 Solution

Writing $p_n=\mathbb{P}(X=n)$, the ordinary probability generating function isIts finite sum can be differentiated termwise, givingThe Dirichlet probability generating function is the different transform $q(z)=\mathbb{E}[X^{-z}]$. Evaluating at $z=-1$ gives the first ordinary moment, whereas differentiation gives $q^{\prime }(z)=-{\sum }_n{p}_n(\log n)n^{-z}$. ThereforeThere is no convergence issue because the support is finite and all its values are positive.