# On the Total Positivity and Accurate Computations of \(r\)-Bell Polynomial Bases
## 3 Bidiagonal Factorization of \(r\)-Stirling Matrices  
Let us recall that given \(n,m\in\mathbb{N}\) with \(m<n\), the (signless) Stirling number of the first kind, usually denoted by \(\big{[} \begin{smallmatrix}n\\ m\end{smallmatrix} \big{]}\), can be defined combinatorially as the number of permutations of the set \(\{1,\ldots,n\}\) having \(m\) cycles. On the other hand, the Stirling number of the second kind, usually denoted by \(\big{\{} \begin{smallmatrix}n\\ m\end{smallmatrix} \big{\}}\), coincides with the number of partitions of the set \(\{1,\ldots,n\}\) into \(m\) non-empty disjoint blocks (cf. [3]). Moreover, the Bell number \(B_{n}\) gives the total number of partitions of the set \(\{1,\ldots,n\}\); that is,  
\[B_{n}:=\sum_{m=0}^{n}\Big{\{} \begin{smallmatrix}n\\ m\end{smallmatrix} \Big{\}}.\]  
The first and second kinds of Stirling numbers can be recursively computed as follows,\[\left[ \begin{array}{c}n\\ k\end{array} \right]=(n-1)\left[ \begin{array}{c}n-1\\ k\end{array} \right]+\left[ \begin{array}{c}n-1\\ k-1\end{array} \right],\quad\left\{ \begin{array}{c}n\\ k\end{array} \right\}=k\left\{ \begin{array}{c}n-1\\ k\end{array} \right\}+\left\{ \begin{array}{c}n-1\\ k-1\end{array} \right\}, \tag{14}\]  
with the initial conditions  
\[\left[ \begin{array}{c}0\\ 0\end{array} \right]:=1,\quad\left[ \begin{array}{c}n\\ 0\end{array} \right]=\left[ \begin{array}{c}0\\ n\end{array} \right]:=0,\qquad\left\{ \begin{array}{c}n\\ n\end{array} \right\}:=1,\quad\left\{ \begin{array}{c}n\\ 0\end{array} \right\}=\left\{ \begin{array}{c}0\\ n\end{array} \right\}:=0,\quad n\in\mathbb{N}. \tag{15}\]  
The \(r\)-Stirling numbers are polynomials in \(r\) generalizing the regular Stirling numbers and they count some restricted permutations and partitions of sets with a given number of elements. For \(r\in\mathbb{N}\), the first kind \(r\)-Stirling number, denoted by \(\left[ \begin{array}{c}n\\ m\end{array} \right]_{r}\), counts the number of permutations of the elements \(\{1,\ldots,n\}\) having \(m\) cycles, such that \(1,2,\ldots,r\) are in distinct cycles. In addition, the second kind \(r\)-Stirling number, denoted by \(\left\{ \begin{array}{c}n\\ m\end{array} \right\}_{r}\), is defined as the number of set partitions of \(\{1,\ldots,n+r\}\) into \(m+r\) blocks such that the first \(r\) elements are in distinct blocks. Thus the \(r\)-Bell number \(B_{n}^{r}\) gives the total number of partitions of the set \(\{1,\ldots,n+r\}\), such that the first \(r\) elements are in distinct blocks; that is,  
\[B_{n}^{r}:=\sum_{m=0}^{n}\left\{ \begin{array}{c}n\\ m\end{array} \right\}_{r}.\]  
Clearly, for \(r=0\), the \(r\)-Stirling numbers coincide with the corresponding regular Stirling numbers. Theorems 1 and 2 of [3] prove that the \(r\)-Stirling numbers satisfy the recurrence relations verified by the regular Stirling numbers (see (14)); that is,  
\[\left[ \begin{array}{c}n\\ k\end{array} \right]_{r}=(n-1)\left[ \begin{array}{c}n-1\\ k\end{array} \right]_{r}+\left[ \begin{array}{c}n-1\\ k-1\end{array} \right]_{r},\quad\left\{ \begin{array}{c}n\\ k\end{array} \right\}_{r}=k\left\{ \begin{array}{c}n-1\\ k\end{array} \right\}_{r}+\left\{ \begin{array}{c}n-1\\ k-1\end{array} \right\}_{r}, \tag{16}\]  
but with different initial conditions,  
\[\left[ \begin{array}{c}n\\ k\end{array} \right]_{r}=0,\quad n<r,\qquad\left[ \begin{array}{c}n\\ k\end{array} \right]_{r}=\delta_{k,r},\quad n=r,\]  
and  
\[\left\{ \begin{array}{c}n\\ k\end{array} \right\}_{r}=0,\quad n<r,\qquad\left\{ \begin{array}{c}n\\ k\end{array} \right\}_{r}=\delta_{k,r},\quad n=r.\]  
Moreover, for some particular values, the corresponding \(r\)-Stirling numbers can be easily obtained:  
\[\left[ \begin{array}{c}n\\ n\end{array} \right]_{r}=\left\{ \begin{array}{c}n\\ n\end{array} \right\}_{r}=1,\quad n\geq r. \tag{17}\] \[\left[ \begin{array}{c}n\\ k\end{array} \right]_{r}=\left\{ \begin{array}{c}n\\ k\end{array} \right\}_{r}=0,\quad k>n,\] (18) \[\left[ \begin{array}{c}n\\ r\end{array}