Re: Splitting sums in mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg126832] Re: Splitting sums in mathematica
• From: Ray Koopman <koopman at sfu.ca>
• Date: Mon, 11 Jun 2012 00:02:27 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jqusrg\$rnm\$1@smc.vnet.net> <jr1dvt\$73t\$1@smc.vnet.net>

```On Jun 9, 11:14 pm, Ray Koopman <koop... at sfu.ca> wrote:
> On Jun 9, 12:09 am, jimbo <james.a.gord... at googlemail.com> wrote:
>> Hello,
>>
>> 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)
>>
>> ...in 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
>> different?
>>
>> 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
>>
>>
>> James
>
> Let a = {a1,a2,...}. Then a.a gives the sum of the terms with p == q,
> and 2*Dot@@Transpose@Subsets[a,{2}] gives the sum of the terms with
> p != q.

If you have to do that many times then you might want to define
{p,q} = Transpose@Subsets[Range@Length@a,{2}]  once, and then
use  2*a[[p]].a[[q]]  to sum the terms whose subscripts differ.

```

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