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dummy indices / variables (was my own sum)

Daniel Lichtblau wrote:
> Thomas Lemm wrote:
> >
> > I want to implement another type of Sum to treat expressions in a
> > "physical" manner. But I need to know how Mathematica tackles the
> > Problem:
> >
> > Sum[F[x],{x,-a,a}]==Sum[F[y],{y,-a,a}]
> >
> > Help welcome
> >
> > Thomas Lemm
> >
> >
> Not sure exactly what you have in mind, but if you want to just have
> dummy summation variables replaced by a standard variable, call it
> 'local', you could do as below.
> In[10]:= mySum[a_,{x_,y_,z_}] := Block[{local},
>         ReleaseHold[Hold[Sum[a, {local,y,z}]] /. x->local]]
> In[11]:= mySum[F[x],{x,-a,a}]
> Out[11]= Sum[F[local], {local, -a, a}]
> In[12]:=  mySum[F[x],{x,-a,a}]==mySum[F[y],{y,-a,a}] Out[12]= True

This touches on a problem that I am tackling at the moment, which is
coming up with a method to let Mathematica recognise (and then perform
simplifications upon) equivalent expressions with dummy indices. 

For instance,
   Sum[f[a],{a,1,m}] + Sum[f[b],{b,1,m}] // FullSimplify

does not recognise that the two sums are identical, and misses what (to
a human being, at least!) is a trivial simplification.

What I am wondering is whether 

a) It is possible to extend Simplify or FullSimplify by specifying rules
that would allow them to spot expressions such as integrals or sums 
which differed only in their dummy index, especially in cases where you
have more than one dummy index, and complicated integrands.

b) Or, for that matter, how would you implement a "MySimplify" which
worked with a small subset of possible simplifications (including
simplifying over dummy indices)?

Any comment most appreciated! 

Richard Easther

PS If anyone is keeping track, I would add the ability to recognise
dummy indices to the wish-list for the next version of mathematica!

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