I encountered two Laurent polynomials
$$\sum_{j=0}^k\sum_{
\mbox{$\begin{array}{c}
i_1,\dots,i_k\geq 0\\
i_0=-N-i_1-\dots-i_k\end{array}$}}
\sum_{m=0}^{i_j}
\begin{pmatrix}
-N\\
i_1,\dots,i_k
\end{pmatrix}
\begin{pmatrix}
i_j\\
m
\end{pmatrix}
x_0^{i_0}\cdots x^{i_j-m}\cdots x_k^{i_k}
$$ and
$$
\sum_{j=0}^k\sum_{N\geq p_1\geq\cdots\geq p_{n}\geq 0}
\sum_{
\mbox{$\begin{array}{c}
i_1,\dots,i_k\geq 0\\
i_0=-N-i_1-\dots-i_k-p_1-\cdots- p_{n}\end{array}$}}
\begin{pmatrix}
N\\
p_1
\end{pmatrix}
\begin{pmatrix}
p_1\\
p_2
\end{pmatrix}\cdots
\begin{pmatrix}
p_{n-1}\\
p_{n}
\end{pmatrix}
\begin{pmatrix}
-(N+p_1+\cdots+p_n)\\
i_1,\dots,i_k
\end{pmatrix}
x_0^{i_0}\cdots x^{i_j+p_1+\cdots+p_n}\cdots x_k^{i_k},
$$
where $N,n,k$ are given positive integers and
$$
\begin{pmatrix}
l\\
i_1,\dots,i_k
\end{pmatrix}=
\begin{pmatrix}
l\\
i_1
\end{pmatrix}
\begin{pmatrix}
l-i_1\\
i_2
\end{pmatrix}\cdots
\begin{pmatrix}
l-i_1-\cdots-i_{k-1}\\
i_k
\end{pmatrix}
$$
for any integer $l$, positive integers $i_1,\dots,i_k$.
I need to prove that they are actually the same.
Can anyone help me? or just give me some hint.
I'll be grateful.
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2
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$\begingroup$ Did you check that they are the same? (Say, by a computer experiment.) Accidentally, what are the parameters? $N$ and $k$? At first sight, your "polynomials" look more like power series. $\endgroup$– Alex DegtyarevMar 14, 2014 at 12:30
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$\begingroup$ There is no $i_0$, so $j$ should run in both sums from 1 to k. Using $\sum_{m=0}^{i} \binom imx^{i-m}=\sum_{m=0}^{i} \binom imx^{m}=(1+x)^i$ with $i=i_j$ repeatedly for each $j$, you should come up with some power of $(k+x)$ for both expressions. I wonder though if your summations are well-defined. $\endgroup$– WolfgangMar 14, 2014 at 18:09
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