Let denote the trigamma function. Prove that
Lemma 1: It holds that .
Lemma 2: It holds that .
Lemma 3: We define with .
Lemma 4: It holds that .
and the result follows.
Lemma 5: It holds that .
Lemma 6: It holds that .
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.