\frac{m_{D}+m_{V}}{2m_{V}}                          A_{1}(q^{2})-
\frac{m_{D}-m_{V}}{2m_{V}}   A_{2}(q^{2})   \label{def}\end{aligned}$$ and with $A_{0}(0)=A_{3}(0)$.

In the limit of vanishing lepton masses, the term proportional to $A$ in eq.(3) does not contribute to the total amplitude and hence to the decay rate. In this limit, the differential $q^{2}$-distribution of the semileptonic $D\rightarrow V \ell \nu_{\ell}$ decay can be written $$\begin{aligned}
\frac{d\Gamma}{dq^{2}}|_{m_{\ell}=0} & = &  \frac{G^{2}}{(2\pi)^{3}} 
|V_{cQ}|^{2} \frac{p_{V} q^{2}}{12 m_{D}^{2}} 
 (|H_{+}|^{2}+|H_{-}|^{2}+|H_{0}|^{2}) \label{dg1}\end{aligned}$$ where $H_{+},H_{-}$ and $H_{0}$ are the partial helicity amplitudes, $$\begin{aligned}
H_{\pm} & = & 2m_{V}A_{0}(q^{2})+ 
\frac{(m^{2}_{D}-m^{2}_{V}-q^{2})A_{2}(q^{2})\mp 
2m_{D}p_{V} V(q^{2})}{m_{D}+m_{V}} \nonumber \\ 
H_{0} & = & \frac{1}{\sqrt{q^{2}}} \left[ (m^{2}_{D}-m^{2}_{V}-q^{2})
A_{0}(q^{2})+\frac{2m_{V}q^{2}}{m_{D}+m_{V}}A_{2}(q^{2}) \right] \; .\end{aligned}$$

We now compare the lepton spectra in the decays $D\rightarrow  \rho
\bar{\ell}   \nu_{\ell}$ and $D\rightarrow   K^{*}  \bar{\ell}
\nu_{\ell}$ at $q^2=0$. In the limit of vanishing lepton mass, the differential decay rate for $D\rightarrow V \ell \nu_{\ell}$ decay is determined by only one form factor $A_{0}$ at $q^{2}=0$:

$$\begin{aligned}
\frac{d\Gamma(D\rightarrow V \ell \nu_{\ell})}{d q^{2}} \mid_{q^{2}\rightarrow0} 
& = & \frac{G^{2}_{F}}{192 \, \pi^{3} \, m^{3}_{D}} \, |V_{cQ}|^{2}
\,  (m^{2}_{D}-m^{2}_{V})^{3} \, |A^{D\rightarrow  V}_{0}(0)|^{2} 
\label{eq:OT0}   \end{aligned}$$

Hence, the ratio of the two distributions at $q^{2}=0$ is $$\begin{aligned}
\frac{[d\Gamma(D\rightarrow \rho \bar{\ell }\nu_{\ell})/dq^{2}]_{q^{2}
\rightarrow    0}}    {[d\Gamma (D\rightarrow  K^{*} \bar{\ell}
\nu_{\ell}   )/dq^{2}]_{q^{2}   \rightarrow   0}}   &  =  &
\frac{|V_{cd}|^{2}}{|V_{cs}|^{2}} \left(\frac{m^{2}_{D}-
m^{2}_{\rho}}{m^{2}_{D}-m^{2}_{K^{*}}}\right)^{3}
\frac{|A^{D\rightarrow \rho}_{0}(0)|^{2}}{|A^{D\rightarrow
K^{*}}_{0}(0)|^{2}} 