tau)$ be a column function defined in the following way %definincion de F_\nu \begin{eqnarray} \lefteqn{ F_\nu(\tau)}\\ &&=\left(% \begin{array}{c} e^{2\pi i(m_1-\nu)\tau} \\ \vdots \\ e^{2\pi i(m_p-\nu)\tau} \\ \end{array}% \right)+\sum_{m=0}^\infty\left(% \begin{array}{c} a_m(\nu ,r,\varepsilon)(1)e^{2 \pi i(m+m_1) \tau}\\ \vdots \\ a_m(\nu ,r,\varepsilon)(p)e^{2 \pi i(m+m_p) \tau} \\ \end{array}% \right)\\ %alkdjfhalskdjfhlaskdjfhkasjldfh &&=\left(% \begin{array}{c} e^{2\pi i(m_1-\nu)\tau} \\ \vdots \\ e^{2\pi i(m_p-\nu)\tau} \\ \end{array}% \right)+\sum_{m=0}^\infty\left(% \begin{array}{ccc} e^{2 \pi i (m+m_1)\tau} & & \\ & \ddots & \\ & & e^{2 \pi i (m+m_p)\tau} \\ \end{array}% \right)\left(% \begin{array}{c} a_m(\nu ,r,\varepsilon)(1)\\ \vdots \\ a_m(\nu ,r,\varepsilon)(p) \\ \end{array}% \right)\\ %;adlkfj;aldkfjas;dlfkjasfd;lkj &&=\left(% \begin{array}{c} e^{2\pi i(m_1-\nu)\tau} \\ \vdots \\ e^{2\pi i(m_p-\nu)\tau} \\ \end{array}% \right)+ 2 \pi \sum_{m=0}^\infty \left(% \begin{array}{ccc} e^{2 \pi i (m+m_1)\tau} & & \\ & \ddots & \\ & & e^{2 \pi i (m+m_p)\tau} \\ \end{array}% \right)\sum_{c=1}^{\infty}\frac{1}{c} \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\varepsilon^{-1}(V_{c,d}) \\ &&\left( \rho^{-1} (V_{c,d})\right)^t\left(% \begin{array}{ccc} e^{2 \pi i m_1 \frac{d'-d}{c}} & & \\ & \ddots & \\ & & e^{2 \pi i m_p \frac{d'-d}{c}} \\ \end{array}% \right) \left(% \begin{array}{c} e^{-2 \pi i \frac{dm+d'\nu}{c}}B_{c,\nu,m, \rho, \varepsilon, r}^1 \\ \vdots \\ e^{-2 \pi i \frac{dm+d'\nu}{c}}B_{c,\nu,m, \rho, \varepsilon, r}^p \\ \end{array}% \right)\\ %;adlkfj;aldkfjas;dlfkjasfd;lkj &&=\left(% \begin{array}{c} e^{2\pi i(m_1-\nu)\tau} \\ \vdots \\ e^{2\pi i(m_p-\nu)\tau} \\ \end{array}% \right)+ 2 \pi \sum_{m=0}^\infty \sum_{c=1}^{\infty}\frac{1}{c} \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\varepsilon^{-1}(V_{c,d})\rho^{-1} (V_{c,d})\left(% \begin{array}{ccc} e^{2 \pi i m_1 \frac{d'-d}{c}} & & \\ & \ddots & \\ & & e^{2 \pi i m_p \frac{d'-d}{c}} \\ \end{array}% \right) \\ && \left(% \begin{array}{ccc} e^{2 \pi i (m+m_1)\tau} & & \\ & \ddots & \\ & & e^{2 \pi i (m+m_p)\tau} \\ \end{array}% \right)\left(% \begin{array}{c} e^{-2 \pi i \frac{dm+d'\nu}{c}}B_{c,\nu,m, \rho, \varepsilon, r}^1 \\ \vdots \\ e^{-2 \pi i \frac{dm+d'\nu}{c}}B_{c,\nu,m, \rho, \varepsilon, r}^p \\ \end{array}% \right)\\ %;adlkfj;aldkfjas;dlfkjasfd;lkj &&=\left(% \begin{array}{c} e^{2\pi i(m_1-\nu)\tau} \\ \vdots \\ e^{2\pi i(m_p-\nu)\tau} \\ \end{array}% \right)+ 2 \pi \sum_{m=0}^\infty \sum_{c=1}^{\infty}\frac{1}{c} \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\varepsilon^{-1}(V_{c,d})\rho^{-1} (V_{c,d}) \\ && \left(% \begin{array}{ccc} e^{2 \pi i m_1 \frac{d'-d}{c}} & & \\ & \ddots & \\ & & e^{2 \pi i m_p \frac{d'-d}{c}} \\ \end{array}% \right) \left(% \begin{array}{c} e^{-2 \pi i \frac{dm+d'\nu}{c}}B_{c,\nu,m, \rho, \varepsilon, r}^1 e^{2 \pi i (m+m_1)\tau}\\ \vdots \\ e^{-2 \pi i \frac{dm+d'\nu}{c}}B_{c,\nu,m, \rho, \varepsilon, r}^p e^{2 \pi i (m+m_p)\tau} \\ \end{array}% \right)\\ \end{eqnarray} By the previous corollary and lemma we have that $F(\tau)(j)$ converges absolutely on $m$ and on $c$, and therefore the above interchanges are justified. Also we can see that $F(\tau)$ can be written in the following form \begin{equation} F(\tau)=\left(% \begin{array}{c} \sum_{\nu=1}^\mu b_\nu(1)F_\nu(\tau)(1) \\ \vdots \\ \sum_{\nu=1}^\mu b_\nu(p)F_\nu(\tau)(p) \\ \end{array}% \right) \end{equation} also since the series converges, $F(\tau)$ is regular in $\mathcal{H}$. We will prove the result for $\tau=iy$ and $y>0$, and by analytic continuation the result will follow for $\tau$ on $\mathcal{H}$. Now let us rewrite the function $F_\nu(\tau)(j)$ in the following manner \begin{eqnarray*} \lefteqn{F_\nu(\tau)(j) - e^{2\pi i(m_j-\nu)\tau}}\\ &&= 2 \pi \sum_{m=0}^\infty \sum_{c=1}^{\infty}\frac{1}{c} \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\varepsilon^{-1}(V_{c,d}) \left( \sum_{s=1}^p x_{js}(c,d) e^{2 \pi i m_s \frac{d'-d}{c}}\right) \\ && \quad \quad e^{-2 \pi i \frac{dm+d'\nu}{c}}B_{c,\nu,m, \rho, \varepsilon, r}^s e^{2\pi i(m+m_s)\tau}\\ &&=2 \pi \sum_{m=0}^\infty \sum_{s=1}^p\sum_{c=1}^{\infty}\frac{1}{c} \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\varepsilon^{-1}(V_{c,d}) x_{js}(c,d) e^{2 \pi i \frac {d'(m_s-\nu)}{c}} \\ && \quad \quad B_{c,\nu,m, \rho, \varepsilon, r}^s e^{2\pi i(m+m_s)(\tau - \frac{d}{c})} \end{eqnarray*} to continue with the proof we will need the Lipschitz summation formula for $p>-1$, $0 \leq m_s<1$ and $\mathcal{R}(t)>0$ \begin{eqnarray} \lefteqn{\sum_{n=0}^{\infty}(n+m_s)^p e^{2 \pi it(n+m_s)}}\\ &&=\frac{\Gamma(p+1)}{(2 \pi)^{p+1}}\sum_{q=- \infty}^{\infty}e^{2 \pi i m_s}(t-q i)^{-p-1}, \quad \mbox{ if } m_s +p>0\\ &&=-\frac{1}{2} + \frac{1}{2 \pi}\lim_{N \rightarrow \infty} \sum_{q=-N}{N}(t+q i)^{-1}, \quad \mbox{ if } m_s =p=0 \end{eqnarray} \begin{lem}If $m_s>0$ we have \begin{equation} \sum_{m=0}^\infty B_{c,\nu,m, \rho, \varepsilon, r}^s e^{2\pi i(m+m_s)(\tau - \frac{d}{c})}= \sum_{q=-\infty}^\infty e^{2\pi i m_s q}\left(-i(c\tau + d -cq) \right)^r \sum_{p=r+1}^\infty \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_s)}{c^2 \tau +c d -cq} \right)^p \end{equation} and if $m_s=0$ we have \begin{equation}\begin{split} \sum_{m=0}^\infty B_{c,\nu,m, \rho, \varepsilon, r}^se^{2\pi i(m+m_s)(\tau - \frac{d}{c})}= \frac{1}{2} \left( \frac{2 \pi \nu}{c}\right)^{r+1} \frac{1}{(r+1)!} + \\ \lim_{N\rightarrow \infty} \sum_{q=-N}^N \left(-i(c\tau + d -cq) \right)^r \sum_{p=r+1}^\infty \frac{1}{p!} \left(\frac{2 \pi i \nu }{c^2 \tau +c d -cq} \right)^p \end{split} \end{equation} \end{lem} \begin{proof}It is a simply application of the Lipschitz formula and the power series expression for $I_{r+1}$, in the same way done by Knopp in [1]. \end{proof} The above result implies that % aqui va la nueva definicion de F_v \begin{eqnarray} \lefteqn{F_\nu(\tau)}\\ &&=\left(% \begin{array}{c} e^{2\pi i(m_1-\nu)\tau} \\ \vdots \\ e^{2\pi i(m_p-\nu)\tau} \\ \end{array}% \right)+K +\sum_{c=1}^{\infty} \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\varepsilon^{-1}(V_{c,d}) \rho^{-1}(V_{c,d})\\ &&\left(% \begin{array}{c} e^{2 \pi i \frac {d'(m_1-\nu)}{c}}\lim_{N\rightarrow \infty} \sum_{q=-N}^N e^{2\pi i m_1 q}\left(-i(c\tau + d -cq) \right)^r \sum_{p=r+1}^\infty \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_1)}{c^2 \tau +c d -cq} \right)^p\\ \vdots \\ e^{2 \pi i \frac {d'(m_p-\nu)}{c}}\lim_{N\rightarrow \infty} \sum_{q=-N}^N e^{2\pi i m_p q}\left(-i(c\tau + d -cq) \right)^r \sum_{p=r+1}^\infty \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_p)}{c^2 \tau +c d -cq} \right)^p \\ \end{array}% \right) \end{eqnarray} where $K$ is a column vector independent of $\tau$ given by \begin{equation} K=\frac{1}{2}\sum_{c=1}^{\infty} \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\varepsilon^{-1}(V_{c,d}) \rho^{-1}(V_{c,d})\left(% \begin{array}{c} e^{2 \pi i \frac {d'(m_1-\nu)}{c}}\varphi(1) \\ \vdots \\ e^{2 \pi i \frac {d'(m_p-\nu)}{c}}\varphi(p) \\ \end{array}% \right) \end{equation} where \begin{eqnarray} \varphi(j)&=&\left(\frac{2 \pi \nu}{c} \right)^{r+1}\frac{1}{(r+1)!} \quad \mbox{ if } m_j=0\\ &=& 0\quad \mbox{ if } m_j>0 \end{eqnarray} Now let $G_\nu(\tau)=F_\nu(\tau)-K$ then \begin{eqnarray} G_\nu(\tau)(j)&=&e^{2\pi i(m_j-\nu)\tau}+\sum_{c=1}^{\infty} \sum_{\tiny{\begin{array}{c} d,c \\ 0 \leq d < c \\ (d,c)=1 \\ \end{array}}}\varepsilon^{-1}(V_{c,d})\sum_{s=1}^p x_{js}(c,d)e^{2 \pi i \frac {d'(m_s-\nu)}{c}}\\ &&\lim_{N\rightarrow \infty} \sum_{q=-N}^N e^{2\pi i m_s q}\left(-i(c\tau + d -cq) \right)^r \sum_{p=r+1}^\infty \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_s)}{c^2 \tau +c d -cq} \right)^p \end{eqnarray} now let $\delta = d-cq$, as $q$ runs through all integers and as $d$ runs through the set \begin{equation} D_c= \left\{ d | \exists \left(% \begin{array}{cc} * & * \\ c & d \\ \end{array}% \right) \in \Gamma(1) \mbox{, with } 0 \leq -d < c \right\} \end{equation} $\delta$ assumes exactly once, each value $\delta \in D^c= \{ \delta \in \mathbb{Z} | (c,\delta)=1\}$, and we can define $V_{c,\delta}=S^{-q}V_{c,d}$, we have \begin{eqnarray} \lefteqn{\varepsilon^{-1}(V{c,d})\rho^{-1}(V_{c,d})}\\ &&=\varepsilon^{-1}(V{c,d}) \left(% \begin{array}{ccc} x_{11}(c,d) & \cdots & x_{1p}(c,d) \\ \vdots & \ddots & \vdots \\ x_{p1}(c,d) & \cdots & x_{pp}(c,d) \\ \end{array}% \right)\\ &&=\varepsilon^{-1}(V_{c,\delta}S^q) \rho^{-1}(V_{c,\delta}S^q)\\ &&=\varepsilon^{-1}(V_{c,\delta}) \left(% \begin{array}{ccc} e^{-2 \pi i m_1 q} & & \\ & \ddots & \\ & & e^{-2 \pi i m_p q} \\ \end{array}% \right)\rho^{-1}(V_{c,\delta})\\ &&=\varepsilon^{-1}(V_{c,\delta}) \left(% \begin{array}{ccc} e^{-2 \pi i m_1 q} & & \\ & \ddots & \\ & & e^{-2 \pi i m_p q} \\ \end{array}% \right)\left(% \begin{array}{ccc} x_{11}(c,\delta) & \cdots & x_{1p}(c,\delta) \\ \vdots & \ddots & \vdots \\ x_{p1}(c,\delta) & \cdots & x_{pp}(c,\delta) \\ \end{array}% \right) \end{eqnarray} and therefore \begin{equation} \varepsilon^{-1}(V_{c,d})x_{js}(c,d)= \varepsilon^{-1}(V_{c,\delta})e^{-2 \pi i m_s q}x_{js}(c,\delta) \end{equation} Note also that $\delta'=d'$, and then we have \begin{equation} e^{-\frac {2 \pi i \delta ' (\nu-m_s)}{c}}=e^{-\frac {2 \pi i d' (\nu-m_s)}{c}} \end{equation} now we can rewrite $G_{\nu}(\tau)(j)$, but instead of writing $\delta$ we write $d$ \begin{eqnarray*} \lefteqn{G_{\nu}(\tau)(j)}\\ &&= e^{2\pi i(m_j-\nu)\tau}+\sum_{s=1}^p\sum_{c=1}^{\infty} \lim_{N\rightarrow \infty} \sum_{\tiny{\begin{array}{c} d=-N \\ d \in D^c \\ \end{array}}}^N\varepsilon^{-1}(V_{c,d}) x_{js}(c,d) e^{2 \pi i \frac {d'(m_s-\nu)}{c}}\\ &&\quad \quad \left(-i(c\tau + d ) \right)^r \sum_{p=r+1}^\infty \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_s) }{c(c \tau + d )} \right)^p\\ %alkjdsfhhlakjdfhalsdfjh &&= e^{2\pi i(m_j-\nu)\tau}+\sum_{s=1}^p\sum_{c=1}^{\infty} \lim_{N\rightarrow \infty} \sum_{\tiny{\begin{array}{c} d=-N \\ d \in D^c \\ \end{array}}}^N\varepsilon^{-1}(V_{c,d}) x_{js}(c,d) e^{2 \pi i \frac {d'(m_s-\nu)}{c}}\\ &&\quad \quad \left(-i(c\tau + d ) \right)^r \frac{1}{(r+1)!