New properties of \(r\)-Stirling series (Q949850)

The Stirling numbers of the first kind denoted \([{{n}\over {m}}]\) count the number of permutations of \(n\) elements with \(m\) cycles, while the Stirling numbers of the second kind \(\{{{n}\over{m}}\}\) count the number of partitions of a set with \(n\) elements into \(m\) nonempty disjoint subsets. For a positive integer \(r\), let \([{{n}\over {m}}]_r\) be the \(r\)-Stirling numbers of the first kind, which count the number of permutations of the set with \(n\) elements in \(m\) disjoint cycles such that the first \(r\) elements are in disjoint cycles, and similarly \(\{{{n}\over {m}}\}_r\) be the \(r\)-Stirling numbers of the second kind which count the number of partitions of a set with \(n\) elements into \(m\) disjoint subsets such that the first \(r\) elements are in disjoint distinct subsets. Putting \[ G_{k,m,q}^{1,r}(z)=\sum_{n=1}^{\infty}\left[{{n+k+r}\over {m+r}}\right]_r{{z^n}\over {n^q n!}} \] and \(G_{k,m,q}^{2,r}(z)\) for the analogous sum where the \(r\)-Stirling number of the first kind are replaced by the \(r\)-Stirling numbers of the second kind, in this paper, the author determines recurrences for the above numbers. For example, \[ G_{0,m,q}^{1,r}(z)={{1}\over {r-1}}\left(G_{0,m,q-1}^{1,r-1}+(r-1)G_{0,m,q}^{1,r-1}-G_{0,m-1,q}^{1,r}\right)(z) \] and for \(k\geq 1\), \[ G_{k,m,q}^{1,r}(z)=G_{k-1,m,q-1}^{1,r}(z)+(k+r-1)G_{k-1,m,q}^{1,r}(z)+G_{k-1,m-1,q}^{1,r}(z) \] (Theorem 1). Similar recurrences are given for \(G_{k,m,q}^{2,r}\) (Theorems 3 and 4). The last section of the paper deals with the asymptotic behaviors of the \(r\)-Stirling numbers of the first and second kind.