In this article Theory of Complex Spectra II Giulio Racah defines $f(m_{1} m_{2} ; jm)$ by
\begin{multline}
\left(m_{1} m_{2} \mid j m\right)
=(-1)^{j_{1}-m_{1}} f\left(m_{1} m_{2} ; j m\right)\left[\left(j_{1}+m_{1}\right) !\left(j_{2}+m_{2}\right) !(j+m) !\right]^{\frac{1}{2}} /\left[\left(j_{1}-m_{1}\right) !\left(j_{2}-m_{2}\right) !(j-m) !\right]^{\frac{1}{2}}
\end{multline}
where $\left(m_{1} m_{2} \mid j m\right)$ are the Clebsch-Gordan coefficients.Then, he shows that
\begin{multline}
f\left(m_{1} m_{2} ; j m-1\right)=\\ \left(j_{2}+m_{2}+1\right)\left(j_{2}-m_{2}\right) f\left(m_{1} m_{2}+1 ; j m\right)-\\ \left(j_{1}+m_{1}+1\right)\left(j_{1}-m_{1}\right) f\left(m_{1}+1 m_{2} ; j m\right) \tag{1}
\end{multline}
Now he claims that
\begin{multline}
f\left(m_{1} m_{2} ; j m\right)=f\left(m_{1} m_{2} ; j j -u\right)=\\ A_{j} \sum_{t}(-1)^{t}\left(\begin{array}{l}
u \\
t
\end{array}\right) \frac{\left(j_{1}+m_{1}+t\right) !\left(j_{1}-m_{1}\right) !\left(j_{2}+m_{2}+u-t\right) !\left(j_{2}-m_{2}\right) !}{\left(j_{1}+m_{1}\right) !\left(j_{1}-m_{1}-t\right) !\left(j_{2}+m_{2}\right) !\left(j_{2}-m_{2}-u+t\right) !}
\tag{2}\end{multline}
The summation parameter takes on all integral values consistent
with the factorial notation, the factorial of a negative number being meaningless.To demonstrate (2) he says that it suffices to verify that it satisfies (1).
This what I tried.We have that
\begin{multline}
\left(j_{1}+m_{1}+1\right)\left(j_{1}-m_{1}\right) f\left(m_{1}+1 m_{2} ; j m\right)=\\ A_{j} \sum_{t}(-1)^{t}\left(\begin{array}{l}
u \\
t
\end{array}\right) \left(j_{1}-m_{1}-t\right)\left(j_{1}+m_{1}+1+t\right) \times \\ \frac{\left(j_{1}+m_{1}+t\right) !\left(j_{1}-m_{1}\right) !\left(j_{2}+m_{2}+u-t\right) !\left(j_{2}-m_{2}\right) !}{\left(j_{1}+m_{1}\right) !\left(j_{1}-m_{1}-t\right) !\left(j_{2}+m_{2}\right) !\left(j_{2}-m_{2}-u+t\right) !}
\end{multline}
\begin{multline} \left(j_{2}+m_{2}+1\right)\left(j_{2}-m_{2}\right) f\left(m_{1} m_{2} +1 ; j m\right)=\\ A_{j} \sum_{t}(-1)^{t}\left(\begin{array}{l} u \\ t \end{array}\right) \left(j_{2}-m_{2}-u+t\right)\left(j_{2}+m_{2}+1+u -t \right) \times \\ \frac{\left(j_{1}+m_{1}+t\right) !\left(j_{1}-m_{1}\right) !\left(j_{2}+m_{2}+u-t\right) !\left(j_{2}-m_{2}\right) !}{\left(j_{1}+m_{1}\right) !\left(j_{1}-m_{1}-t\right) !\left(j_{2}+m_{2}\right) !\left(j_{2}-m_{2}-u+t\right) !} \end{multline}
and \begin{multline} f\left(m_{1} m_{2}; j m-1\right)=\\ A_{j} \sum_{t}(-1)^{t}\left(\begin{array}{l} u \\ t \end{array}\right) \times \\ \frac{u+1}{u+1-t}\left(j_{2}+m_{2}+u+1-t\right)\left(j_{2}-m_{2}-u+t\right) \times \\ \frac{\left(j_{1}+m_{1}+t\right) !\left(j_{1}-m_{1}\right) !\left(j_{2}+m_{2}+u-t\right) !\left(j_{2}-m_{2}\right) !}{\left(j_{1}+m_{1}\right) !\left(j_{1}-m_{1}-t\right) !\left(j_{2}+m_{2}\right) !\left(j_{2}-m_{2}-u+t\right) !} \end{multline} and so we arrive at the following \begin{multline} \frac{(j_{1}+m_{1}+u+1)!(j_{1}-m_{1})!}{(j_{1}-m_{1}-u-1)!(j_{1}+m_{1})!}A_{j}(-1)^{u+1}+\\ A_{j} \sum_{t}(-1)^{t}\left(\begin{array}{l} u \\ t \end{array}\right) \frac{\left(j_{1}+m_{1}+t\right) !\left(j_{1}-m_{1}\right) !\left(j_{2}+m_{2}+u-t\right) !\left(j_{2}-m_{2}\right) !}{\left(j_{1}+m_{1}\right) !\left(j_{1}-m_{1}-t\right) !\left(j_{2}+m_{2}\right) !\left(j_{2}-m_{2}-u+t\right) !}\times \\ \Bigg[ \frac{u+1}{u+1-t}\left(j_{2}+m_{2}+u+1-t\right)\left(j_{2}-m_{2}-u+t\right)+\left(j_{1}-m_{1}-t\right)\left(j_{1}+m_{1}+1+t\right) \\-\left(j_{2}-m_{2}-u+t\right)\left(j_{2}+m_{2}+1+u -t \right) \Bigg] =0 /tag{3} \end{multline}
where $m=m_1+m_2$ , $u=j-m$ and $j=j_1+j_2$
Now $j_1-m_1=j_1-(m-m_2)=j_1-(j-u-m_2)=-j_2+m_2+u$ and so $-\left(j_{1}-m_{1}-t\right)=(j_2 -m_2-u +t)$
After some algebra last equation becomes \begin{multline} \frac{(j_{1}+m_{1}+u+1)!(j_{1}-m_{1})!}{(j_{1}-m_{1}-u-1)!(j_{1}+m_{1})!}A_{j}(-1)^{u+1}+ \\ A_{j} \sum_{t}(-1)^{t}\left(\begin{array}{l} u \\ t \end{array}\right) \frac{\left(j_{1}+m_{1}+t\right) !\left(j_{1}-m_{1}\right) !\left(j_{2}+m_{2}+u-t\right) !\left(j_{2}-m_{2}\right) !}{\left(j_{1}+m_{1}\right) !\left(j_{1}-m_{1}-t\right) !\left(j_{2}+m_{2}\right) !\left(j_{2}-m_{2}-u+t\right) !}\times \\ (j_2 -m_2-u +t)\Bigg[ \frac{u+1}{u+1-t}\left(j_{2}+m_{2}\right) -j-m-1 \Bigg] =0 \end{multline}
Can anyone tell why this last expression is zero?