} \left(\frac{2 \pi i (\nu -m_s) }{c(c \tau + d )} \right)^{r+1}\\ %lasfdsahdfhjhfhf &&\quad \quad + \sum_{s=1}^p\sum_{c=1}^{\infty} \lim_{N\rightarrow \infty} \sum_{\tiny{\begin{array}{c} d=-N \\ d \in D^c \\ \end{array}}}^N\varepsilon^{-1}(V_{c,d}) x_{js}(c,d) e^{2 \pi i \frac {d'(m_s-\nu)}{c}}\\ &&\quad \quad \left(-i(c\tau + d ) \right)^r \sum_{p=r+2}^\infty \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_s) }{c(c \tau + d )} \right)^p \end{eqnarray*} The separation into two sums is justified since the first converges by Lemma (2.10) in [1], which is applicable since $\tau=iy$ with $y>0$, and the second is absolutely convergent triple sum, which can be proven by using Lemma (2.5) in [1], and therefore the second sum can be rearranged in any manner. Now let \begin{equation} V=\left(% \begin{array}{cc} \alpha & \beta \\ \gamma & \delta \\ \end{array}% \right) \in \Gamma(1), \quad \alpha >0, \quad \beta<0 \quad \delta > \gamma >0 \end{equation} and $t=(\gamma - \frac{\beta}{2})\delta^{-1}$, and define $\mathcal{J}_V(K)$ to be the trapezoid in the c-d plane bounded by the lines \begin{equation} c=0, \quad \alpha c + \gamma d=tK, \quad \delta d + \beta c= \pm K \end{equation} we obtain \begin{eqnarray*} \lefteqn{G_{\nu}(\tau)(j)}\\ %alkjdsfhhlakjdfhalsdfjh &&= e^{2\pi i(m_j-\nu)\tau}+\sum_{s=1}^p\lim_{K\rightarrow \infty}\sum_{c=1}^{\infty} \sum_{\tiny{\begin{array}{c} (c,d) \in \mathcal{J}_V(K) \\ d \in D^c \\ \end{array}}}\varepsilon^{-1}(V_{c,d}) x_{js}(c,d) e^{2 \pi i \frac {d'(m_s-\nu)}{c}}\\ &&\quad \quad \left(-i(c\tau + d ) \right)^r \frac{1}{(r+1)!} \left(\frac{2 \pi i (\nu -m_s) }{c(c \tau + d )} \right)^{r+1}\\ %lasfdsahdfhjhfhf &&\quad \quad + \sum_{s=1}^p\lim_{K\rightarrow \infty}\sum_{c=1}^{\infty} \sum_{\tiny{\begin{array}{c} (c,d) \in \mathcal{J}_V(K) \\ d \in D^c \\ \end{array}}}\varepsilon^{-1}(V_{c,d}) x_{js}(c,d) e^{2 \pi i \frac {d'(m_s-\nu)}{c}}\\ &&\quad \quad \left(-i(c\tau + d ) \right)^r \sum_{p=r+2}^\infty \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_s) }{c(c \tau + d )} \right)^p\\ %aladjkdfsalkjhadfskjlfdsaheasf && =e^{2\pi i(m_j-\nu)\tau}+\sum_{s=1}^p\lim_{K\rightarrow \infty}\sum_{c=1}^{\infty} \sum_{\tiny{\begin{array}{c} (c,d) \in \mathcal{J}_V(K) \\ d \in D^c \\ \end{array}}}\varepsilon^{-1}(V_{c,d}) x_{js}(c,d) e^{2 \pi i \frac {d'(m_s-\nu)}{c}}\\ &&\quad \quad \left(-i(c\tau + d ) \right)^r \left(e^{\frac{2 \pi i (\nu -m_s) }{c(c \tau + d )}}-\sum_{p=0}^r \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_s) }{c(c \tau + d )} \right)^p \right) \end{eqnarray*} Now let \begin{eqnarray} \lefteqn{S_{\nu,K}(\tau)}\\ && =\left(% \begin{array}{c} e^{2\pi i(m_1-\nu)\tau} \\ \vdots \\ e^{2\pi i(m_p-\nu)\tau} \\ \end{array}% \right)+\sum_{c=1}^{\infty} \sum_{\tiny{\begin{array}{c} (c,d) \in \mathcal{J}_V(K) \\ d \in D^c \\ \end{array}}}\varepsilon^{-1}(V_{c,d})\rho^{-1}(V_{c,d})\left(-i(c\tau + d ) \right)^r\\ && \left(% \begin{array}{c} e^{2 \pi i \frac {d'(m_1-\nu)}{c}}e^{\frac{2 \pi i (\nu -m_1) }{c(c \tau + d )}} \\ \vdots \\ e^{2 \pi i \frac {d'(m_p-\nu)}{c}} e^{\frac{2 \pi i (\nu -m_p) }{c(c \tau + d )}}\\ \end{array}% \right) \end{eqnarray} and since $V_{c,d} \in SL(2,\mathbb{Z})$ we can see that \begin{equation} e^{2 \pi i \frac {d'(m_s-\nu)}{c}}e^{\frac{2 \pi i (\nu -m_s) }{c(c \tau + d )}}=e^{2 \pi i (m_s- \nu)V_{c,d}\tau} \end{equation} and therefore \begin{eqnarray} \lefteqn{S_{\nu,K}(\tau)}\\ && =\left(% \begin{array}{c} e^{2\pi i(m_1-\nu)\tau} \\ \vdots \\ e^{2\pi i(m_p-\nu)\tau} \\ \end{array}% \right)+\sum_{c=1}^{\infty} \sum_{\tiny{\begin{array}{c} (c,d) \in \mathcal{J}_V(K) \\ d \in D^c \\ \end{array}}}\varepsilon^{-1}(V_{c,d})\rho^{-1}(V_{c,d})\left(-i(c\tau + d ) \right)^r\\ && \left(% \begin{array}{c} e^{2 \pi i (m_1- \nu)V_{c,d}\tau} \\ \vdots \\ e^{2 \pi i (m_p- \nu)V_{c,d}\tau}\\ \end{array}% \right) \end{eqnarray} now we can include the first term in the summation, since $\varepsilon(I)\rho(I)(-i)^{-r}=I$ and we have that \begin{equation} \left(% \begin{array}{c} e^{2\pi i(m_1-\nu)\tau} \\ \vdots \\ e^{2\pi i(m_p-\nu)\tau} \\ \end{array}% \right)=\varepsilon^{-1}(I)\rho^{-1}(I)(-i)^r\left(% \begin{array}{c} e^{2 \pi i (m_1- \nu)I\tau} \\ \vdots \\ e^{2 \pi i (m_p- \nu)I\tau} \\ \end{array}% \right) \end{equation} now we can include the pair $(c,d)=(0,1)$ and therefore we get \begin{eqnarray} S_{\nu,K}(\tau)&=&\sum_{c=0}^{\infty} \sum_{\tiny{\begin{array}{c} (c,d) \in \mathcal{J}_V(K) \\ d \in D^c \\ (c,d)\neq (0,-1)\\ \end{array}}}\varepsilon^{-1}(V_{c,d})\rho^{-1}(V_{c,d})\left(-i(c\tau + d ) \right)^r\\ && \left(% \begin{array}{c} e^{2 \pi i (m_1- \nu)V_{c,d}\tau} \\ \vdots \\ e^{2 \pi i (m_p- \nu)V_{c,d}\tau}\\ \end{array}% \right) \end{eqnarray} we can now extend the region $\mathcal{J}_V(K)$ to $\mathcal{P}_V(K)$, consisting in reflecting $\mathcal{J}_V(K)$ through the origin by including with $V_{c,d}$ and $-V_{c,d}$, and since we have that \begin{equation} \varepsilon^{-1}(V_{c,d})\rho^{-1}(V_{c,d})\left(-i(c\tau + d ) \right)^r =\varepsilon^{-1}(-V_{c,d})\rho^{-1}(-V_{c,d})\left(-i(-c\tau - d ) \right)^r \end{equation} and \begin{equation} e^{2 \pi i (m_j- \nu)V_{c,d}\tau}=e^{2 \pi i (m_j- \nu)-V_{c,d}\tau} \end{equation} we see that if we make the summation over $\mathcal{P}_V(K)$, every term of $S_{\nu,K}(\tau)$ occurs twice and therefore \begin{eqnarray} S_{\nu,K} (\tau)&=&\frac{1}{2}\sum_{c \in \mathbb{Z}} \sum_{\tiny{\begin{array}{c} (c,d) \in \mathcal{P}_V(K) \\ d \in D^c \\ \end{array}}}\varepsilon^{-1}(V_{c,d})\rho^{-1}(V_{c,d})\left(-i(c\tau + d ) \right)^r\\ && \left(% \begin{array}{c} e^{2 \pi i (m_1- \nu)V_{c,d}\tau} \\ \vdots \\ e^{2 \pi i (m_p- \nu)V_{c,d}\tau}\\ \end{array}% \right) \end{eqnarray} now $\mathcal{P}_V(K)$ is bounded by \begin{equation} \alpha c + \gamma d=\pm tK, \quad \delta d + \beta c= \pm K \end{equation} Therefore we get \begin{eqnarray} \lefteqn{\varepsilon^{-1}(V) \rho^{-1}(V) \left(-i(\gamma \tau + \delta ) \right)^r S_{\nu,K}(V\tau)}\\ &&=\frac{1}{2}\sum_{c \in \mathbb{Z}} \sum_{\tiny{\begin{array}{c} (c,d) \in \mathcal{P}_V(K) \\ d \in D^c \\ \end{array}}}\varepsilon^{-1}(V)\varepsilon^{-1}(V_{c,d})\rho^{-1}(V)\rho^{-1}(V_{c,d})\\ &&\left(-i(\gamma \tau + \delta ) \right)^r \left(-i(cV\tau + d ) \right)^r \left(% \begin{array}{c} e^{2 \pi i (m_1- \nu)V_{c,d}V\tau} \\ \vdots \\ e^{2 \pi i (m_p- \nu)V_{c,d}V\tau}\\ \end{array}% \right)\\ %alkjdsfhlasdkjfhlasdjkfhlakjdsfh &&=\frac{1}{2}\sum_{c \in \mathbb{Z}} \sum_{\tiny{\begin{array}{c} (c,d) \in \mathcal{P}_V(K) \\ d \in D^c \\ \end{array}}}\varepsilon^{-1}(V_{c,d}V)\rho^{-1}(V_{c,d}V)\\ && \left(-i((\alpha c+\gamma d)\tau + (\beta c +\delta d ) \right)^r \left(% \begin{array}{c} e^{2 \pi i (m_1- \nu)V_{c,d}V\tau} \\ \vdots \\ e^{2 \pi i (m_p- \nu)V_{c,d}V\tau}\\ \end{array}% \right) \end{eqnarray} now we can make the transformation $c'=\alpha c+\gamma d$ and $d'=\beta c +\delta d $ which is a 1-1 transformation from $\mathcal{P}_V(K)$ to the rectangle \begin{equation} |c'|\leq tK, \quad \quad |d'| \leq K \end{equation} also this map is a 1-1 correspondence between the pairs $\{(c,d) | c \in \mathbb{Z}, d \in D^c\}$ and the pairs $\{(c',d') | c' \in \mathbb{Z}, d' \in D^{c'}\}$, and then \begin{eqnarray} \lefteqn{\varepsilon^{-1}(V) \rho^{-1}(V) \left(-i(\gamma \tau + \delta ) \right)^r S_{\nu,K}(V\tau)}\\ &&=\frac{1}{2}\sum_{\tiny{\begin{array}{c} c' \in \mathbb{Z} \\ |c'|\leq tK \\ \end{array}}} \sum_{\tiny{\begin{array}{c} d' \in D^c \\ |d'| \leq K\\ \end{array}}}\varepsilon^{-1}(V_{c',d'})\rho^{-1}(V_{c',d'})\\ && \left(-i(c'\tau + d' ) \right)^r \left(% \begin{array}{c} e^{2 \pi i (m_1- \nu)V_{c',d'}\tau} \\ \vdots \\ e^{2 \pi i (m_p- \nu)V_{c',d'}\tau}\\ \end{array}% \right)\\ %alkjdsfhlasdkjfhlasdjkfhlakjdsfh &&=\left(% \begin{array}{c} e^{2 \pi i (m_1- \nu)\tau} \\ \vdots \\ e^{2 \pi i (m_p- \nu)\tau}\\ \end{array}% \right) + \sum_{\tiny{\begin{array}{c} c \in \mathbb{Z} \\ 0< c\leq tK \\ \end{array}}} \sum_{\tiny{\begin{array}{c} d \in D^c \\ |d| \leq K\\ \end{array}}}\varepsilon^{-1}(V_{c,d})\rho^{-1}(V_{c,d})\\ && \left(-i(c\tau + d ) \right)^r \left(% \begin{array}{c} e^{2 \pi i (m_1- \nu)V_{c,d}\tau} \\ \vdots \\ e^{2 \pi i (m_p- \nu)V_{c,d}\tau}\\ \end{array}% \right) \end{eqnarray} now \begin{eqnarray} \lefteqn{\varepsilon^{-1}(V) \rho^{-1}(V) \left(-i(\gamma \tau + \delta ) \right)^r G_{\nu}(V\tau)}\\ &&=\lim_{k\rightarrow \infty} (\varepsilon^{-1}(V) \rho^{-1}(V) \left(-i(\gamma \tau + \delta ) \right)^r S_{\nu,K}(V\tau)\\ &&- \sum_{c=1}^{\infty} \sum_{\tiny{\begin{array}{c} (c,d) \in \mathcal{J}_V(K) \\ d \in D^c \\ \end{array}}}\varepsilon^{-1}(V)\varepsilon^{-1}(V_{c,d})\rho^{-1}(V)\rho^{-1}(V_{c,d})\left(-i(\gamma \tau + \delta ) \right)^r\\ && \left(-i(cV\tau + d ) \right)^r \left(% \begin{array}{c} e^{2 \pi i \frac {d'(m_1-\nu)}{c}}\sum_{p=0}^r \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_1) }{c(c V\tau + d )} \right)^p \\ \vdots \\ e^{2 \pi i \frac {d'(m_p-\nu)}{c}} \sum_{p=0}^r \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_p) }{c(c V\tau + d )} \right)^p \\ \end{array}% \right))\\ &&=\left(% \begin{array}{c} e^{2 \pi i (m_1- \nu)\tau} \\ \vdots \\ e^{2 \pi i (m_p- \nu)\tau}\\ \end{array}% \right) +\lim_{k\rightarrow \infty} (\sum_{\tiny{\begin{array}{c} c \in \mathbb{Z} \\ 0< c\leq tK \\ \end{array}}} \sum_{\tiny{\begin{array}{c} d \in D^c \\ |d| \leq K\\ \end{array}}}\varepsilon^{-1}(V_{c,d})\rho^{-1}(V_{c,d})\left(-i(c\tau + d ) \right)^r \left(% \begin{array}{c} e^{2 \pi i (m_1- \nu)V_{c,d}\tau} \\ \vdots \\ e^{2 \pi i (m_p- \nu)V_{c,d}\tau}\\ \end{array}% \right)\\ && - \sum_{c=1}^{\infty} \sum_{\tiny{\begin{array}{c} (c,d) \in \mathcal{J}_V(K) \\ d \in D^c \\ \end{array}}}\varepsilon^{-1}(V_{c,d}V)\rho^{-1}(V_{c,d}V)\left(-i(\gamma \tau + \delta ) \right)^r \left(-i(cV\tau + d ) \right)^r\\ && \left(% \begin{array}{c} e^{2 \pi i \frac {d'(m_1-\nu)}{c}}\sum_{p=0}^r \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_1) }{c(c V\tau + d )} \right)^p \\ \vdots \\ e^{2 \pi i \frac {d'(m_p-\nu)}{c}} \sum_{p=0}^r \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_p) }{c(c V\tau + d )} \right)^p \\ \end{array}% \right)) \end{eqnarray} also using lemma (2.13) in [1]. \begin{eqnarray} \lefteqn{ G_{\nu}(\tau)}\\ &&=\left(% \begin{array}{c} e^{2 \pi i (m_1- \nu)\tau} \\ \vdots \\ e^{2 \pi i (m_p- \nu)\tau}\\ \end{array}% \right) +\lim_{k\rightarrow \infty} (\sum_{\tiny{\begin{array}{c} c \in \mathbb{Z} \\ 0< c\leq tK \\ \end{array}}} \sum_{\tiny{\begin{array}{c} d \in D^c \\ |d| \leq K\\ \end{array}}}\varepsilon^{-1}(V_{c,d})\rho^{-1}(V_{c,d})\left(-i(c\tau + d ) \right)^r \left(% \begin{array}{c} e^{2 \pi i (m_1- \nu)V_{c,d}\tau} \\ \vdots \\ e^{2 \pi i (m_p- \nu)V_{c,d}\tau}\\ \end{array}% \right)\\ && - \sum_{\tiny{\begin{array}{c} c \in \mathbb{Z} \\ 0< c\leq tK \\ \end{array}}} \sum_{\tiny{\begin{array}{c} d \in D^c \\ |d| \leq K\\ \end{array}}}\varepsilon^{-1}(V_{c,d})\rho^{-1}(V_{c,d}) \left(-i(c\tau + d ) \right)^r \left(% \begin{array}{c} e^{2 \pi i \frac {d'(m_1-\nu)}{c}}\sum_{p=0}^r \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_1) }{c(c \tau + d )} \right)^p \\ \vdots \\ e^{2 \pi i \frac {d'(m_p-\nu)}{c}} \sum_{p=0}^r \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_p) }{c(c \tau + d )} \right)^p \\ \end{array}% \right)) \end{eqnarray} therefore \begin{eqnarray} \lefteqn{ G_{\nu}(\tau)-\varepsilon^{-1}(V) \rho^{-1}(V) \left(-i(\gamma \tau + \delta ) \right)^r G_{\nu}(V\tau)}\\ &&=\lim_{k\rightarrow \infty} (\sum_{c=1}^{\infty} \sum_{\tiny{\begin{array}{c} (c,d) \in \mathcal{J}_V(K) \\ d \in D^c \\ \end{array}}}\varepsilon^{-1}(V_{c,d}V)\rho^{-1}(V_{c,d}V)\left(-i(\gamma \tau + \delta ) \right)^r \left(-i(cV\tau + d ) \right)^r\\ && \left(% \begin{array}{c} e^{2 \pi i \frac {d'(m_1-\nu)}{c}}\sum_{p=0}^r \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_1) }{c(c V\tau + d )} \right)^p \\ \vdots \\ e^{2 \pi i \frac {d'(m_p-\nu)}{c}} \sum_{p=0}^r \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_p) }{c(c V\tau + d )} \right)^p \\ \end{array}% \right)\\ && - \sum_{\tiny{\begin{array}{c} c \in \mathbb{Z} \\ 0< c\leq tK \\ \end{array}}} \sum_{\tiny{\begin{array}{c} d \in D^c \\ |d| \leq K\\ \end{array}}}\varepsilon^{-1}(V_{c,d})\rho^{-1}(V_{c,d}) \left(-i(c\tau + d ) \right)^r \left(% \begin{array}{c} e^{2 \pi i \frac {d'(m_1-\nu)}{c}}\sum_{p=0}^r \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_1) }{c(c \tau + d )} \right)^p \\ \vdots \\ e^{2 \pi i \frac {d'(m_p-\nu)}{c}} \sum_{p=0}^r \frac{1}{p!} \left(\frac{2 \pi i (\nu -m_p) }{c(c \tau + d )} \right)^p \\ \end{array}% \right)) \end{eqnarray} now since the factor$(\gamma \tau + \delta )^r$ combine with $(c V\tau + d )^{r-p}$ produces a polynomial of degree at most $r$. On the other hand the limit of a sequence of polynomials of degree at most $r$ converging at $r+1$ points is a polynomial of degree at most $r$. Now since $F_\nu(\tau) = G_\nu(\tau)+K$, then we have that $F_\nu(V\tau) = G_\nu(V\tau)+K$, therefore we have that \begin{equation} F_{\nu}(\tau)-\varepsilon^{-1}(V) \rho^{-1}(V) \left(-i(\gamma \tau + \delta ) \right)^r F_{\nu}(V\tau)=p_{V,\nu}(\tau, r, \varepsilon, \rho) \end{equation} for \begin{equation} V=\left(% \begin{array}{cc} \alpha & \beta \\ \gamma & \delta \\ \end{array}% \right) \in \Gamma(1), \quad \alpha >0, \quad \beta<0 \quad \delta > \gamma >0 \end{equation} to prove it for every $M=\left(% \begin{array}{cc} a & b \\ c & d \\ \end{array}% \right) \in \Gamma(1)$, first we see that if $M=S^q$, we have \begin{equation} F_\nu(S^q \tau) =\left(% \begin{array}{ccc} e^{2 \pi i q m_1} & & \\ & \ddots & \\ & & e^{2 \pi i q m_p} \\ \end{array}% \right)F_\nu(\tau) \end{equation} which follows from the fourier expansion of $F_\nu(\tau)$, otherwise it is easy to show that exist $m,n \in \mathbb{Z}$ and $V=\left(% \begin{array}{cc} \alpha & \beta \\ \gamma & \delta \\ \end{array}% \right) \in \Gamma(1)$ such that \begin{equation} M=S^kVS^n, \quad \alpha >0, \quad \beta<0 \quad \delta > \gamma >0 \quad \gamma=c,\quad \delta= d-cn \end{equation} and therefore \begin{eqnarray} F_\nu(M\tau)&=& F_\nu(S^kVS^n\tau)=\left(% \begin{array}{ccc} e^{2 \pi i k m_1} & & \\ & \ddots & \\ & & e^{2 \pi i k m_p} \\ \end{array}% \right)F_\nu(VS^n\tau)\\ &&=\left(% \begin{array}{ccc} e^{2 \pi i k m_1} & & \\ & \ddots & \\ & & e^{2 \pi i k m_p} \\ \end{array}% \right)\varepsilon(V)\rho(V)\left(-i(\gamma S^n\tau + \delta ) \right)^r\\ &&\left(F_\nu(S^n\tau)-p_{V,\nu}(S^n\tau,\varepsilon,\rho)\right) \end{eqnarray} and since $\left(-i(\gamma S^n\tau + \delta ) \right)^r=\left(-i(c\tau + d ) \right)^r$, we have \begin{eqnarray} \lefteqn{F_\nu(\tau)-\left(% \begin{array}{ccc} e^{-2 \pi i n m_1} & & \\ & \ddots & \\ & & e^{-2 \pi i n m_p} \\ \end{array}% \right)p_{V,\nu}(S^n\tau,\varepsilon,\rho)}\\ &&=\varepsilon^{-1}(V)\left(-i(c\tau + d ) \right)^r\left(% \begin{array}{ccc} e^{-2 \pi i n m_1} & & \\ & \ddots & \\ & & e^{-2 \pi i n m_p} \\ \end{array}% \right)\\ &&\rho^{-1}(V)\left(% \begin{array}{ccc} e^{-2 \pi i k m_1} & & \\ & \ddots & \\ & & e^{-2 \pi i k m_p} \\ \end{array}% \right)F_\nu(M\tau)\\ &&=\varepsilon^{-1}(M)\rho^{-1}(M)\left(-i(c\tau + d ) \right)^rF_\nu(M\tau) \end{eqnarray} and since \begin{equation} \left(% \begin{array}{ccc} e^{-2 \pi i n m_1} & & \\ & \ddots & \\ & & e^{-2 \pi i n m_p} \\ \end{array}% \right)p_{V,\nu}(S^n\tau,\varepsilon,\rho) \end{equation} is a column vector of degree at most $r$ the proof is complete \end{proof} \begin{equation} \end{equation} \begin{eqnarray} \lefteqn{} \end{eqnarray} \begin{thebibliography}{99} \bibitem{[12]} M. ~Knopp, Automorphic Forms of Nonnegative dimension and exponential sums, Michigan Mathematical Journal. 7 (1960) 257-287. \bibitem{[3]} M. ~Knopp and G. Mason, On vector-valued modular forms and their Fourier coefficients, Acta Arith. 110 (2003), no. 2, 117–124. \bibitem{[4]} M. ~Knopp and G. Mason, Vector-Valued Modular forms and Poincare Series, Illinois Journal of Mathematics 48 (2004), 1345-1366. \bibitem{[8]} H. ~Rademacher, The Fourier series and the functional equation of the absolute modular invariant $J(\tau)$. Am. J. Math. 61 (1939), 237-248. \bibitem{[11]} H. ~Rademacher and H. Zuckerman, On the the Fourier coefficients of certain modular forms of positive dimension. Ann. Math. 39 (1938), 433-462. \end{thebibliography} \end{document}