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Splitting sums in mathematica


Does anyone know of a way to algebraically split sums (if that is the
correct name for it) in mathematica? I need to do the following (and
higher-order versions of the same)...

Sum_over_p [  Sum_over_q (  a_p * a_q  )   ] = Sum_over_p_equal_q
(a_p^2) + Sum_over_p_not_equal_q (a_p*a_q) other words, turn a nested sum running over two indices into two
sum, one for when the indices are equal, and one for when they are

I then need to make some substitutions once there are no nested sums
and simplify my answer. I have an equation with many hundreds of terms
in it, so doing it by hand is not an option. I can send a typeset
example if anyone needs clarification

I have version 5.2 for students

Thanks in advance


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