Series with binomial coefficients

AI Thread Summary
The discussion centers around a complex equation involving binomial coefficients, presented in Mathematica code, which simplifies to an identity according to Mathematica's FullSimplify command. The user seeks an analytical explanation for why this equation holds true, despite the output confirming its validity. All parameters in the equation are stated to be weakly positive and real. The user also attempts to express the equation in LaTeX format but struggles with the formatting. The conversation highlights the challenge of understanding the analytical reasoning behind the identity, despite computational verification.
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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!
 
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Try translating the equation into latex form.
 
\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]
 
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
 

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