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I have an expression that is a product of two symbolic sums.

Sum[a[k], {k, 1, n}]*Sum[b[k], {k, 1, n}]

How can I expand this expression? I want to see something like this:

Sum[Sum[a[k1]*b[k2], {k2, 1, n}], {k1, 1, n}]

If the running indices weren't the same, I would try to fashion a transformation rule to match the situation above. But here, I'd have to create a new symbol, so I am unsure where to even start.

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  • $\begingroup$ "I'd have to create a new symbol" - this is where the built-in dummy variable K comes in handy; you can just systematically replace indices with K[1], K[2], etc. $\endgroup$
    – J. M.'s eventual burnout ♦
    Jan 4, 2021 at 14:21

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Note, a double sum can be written with a single Sum. With this we make a replacement:

   Sum[a[k], {k, 1, n}]*Sum[b[k], {k, 1, n}] /. 
     Sum[a1_, {a2_, 1, n}] Sum[b1_, {b2_, 1, n}] :> 
      Sum[(a1 /. a2 -> k1) (b1 /. b2 -> k2), {k1, 1, n}, {k2, 1, n}]

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