Asymptotic approximation for the r-associated Stirling numbers of the second kind

Question

It is well know that for fixed $k$ the asymptotic approximation for the Stirling numbers of the second kind is given by $\frac{k^n}{k!}$. Does such simple asymptotic expression also exist for the r-associated Stirling numbers of the second kind? Thank you,

Answer 1

From the recurrence identity $S_2(n,k)=n! \sum_{j=0}^k \frac{(-1)^jS(n-j-k-j)}{j!(n-j)!}$ given by Howard and the approximation for the Stirling number of the second kind we can get that:

$S_2(n,k) \sim \frac{k^n}{k!}$ and maybe more interesting that $S(n,k)-S_2(n,k) \sim \frac{n(k-1)^{n-1}}{(k-1)!}$

Higher order associated Stirling of the second kind can be derived in a similar way by analyzing the general recurrence identity given at Howard.