Nested Sums to multiple sum

Question

I would like to automatically "move nested sums to the left". I mean, just take out of an expression all the summations and go from a nested Sum to a multiple sum. Something like starting with:

Sum[f[y] Sum[g[x],x],y]

And changing it into:

Sum[ f[y] g[x],y,x]

But I can't think of a way to do it.

Thanks!

Answer 1

Since my comment seamed to answer the question I'll expand it to an answer. The general notion is simply that when you want to manipulate an expression you can use patterns and replacement to achieve that goal. Here I simply took almost verbatim the replacement you wanted and inserted some patterns to match. The reason x___ and y___ are used is simply so the rule will work for any number of nested sum:

Sum[f[y] Sum[g[x], x], y] //. Sum[a_ Sum[b_, x___], y___] :> Sum[a b, x, y]
(* Sum[f[y]*g[x], x, y] *)

Sum[ m[d] Sum[f[y] Sum[g[x], x], y], d] //. Sum[a_ Sum[b_, x___], y___] :> Sum[a b, x, y]
(* Sum[f[y]*g[x]*m[d], x, y, d] *)