Approximation of the moments of arbitrary integer orders of generalized factorial powers

For non-negative integer random variables $\xi$, we consider approximations of the moments $\boldsymbol{\mathsf E}\xi^m$, where $m$ are integers, including negative integers. We find estimates of the difference
$$ \boldsymbol{\mathsf E}\xi^m - \sum_{k=0}^s\genfrac{\{}{\}}{0mm}{}m{m-k}\boldsymbol{\mathsf E}\xi^{\underline {m-k}}, $$
where $\genfrac{\{}{\}}{0mm}{}m{m-k}$ are extensions to all integers $m$ of Stirling numbers of the second kind, the functions $ x^{\underline m}$ are the generalised factorial powers, and $s$ is a positive integer.