I'm trying to calculate a pretty complicated sum, $ S_n $, which in the end satisfies the relationship. $ (S_n + T_n) m ^ 3 = ( frac {(n + 1) ^ 2 – 1} {n + 1}) m ^ 3 $. I should keep in mind that $ T_n $ It is also unknown. There are many intermediate sums in this calculation, for example:

```
summand = ((3 + i + 8 j + 4 j ^ 2 - m) (-2 + 2 j - n) (5 i + 8 i j + 2 i j ^ 2 -
5 m - 8 j m - 2 j ^ 2 m - i n - i j n + m n + j m n)) / (4 (1 + 2 j) ^ 2 (3 + 2 j) ^ 2)
```

```
Sum[summand, {j, r, n/2-3,2}, Assumptions-> Mod[n,4]== 0 && Mod[r,2]== 1]// FullSimplify
```

The output is a collection of PolyGamma functions:

```
- (1/512) (i - m) (-32 - 32 n + 24 n ^ 2 - 64 n r + 32 r ^ 2 +
8 (9 - 3 m - n (2 m + n) + i (3 + 2 n)) PolyGamma[0,
1/4 (-1 + n)] -
8 (5 i - 5 (5 + m) + 2 i n - 2 m n + n ^ 2) PolyGamma[0, (1 + n)/
4] + 8 (-9 - 3 i + 3 m - 2 i n + 2 m n + n ^ 2) PolyGamma[0,
1/4 + r/2] +
8 (5 i - 5 (5 + m) + 2 i n - 2 m n + n ^ 2) PolyGamma[0,
3/4 + r/2] + (i - m) (-9 + n ^ 2) PolyGamma[1,
1/4 (-1 + n)] + (-i + m) (-25 + n ^ 2) PolyGamma[1, (1 + n)/
4] + (-i + m) (-9 + n ^ 2) PolyGamma[1,
1/4 + r/2] + (i - m) (-25 + n ^ 2) PolyGamma[1, 3/4 + r/2])
```

Unfortunately there are many of these sums on the way to computing $ S_n $ and, finally, tracking the PolyGamma term collection becomes too difficult for Mathematica to handle and computing stops.

I am curious about the possibility that there are other methods to evaluate definite sums of rational functions like mine that can produce cleaner results. I've been looking at some of the software libraries that use the Gosper algorithm and the telescopic creative libraries described here, https://www3.risc.jku.at/research/combinat/software/ergosum/RISC/fastZeil.html, but until now It has not been possible to produce a solution in a closed form.

I would expect a *rational* expression of closed form exists for $ S_n $ Based on the previous relationship, then, is there a way to find a rational expression for these intermediate sums?