Let denote the trigamma function. Prove that

*(Seraphim Tsipelis)*

**Solution**

**Lemma 1: **It holds that .

*Proof:*

**Lemma 2: **It holds that .

*Proof:*

**Lemma 3: **We define with .

**Lemma 4: **It holds that .

*Proof:*

and the result follows.

**Lemma 5: **It holds that .

**Lemma 6: **It holds that .

*Proof:*

The proof relies on Multiple Zeta functions. Since it holds that

then we have that

Now returning to the question in hand we have that

and what remains in order to complete the exercise is the evaluation of those two last Euler sums. We proceed with the second one.

while for the first one

Combining the results we get what we want.