Derivation of formula in Theory of Complex Spectra. II of Giulio Racah

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 mright)
=(-1)^{j_{1}-m_{1}} fleft(m_{1} m_{2} ; j mright)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 $left(m_{1} m_{2} mid j mright)$ are the Clebsch-Gordan coefficients.Then, he shows that
begin{multline}
fleft(m_{1} m_{2} ; j m-1right)= left(j_{2}+m_{2}+1right)left(j_{2}-m_{2}right) fleft(m_{1} m_{2}+1 ; j mright)- left(j_{1}+m_{1}+1right)left(j_{1}-m_{1}right) fleft(m_{1}+1 m_{2} ; j mright) tag 1
end{multline}
Now he claims that
begin{multline}
fleft(m_{1} m_{2} ; j mright)=fleft(m_{1} m_{2} ; j -uright)= A_{j} sum_{t}(-1)^{t}left(begin{array}{l}
u
t
end{array}right) frac{left(j_{1}+m_{1}+tright) !left(j_{1}-m_{1}right) !left(j_{2}+m_{2}+u-tright) !left(j_{2}-m_{2}right) !}{left(j_{1}+m_{1}right) !left(j_{1}-m_{1}-tright) !left(j_{2}+m_{2}right) !left(j_{2}-m_{2}-u+tright) !}
tag 2end{multline}
Where 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}+1right)left(j_{1}-m_{1}right) fleft(m_{1}+1 m_{2} ; j mright)= A_{j} sum_{t}(-1)^{t}left(begin{array}{l}
u
t
end{array}right) left(j_{1}-m_{1}-tright)left(j_{1}+m_{1}+1+tright) times frac{left(j_{1}+m_{1}+tright) !left(j_{1}-m_{1}right) !left(j_{2}+m_{2}+u-tright) !left(j_{2}-m_{2}right) !}{left(j_{1}+m_{1}right) !left(j_{1}-m_{1}-tright) !left(j_{2}+m_{2}right) !left(j_{2}-m_{2}-u+tright) !}
end{multline}

begin{multline}
left(j_{2}+m_{2}+1right)left(j_{2}-m_{2}right) fleft(m_{1} m_{2} +1 ; j mright)= A_{j} sum_{t}(-1)^{t}left(begin{array}{l}
u
t
end{array}right) left(j_{2}-m_{2}-u+tright)left(j_{2}+m_{2}+1+u -t right) times frac{left(j_{1}+m_{1}+tright) !left(j_{1}-m_{1}right) !left(j_{2}+m_{2}+u-tright) !left(j_{2}-m_{2}right) !}{left(j_{1}+m_{1}right) !left(j_{1}-m_{1}-tright) !left(j_{2}+m_{2}right) !left(j_{2}-m_{2}-u+tright) !}
end{multline}

and
begin{multline}
fleft(m_{1} m_{2}; j m-1right)= 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-tright)left(j_{2}-m_{2}-u+tright) times frac{left(j_{1}+m_{1}+tright) !left(j_{1}-m_{1}right) !left(j_{2}+m_{2}+u-tright) !left(j_{2}-m_{2}right) !}{left(j_{1}+m_{1}right) !left(j_{1}-m_{1}-tright) !left(j_{2}+m_{2}right) !left(j_{2}-m_{2}-u+tright) !}
end{multline}
and so we arrive at the following
begin{multline}
fleft(m_{1} m_{2}; j m-1right)= A_{j} sum_{t}(-1)^{t}left(begin{array}{l}
u
t
end{array}right) frac{left(j_{1}+m_{1}+tright) !left(j_{1}-m_{1}right) !left(j_{2}+m_{2}+u-tright) !left(j_{2}-m_{2}right) !}{left(j_{1}+m_{1}right) !left(j_{1}-m_{1}-tright) !left(j_{2}+m_{2}right) !left(j_{2}-m_{2}-u+tright) !}times
Bigg[ frac{u+1}{u+1-t}left(j_{2}+m_{2}+u+1-tright)left(j_{2}-m_{2}-u+tright)+left(j_{1}-m_{1}-tright)left(j_{1}+m_{1}+1+tright) -left(j_{2}-m_{2}-u+tright)left(j_{2}+m_{2}+1+u -t right) Bigg] =0
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
$-left(j_{1}-m_{1}-tright)=(j_2 -m_2-u +t)$

After some algebra last equation becomes
begin{multline}
fleft(m_{1} m_{2}; j m-1right)= A_{j} sum_{t}(-1)^{t}left(begin{array}{l}
u
t
end{array}right) frac{left(j_{1}+m_{1}+tright) !left(j_{1}-m_{1}right) !left(j_{2}+m_{2}+u-tright) !left(j_{2}-m_{2}right) !}{left(j_{1}+m_{1}right) !left(j_{1}-m_{1}-tright) !left(j_{2}+m_{2}right) !left(j_{2}-m_{2}-u+tright) !}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 equation is true?