Advanced Search

Question: Splitting nested sums into a set of single sums

Question:Splitting nested sums into a set of single sums

jimbo_cambridge 5 Maple
sum
October 17 2011
1

Hello,

 

I have a derivation to do for my PhD thesis, and it is long and tedious. It is essentially a case of splitting sums, making substitutions and then simplifying. I believe maple can do this for me, but have not found how yet. Any suggestions would be more than welcome. I have done this for a third order problem (three nested sums) and the derivation runs to many pages. I now need to do so for the 4th order problem, and it is a more than daunting prospect

So, If I have a double sum such that Sum( Sum( x[i]*x[j] ) )

S:=Sum(Sum(x[i]*x[j] ,j=1 .. n), i = 1 .. n);

I can split this sum into two parts, one where i=j (which I will call k such that Sum( x[k]^2 ) ), and one where i≠j. This i≠j term can be written in terms of two single sums multiplied together such that Sum(x[i])*Sum(x[j])

S: = Sum(x[k]^2,k = 1 .. n) + Sum(x[i] , i = 1 .. n) * Sum(x[j] , j = 1 .. n);

which allows me to make some substitutions to work the derivation. My question is, for more complicated nested sums such as

S:=Sum(Sum(Sum(x[p]*x[q]*x[r],p=1 .. n),q=1 .. n)r=1 .. n);

can I have maple return me the single sum series, ie. an sum(x^3) term, 3 terms with sum(x^2)*sum(x) and one term with sum(x)*sum(x)*sum(x)? I would then have maple make a substitution (swap the Sum(x) for Sum(y)) and then do some simplification as a lot of terms will cancel. I should point out that it is not possible to make the substitutions before expanding the sums as the moments 'y' are not the same as 'x' i.e. x=y but x^3 ≠y^3.

Thanks in advance, I hope this is trivial for maple gurus.

 

Jimbo
Please Wait...