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If I have an expression such as :

$$\prod_{k=0}^K \left \{ \sum_{n=0}^N a_{n,k} \right \} $$

If we expand a bit, we get:

$$\prod_{k=0}^M \left \{ a_{0,k} +a_{1,k}+...+a_{N-1,k}+ a_{N,k} \right \}$$ $$(a_{0,0} +a_{1,0}+...+a_{N-1,0}+ a_{N,0})(a_{0,1} +a_{1,1}+...+a_{N-1,1}+ a_{N,1})...(a_{0,M} +a_{1,M}+...+a_{N-1,M}+ a_{N,M})$$ How can this be represented concisely in matrix form? Any other interesting properties/theorems on this (i.e. convergence for when M and/or N $\rightarrow \infty$...etc.)?

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Let $u=(1,1,...,1)^T$ be the $N$-dimensional vector with $1$ as its components, and $a_k=(a_{0,k}, a_{1,k},...,a_{N,k})^T$.

If $K=0$, then the product is $\langle a_0, u\rangle=a_0^Tu=u^Ta_0$.

If $K=1$, then the product is $a_0^Tuu^Ta_1=a_0^TMa_1$, where $M=u^Tu=u\otimes u$ is the $N\times N$ matrix full of $1$s.

For $K>1$, then the product cannot be simply expressed with matrices (unless you want to keep the product somewhere), but you can use the tensor $M=u\otimes u\otimes \ ... \, \otimes \, u$ with $1$s as entries.

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