= Dirichlet probability generating function
{c}
{title2=$q_X(z)=\mathbb E[X^{-z}]$}
For a positive integer-valued <random variable> of finite <support of a probability distribution>[support], this transform is a finite <Dirichlet series>. Its values encode ordinary <moments>, $q_X(-r)=\mathbb E[X^r]$, and its <derivatives> at zero encode <moment>[logarithmic moments]: $(-1)^rq_X^{(r)}(0)=\mathbb E[(\log X)^r]$. Positive <support of a probability distribution>[support] is essential to these definitions; the transform is distinct from the ordinary <probability generating function> $\mathbb E[s^X]$.
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