My question is about second kind r-Stirling numbers. Here are two important papers about it. https://www.sciencedirect.com/science/article/pii/0012365X84901614#:~:text=The%20r%2DStirling%20numbers%E2%98%86&text=The%20r%2DStirling%20numbers%20of,cycles%20and%20respectively%20distinct%20subsets. Broder/ r-Stirling numbers. https://www.sciencedirect.com/science/article/pii/S0012365X14001241#:~:text=3.&text=%2DLah%20numbers%20The%20%2DLah%20numbers,be%20in%20distinct%20ordered%20blocks. Ryul and Nacz / r-Lah numbers.
In Broder / r-Stirling numbers article, r-Stirling numbers of second kind are defined; $\left\{\begin{array}{l}n \\ m\end{array}\right\}_{r}=$ The number of partitions of the set $\{1, \ldots, n\}$ into $m$ non-empty disjoint subsets ,such that the numbers $1,2,3...,r$ are in distinct subsets.
I try to understand the definition of second kind r-Stirling numbers. For r-Stirling numbers r is a natural number. Can the range of r be extended to rational numbers or complex numbers? Thanks.
Not sure what are your intentions, but you can use the known expression $${n\brace k}_r=\sum _{i=0}^n\binom{n}{i}{i\brace k}r^{n-i}$$ to extend it. That comes from the fact that you can choose the numbers that are going to be in the first $r$ subsets(say $n-i$ of them) in $\binom{n}{n-i}$ ways, and the rest elements have to be in $k$ blocks in ${i\brace k}$ ways. To distribute the $n-i$ elements in the $r$ first blocks, you can use any function in $r^{n-i}$ ways.
For example, for $r<0$ and $n,k$ even, one can express ${n\brace k}_r$ using Stirling numbers of higher level (see remark 3.1 here).