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So I start with a sum-of-products:

$$f = \sum_{k=a}^b{\prod_{j=c}^d{g(j,k)}}$$

I'm wondering if we can somehow convert this into a sum of sums, ie:

$$f = \sum_{k=a}^b{\sum_{j=c}^d{h(j,k)}}$$

It's important to note that THE SUMS SHOULD BE ON THE OUTSIDE of the right hand side of the equation.


Using the identity/definition of the discrete product, found here:

$$\prod_x{f(x)} = E^{\displaystyle\sum_x{\log(f(x))}}$$

...we can convert a product to a sum. Now, my guess is, that if we somehow take the logarithm, or multiple logarithms of this, that we should somehow be able to convert this into a sum of sums. However, for the moment, the method escapes me.

Can we convert this into a sum of sums?

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