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I wish to compute compute terms like Sum[f[t[j[1]],t[j[2]],...],{j[1],m},{j[2],n},...] for arbitrary positive integer n and any number of indices j[1],...,j[m]. I also wish to able to do differentiations like D[f,t[j[k]]] and obtain the correct result.

It seems you cannot do the folowing:

f = Sum[1/(1 - x Exp[I (t[j[1]] - t[j[2]])]),Evaluate[Sequence @@ Table[{j[k], n}, {k, 2}]]]
t /: D[t[a_], t[b_], NonConstants -> {t}] := KroneckerDelta[a, b]
D[f, t[m], NonConstants -> t]

The kernel keeps running without simplifying the output.

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