# On the Total Positivity and Accurate Computations of \(r\)-Bell Polynomial Bases
## 3 Bidiagonal Factorization of \(r\)-Stirling Matrices
\right]_{r}=\vec{r^{n-r}}:=(n-1)(n-2)\ldots r,\quad k>n,\] (19) \[\left\{ \begin{array}{c}n\\ r\end{array} \right\}_{r}=r^{n-r}. \tag{20}\]  
Finally, let us recall a \"cross\" recurrence relating \(r\)-Stirling numbers of the second kind with a different \(r\) that is going to be used in this paper (see Theorem 4 of [3]). The r-Stirling numbers of the second kind satisfy  
\[\left\{ \begin{array}{c}n\\ k\end{array} \right\}=\left\{ \begin{array}{c}n\\ k\end{array} \right\}_{r-1}-(r-1)\left\{ \begin{array}{c}n-1\\ k\end{array} \right\}_{r-1},\quad 1\leq r\leq n. \tag{21}\]  
Now, we are going to define triangular matrices whose entries are given in terms of \(r\)-Stirling numbers. We shall analyze their Neville elimination and derive their bidiagonal factorization (5). **Definition 1**. : _For \(r,n\in\mathbb{N}\) with \(0\leq r\leq n\), the \((n+1)\times(n+1)\) the first kind of \(r\)-Stirling matrix is the lower triangular matrix \(L_{n}^{r}=(l_{i,j}^{r})_{1\leq i,j\leq n+1}\) with_  
\[l_{i,j}^{r}:= \begin{bmatrix}i+r-1\\ j+r-1\end{bmatrix} _{r},\quad 1\leq i,j\leq n+1. \tag{22}\]  
_On the other hand, the \((n+1)\times(n+1)\) second kind of \(r\)-Stirling matrix, is the lower triangular matrix \(L_{n}^{r}=(\bar{l}_{i,j})_{1\leq i,j\leq n+1}\) with_  
\[\bar{l}_{i,j}^{r}:= \begin{cases}i+r-1\\ j+r-1\end{cases} _{r},\quad 1\leq i,j\leq n+1. \tag{23}\]  
The following results provide the expression of the pivots and multipliers of the Neville elimination of \(r\)-Stirling matrices and their inverses. Then, their bidiagonal factorization (5) will be deduced and their total positivity property analyzed. **Theorem 2**. : _For \(r,n\in\mathbb{N}\) with \(r\leq n\), the first kind of \(r\)-Stirling matrix \(L_{n}^{r}\in\mathbb{R}^{(n+1)\times(n+1)}\) is totally positive and admits the following factorization_  
\[L_{n}^{r}=F_{n}\ldots F_{1}, \tag{24}\]  
_where \(F_{i}\), \(i=1,\ldots,n\), are the lower triangular bidiagonal matrices of the form (6), whose off-diagonal entries are given by_  
\[m_{i,j}=i+r-j-1,\quad 1\leq j<i\leq n+1. \tag{25}\]  
_Moreover,_  
\[(L_{n}^{r})^{-1}=\widehat{F}_{n}\ldots\widehat{F}_{1}, \tag{26}\]  
_where \(\widehat{F}_{i}\), \(i=1,\ldots,n\), are the lower triangular bidiagonal matrices of the form (6), whose off-diagonal entries are given by_  
\[m_{i,j}=-(r+j-1),\quad 1\leq j<i\leq n+1. \tag{27}\]  
Proof. : Let us define \(L^{(1)}:=L_{n}^{r}\) and \(L^{(k)}=(l_{ij}^{(k)})_{1\leq i,j\leq n+1}\), \(k=2,\ldots,n\), the matrix obtained after \(k-1\) steps of the Neville elimination of \(L_{n}^{r}\). We shall deduce by induction on \(k\in\{1,\ldots,n+1\}\) that  
\[l_{i,j}^{(k)}= \begin{bmatrix}i+r-k\\ j+r-k\end{bmatrix} _{r}. \tag{28}\]  
For \(k=1\), identities (28) follow from (22). Now, suppose that (28) is verified for some \(k\in\{1,\ldots,n\}\). Then, taking into account (19), we can write  
\[l_{i,k}^{(k)}/l_{i-1,k}^{(k)}= \begin{bmatrix}i+r-k\\ r\end{bmatrix} _{r}/ \begin{bmatrix}i+r-k-1\\ r\end{bmatrix} _{r}=r^{\overline{l-k}}/r^{\overline{l-k-1}}=i+r-k-1. \tag{29}\]  
By (1), we have \(l_{i,j}^{(k+1)}=l_{i,j}^{(k)}- \begin{pmatrix}l_{i,k}^{(k)}/l_{i-1,k}^{(k)} \end{pmatrix} l_{i-1,j}^{(k)}\) and, using (16), (28), and (29), we derive  
\[l_{i,j}^{(k+1)}= \begin{bmatrix}i+r-k\\ j+r-k\end{bmatrix} _{r}-(i+r-k-1) \begin{bmatrix}i+r-k-1\\ j+r-k\end{bmatrix} _{r}= \begin{bmatrix}i+r-k-1\\ j+r-k-1\end{bmatrix} _{r},\]  
corresponding to the identity (35) for \(k+1\). By (2), (19), and (28), we deduce that the pivots of the Neville elimination of \(L_{n}^{r}\) satisfy  
\[p_{i,j}=l_{i,j}^{(j)}= \begin{bmatrix}i+r-j\\ r\end{bmatrix} _{r}.\]and so the diagonal pivots are \(p_{i,i}=\left[ \begin{smallmatrix}r\\ r\end{smallmatrix} \right]_{r}=1\) for \(i=1,\ldots,n+1\) (see (17)). Moreover, the following expression for the multipliers can be deduced  
\[m_{i,j}=p_{i,j}/p_{i-1,j}=\left[ \begin{smallmatrix}i+r-j\\ r\end{smallmatrix} \right]_{r}/\left[ \begin{smallmatrix}i+r-j-1\\ r\end{smallmatrix}