Evaluating moments of length of Pitman partition

The Pitman sampling formula has been intensively studied as a distribution of random partitions. One of the objects of interest is the length $K (= K_{n,\theta,\alpha})$ of a random partition that follows the Pitman sampling formula, where $n\in\mathbb{N}$, $\alpha\in(0,\infty)$ and $\theta > -\alpha$ are parameters. This paper presents asymptotic evaluations of its $r$-th moment $\mathsf{E}[K^r]$ ($r=1,2,\ldots$) under two asymptotic regimes. In particular, the goals of this study are to provide a finer approximate evaluation of $\mathsf{E}[K^r]$ as $n\to\infty$ than has previously been developed and to provide an approximate evaluation of $\mathsf{E}[K^r]$ as the parameters $n$ and $\theta$ simultaneously tend to infinity with $\theta/n \to 0$. The results presented in this paper will provide a more accurate understanding of the asymptotic behavior of $K$.