than $q=[(m-1)/2]$ times, we have $$\begin{aligned}
\label{2eq5}
\mathcal{G}^{i}{}_{j}&=&\sum_{k=0}^{p}\frac{c_{k}}{2^{k+1}}\delta^{i
\mu_1\cdots \mu_k\nu_1\cdots\nu_k}_{j\lambda_1\cdots
\lambda_k\sigma_1\cdots\sigma_k}R^{\lambda_1\sigma_1}_{~~~~
\mu_1\nu_1}\cdots R^{\lambda_k\sigma_k}_{~~~~ \mu_k\nu_k}\nonumber \\
&=&\sum_{t}^{q}\sum_{k=t}^{p} {k \choose t} \frac{c_{k}}{2^{k+1}}
\delta_{j m_{1} n_{1}\cdots m_{t} n_{t}  e_{1} \cdots e_{k-t}
f_{1}\cdots f_{k-t}}^{i k_1 l_1 \cdots k_t l_t  c_{1}\cdots c_{k-t}
d_{1} \cdots d_{k-t}}
R_{k_1 l_1}{}{}^{m_{1} n_{1}}\cdots R_{k_t l_t}{}{}^{m_{t} n_{t}}\nonumber \\
&& \times R_{c_{1}d_{1}}{}{}^{e_{1}f_{1}}\cdots
R_{c_{k-t}d_{k-t}}{}{}^{e_{k-t}f_{k-t}}\, .\end{aligned}$$ Substituting Eqs. (\[riemann1\]) and (\[riemann2\]) into (\[2eq5\]) and using the identity $$\delta_{\nu_{1}\cdots \nu_{p-1}\nu_{p}}^{\mu_{1}\cdots \mu_{p-1}
\mu_{p}}\delta_{\mu_{p-1}\mu_{p}}^{\nu_{p-1}\nu_{p}}
=2\left[r-(p-1)\right]\left[r-(p-2)\right]\delta_{\nu_{1}\cdots
\nu_{p-2}}^{\mu_{1}\cdots \mu_{p-2}}\, , \label{idenKD}$$ where $r$ denotes the range of the index ($r=m$ for $\mathcal{M}^m$ and $r=n$ for $\mathcal{N}^n$), we have $$\begin{aligned}
\label{emm}
\mathcal{G}^{i}{}_{j}&=&\sum_{t=0}^{q}\left[\sum_{k=t}^{p} {k
\choose t} c_k \frac{(D-m)!}{(D-m-2(k-t))!}
\left(\frac{\bar{k}}{r_{0}^2 }\right)^{k-t}
\right]\overline{\mathcal{G}}^{i}_{(t)j}\, ,\end{aligned}$$ where $$\overline{\mathcal{G}}^{i}_{(t)j}=\frac{1}{2^{t+1}}\delta^{ik_1 l_1
\cdots k_{t} l_{t}}_{jm_{1} n_{1}\cdots m_{t} n_{t}}R_{k_1
l_1}{}{}^{m_{1} n_{1}}\cdots R_{k_{t} l_{t}}{}{}^{m_{t} n_{t}}\, .$$ Similarly, we have $$\begin{aligned}
\mathcal{G}^{a}{}_{b}&=&\sum_{k=0}^{p}\frac{c_{k}}{2^{k+1}}\delta^{a
\mu_1\cdots \mu_k\nu_1\cdots\nu_k}_{b\lambda_1\cdots
\lambda_k\sigma_1\cdots\sigma_k}R^{\lambda_1\sigma_1}_{~~~~
\mu_1\nu_1}\cdots R^{\lambda_k\sigma_k}_{~~~~ \mu_k\nu_k}\nonumber \\
&=&\sum_{t=0}^{s}\sum_{k=t}^{p} {k \choose t} \frac{c_{k}}{2^{k+1}}
\delta_{b m_{1} n_{1}\cdots m_{t} n_{t} e_{1} \cdots e_{k-t}
f_{1}\cdots f_{k-t}}^{a k_1 l_1 \cdots k_t l_s  c_{1}\cdots c_{k-t}
d_{1} \cdots d_{k-t}}
R_{k_1 l_1}{}{}^{m_{1} n_{1}}\cdots R_{k_t l_t}{}{}^{m_{t} n_{t}}\nonumber \\
&&\times R_{c_{1}d_{1}}{}{}^{e_{1}f_{1}}\cdots
R_{c_{k-t}d_{k-t}}{}{}^{e_{k-t}f_{k-t}}\, ,\end{aligned}$$ where $s=[m/2]$. Substituting Eqs. (\[riemann1\]), (\[riemann2\]) and (\[idenKD\]) into the above equation, we can express it as $$\begin{aligned}
\label{enn} \mathcal{G}^{a}{}_{b}&=& \frac{1}{2}\delta^{a}{}_{b}
\Bigg\{ \sum_{t=0}^{s}\left[\sum_{k=t}^{p} {k \choose t} c_k
\frac{(D-m-1)!}{(D-m-1-2(k-t))!} \left(\frac{\bar{k}}{r_{0}^2
}\right)^{k-t}\right]\bar{L}_t \Bigg\}\, ,\end{aligned}$$ where $$\bar{L}_t =\frac{1}{2^t}\delta^{k_1l_1\cdots
k_{t}l_{t}}_{m_1n_1\cdots 