COMPUTING SOME INFINITE SUMS WITH THE AID OF POLYGAMMA FUNCTION
The initial goal of this post was to proof the following relations seen in this Twitter post . However, since I wanted to show all the details of the computation, we will only proof the third and fourth relations, and leave the other four for a future post. So, in order to proof these two results we will first proof the well known Legendre duplication formula for The Gamma function and then compute some special values of the Quadrigamma(?) function. After these results are established we will be able to proof the two above mentioned relations. As a bonus, in the end of the post we will also compute the value of the Dirichlet beta function of order 3 that comes easily as a consequence of the evaluation of these two infinite sums. It´s a long post, but I think it´s worth. Hope you enjoy it! Legendre Duplication Formula for the Gamma Function Recall the Beta function Let (1) let in the second integral, then Now let Then Note also that and therefo