# Problem with infinite sum

I am facing a problem in evaluating infinite sums of the following form.

As a first thing define the following function

Z[n_] := Product[1/(1 - t^i), {i, 1, n}]

P[list_] := (num = Union[list]; Product[Z[Count[list, num[[i]]]], {i, 1, Length[num]}])


Then consider a sum of the following kind

Sum[P[{i, j}]*t^(-i - j), {i, 0, ∞}, {j, 0, ∞}]


Now, Mathematica behaves in a strange way: it considers the P[{i,j}] factor as if i is always different than j. This does not make sense, since for example I am summing over the couples (2,2) and (3,3).

I believe the problem relies in the way the infinity is considered, since everything works fine with arbitrary high finite sums.

What could I do?

• You are attempting to have a symbolic summation understand procedurally computed arguments (that is, they require Union and Count). I cannot imagine how one might get this to work in its current form. Commented Sep 9, 2014 at 21:26

a bit of a cheat maybe:

 Z[n_] := Product[1/(1 - t^i), {i, 1, n}]
P1[list_] := (Z[1]^2)
P2[list_] := Z[2]
Sum[P1[{i, j}]*t^(-i - j), {i, 0, j - 1}, {j, 0, Infinity}] +
Sum[P2[{j, j}]*t^(-2 j), {j, 0, Infinity}] +
Sum[P1[{i, j}]*t^(-i - j), {i, j + 1, Infinity}, {j, 0, Infinity}] // Simplify


(t (-(-1 + t) t^-j + (t^2 (3 + t))/(1 + t)^2))/(-1 + t)^4

• This does work, but it doesn't solve the problem at all. Probably it's my fault that I have not explained well. I would like P[list_], defined as it is, to work for any list and not only a list of 2 elements. If for example I have 894 elements in the list, your method of splitting the sum by hand, in order to insert the right factor, will be quite annoying to carry out. Commented Jul 12, 2014 at 17:38