This is just the long-form derivation of the formula for Wigner’s $\mathfrak{D}$ matrices, discussed more on this page.

\begin{align*} \mathbf{e}_{(\emprime)}(\quat{R}\, \quat{Q}) &= \frac{(\quat{R}\, \quat{Q})_{a}^{\ell+\emprime}\, (\quat{R}\, \quat{Q})_{b}^{\ell-\emprime}} {\sqrt{ (\ell+\emprime)!\, (\ell-\emprime)! }} \\\
&= \frac{(\quat{R}_a\, \quat{Q}_a - \co{\quat{R}}_b\, \quat{Q}_b)^{\ell+\emprime}\, (\quat{R}\, \quat{Q})_{b}^{\ell-\emprime}} {\sqrt{ (\ell+\emprime)!\, (\ell-\emprime)! }} \\\
&= \sum_{\rho} \binom{\ell+m’} {\rho} \frac{(\quat{R}_a\, \quat{Q}_a)^{\ell+\emprime-\rho} (- \co{\quat{R}}_b\, \quat{Q}_b)^{\rho}\, (\quat{R}_b\, \quat{Q}_a + \co{\quat{R}}_a\, \quat{Q}_b)^{\ell-\emprime}} {\sqrt{ (\ell+\emprime)!\, (\ell-\emprime)! }} \\\
&= \sum_{\rho,\rho’} \binom{\ell+m’} {\rho} \binom{\ell-m’} {\rho’} \frac{(\quat{R}_a\, \quat{Q}_a)^{\ell+\emprime-\rho} (- \co{\quat{R}}_b\, \quat{Q}_b)^{\rho}\, (\quat{R}_b\, \quat{Q}_a)^{\ell-\emprime-\rho’} (\co{\quat{R}}_a\, \quat{Q}_b)^{\rho’}} {\sqrt{ (\ell+\emprime)!\, (\ell-\emprime)! }} \\\
&= \sum_{\rho,\em} \binom{\ell+m’} {\rho} \binom{\ell-m’} {\ell-\em-\rho} \frac{(\quat{R}_a\, \quat{Q}_a)^{\ell+\emprime-\rho} (- \co{\quat{R}}_b\, \quat{Q}_b)^{\rho}\, (\quat{R}_b\, \quat{Q}_a)^{\em-\emprime+\rho} (\co{\quat{R}}_a\, \quat{Q}_b)^{\ell-\em-\rho}} {\sqrt{ (\ell+\emprime)!\, (\ell-\emprime)! }} \\\
&= \sum_{\rho,\em} \binom{\ell+m’} {\rho} \binom{\ell-m’} {\ell-\em-\rho} \quat{R}_a^{\ell+\emprime-\rho} (- \co{\quat{R}}_b)^{\rho}\, \quat{R}_b^{\em-\emprime+\rho} \co{\quat{R}}_a^{\ell-\em-\rho} \frac{\quat{Q}_a^{\ell+\em} \quat{Q}_b^{\ell-\em}} {\sqrt{ (\ell+\emprime)!\, (\ell-\emprime)! }} \\\
&= \sum_{\em} \mathbf{e}_{(\em)}(\quat{Q}) \sum_{\rho} \binom{\ell+m’} {\rho} \binom{\ell-m’} {\ell-\em-\rho} \quat{R}_a^{\ell+\emprime-\rho} (- \co{\quat{R}}_b)^{\rho}\, \quat{R}_b^{\em-\emprime+\rho} \co{\quat{R}}_a^{\ell-\em-\rho} \frac{\sqrt{ (\ell+\em)!\, (\ell-\em)! }} {\sqrt{ (\ell+\emprime)!\, (\ell-\emprime)! }} \\\
\end{align*}

We have introduced a new summation variable $\em$ and used the substitution $\rho’ \mapsto \ell-\em-\rho$ to bring this into the form we need to express Wigner’s $\mathfrak{D}$ matrix. Alternatively, we could have made an equivalent substitution for $\rho$, so that $\mathfrak{D}$ would be given as a sum over $\rho’$. This would have the effect of reversing the roles of $a$ and $b$, which is what we do on this page when $\lvert \quat{R}_a \rvert < \lvert \quat{R}_b \rvert$.