Series with binomial coefficients

[ydydry]

Hi all, I have an apparently simple equation. I copy here its Mathematica code:

Sum[(p/(1 - p))^s*(q/(1 - q))^s*Binomial[n, s]*(Binomial[m - 1, s]*(p*q*(m + n) + (2*m - 1)*(-p - q + 1))), {s, 0, n}] == Sum[(p/(1 - p))^s*(q/(1 - q))^s*Binomial[n, s]*((-(-p - q + 1))*Binomial[m - 2, s] + m*p*q*Binomial[m, s] + m*(-p - q + 1)*(Binomial[m - 2, s] + Binomial[m, s])), {s, 0, n}]

Mathematica's FullSimplify command immediately tells me that it is an identity, giving me "True" as output, but I fail to see the analytical reason.

All parameters are weakly positive and reals, although I do not need to assume anything for Mathematica to tell me that it is indeed an identity.

Thanks a lot!

 

[mathman]

Try translating the equation into latex form.

 

[ydydry]

\overset{n}{\underset{s=0}{\sum }}\left( \frac{pq}{(1-p)(1-q)}\right)
^{s}\left( \begin{array}{c}n \\
s%
\end{array}%
\right) \left[ \left( \begin{array}{c}m-1 \\
s%
\end{array}%
\right) (pq(m+n)+(2m-1)(1-p-q))\right] =\overset{n}{\underset{s=0}{\sum }}%
\left( \frac{pq}{(1-p)(1-q)}\right) ^{s}\left( \begin{array}{c}n \\
s%
\end{array}%
\right) \left[ (m-1)\left( \begin{array}{c}m-2 \\
s%
\end{array}%
\right) (1-p-q)+\left( \begin{array}{c}m \\
s%
\end{array}%
\right) mpq+\left( \begin{array}{c}m \\
s%
\end{array}%
\right) m(1-p-q))\right]

 

[ydydry]

I apologize for the last attempt to write the code in Latex. I am not familiar with the software, and I clearly failed. I enclose a picture of the expression, which should be more readable than the non-sense above code

 

[mathman]

It looks messy enough. Try looking at the expression to the right of nCs on both sides and see if they are equal.