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I have a question about combinatorics/gamma functions. I would like some help with either disproving or proving the following statement:

Given $k\in\mathbb{N}, m_{1},...,m_{n}\in\mathbb{N}$ such that $m_{1}+...+m_{n}=k$ and $a_{1},...,a_{n} $ positive real numbers and $r_{1},...,r_{n}$ also positive real numbers.


$$\begin{eqnarray}\underset{m_{1}+...+m_{n}=k}{\sum}&\frac{\overset{n}{\underset{i=1}{\prod}}\left(\begin{array}{c} m_{i}+r_{i}\\ m_{i}\end{array}\right)a_{i}^{m_{i}}}{\left(\begin{array}{c} \underset{i=1}{\overset{n}{\sum}}r_{i}+k-1\\ k\end{array}\right)}=\frac{\left(a_{1}r_{1}+...+a_{n}r_{n}\right)^{k}}{\left(\underset{i=1}{\overset{n}{\sum}}r_{i}\right)^{k}}, \end{eqnarray}$$ where the sum is taken over all natural numbers such that $m_{1} + ... + m_{n}=k$.

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Where did this come from? Is it part of some more extensive development of combinatorial identities? Seeing the context may help someone to approach the result. –  Gerry Myerson Jan 19 '12 at 21:41

1 Answer 1

This statement is false. For example, if $k=2$ and $n=1$, it becomes $$ \frac{{r_1+2 \choose 2}a_1^2}{{r_1+1 \choose 2}}=\frac{(a_1 r_1)^2}{r_1^2}, $$ which is never true if $a_1$ and $r_1$ are positive real.

If you take the right-hand side of the proposed statement and expand it as a sum of monomials $C_{(m_1,\ldots,m_n)} \prod_i a_i^{m_i}$, you get $$ \sum_{m_1+\cdots+m_n=k} \frac{{k\choose m_1\ \cdots\ m_n}\prod_i r_i^{m_i}}{(\sum_i r_i)^k}\prod_i {a_i}^{m_i}=\frac{(\sum_i a_i r_i)^k}{(\sum_i r_i)^k}. $$ This is similar to the proposed statement, but contains ordinary powers of the $r_i$'s and of $\sum_i r_i$ instead of rising powers.

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