# On the Total Positivity and Accurate Computations of \(r\)-Bell Polynomial Bases
## 4 Accurate Computations with \(r\)-Bell Bases  
Let \(\mathbf{P}^{n}(I)\) be the (n+1)-dimensional linear space formed by all polynomials in the variable \(x\) defined on a real interval \(I\) and whose degree is not greater than \(n\); that is,  
\[\mathbf{P}^{n}(I):=\text{span}\{1,x,\ldots,x^{n}\},\quad x\in I.\]  
For \(n\in\mathbb{N}\), the \(n\) degree \(r\)-Bell polynomial of the first kind is defined as  
\[B_{n}^{r}(x):=\sum_{k=0}^{n}\genfrac{[}{]}{0.0 pt}{}{n+r}{k+r}_{r}x^{k}. \tag{38}\]  
Then we can define a system \((B_{0}^{r},\ldots,B_{n}^{r})\) of \(r\)-Bell polynomials of the first kind that satisfy  
\[(B_{0}^{r},\ldots,B_{n}^{r})^{T}=L_{n}^{r}(m_{0},\ldots,m_{n})^{T}, \tag{39}\]  
where \(m_{i}(x):=x^{i}\), \(i=0,\ldots,n\) and \(L_{n}^{r}\) is the \((n+1)\times(n+1)\) first kind of \(r\)-Stirling matrix defined by (22). By (24), \(\det L_{n}^{r}=1\) and we can guarantee that \((B_{0}^{r},\ldots,B_{n}^{r})^{T}\) is a basis of \(\mathbf{P}^{n}(\mathbb{R})\). Definition 2: We say that \((B_{0}^{r},\ldots,B_{n}^{r})\) is the \(r\)-Bell basis of the first kind of the polynomial space \(\mathbf{P}^{n}(\mathbb{R})\). On the other hand, the \(r\)-Bell polynomials of the second kind \(\{\bar{B}_{n}^{r}\}_{n\geq 0}\) in [3; 27] are called \(r\)-Bell polynomials, which are defined by their generating function  
\[\sum_{n\geq 0}\bar{B}_{n}^{r}(x)\frac{t^{n}}{n!}=\exp\big{(}x(e^{t}-1)\!+\!rt \big{)}.\]  
They can also be written in terms of the monomial basis as follows,  
\[B_{n}^{r}(x)=\sum_{k=0}^{n}\genfrac{[}{]}{0.0 pt}{}{n+r}{k+r}_{r}x^{k},\quad n \in\mathbb{N}. \tag{40}\]  
and then  
\[(\bar{B}_{0}^{r},\ldots,\bar{B}_{n}^{r})^{T}=L_{n}^{r}(m_{0},\ldots,m_{n})^{T}, \tag{41}\]  
where \(L_{n}^{r}\) is the \((n+1)\times(n+1)\) second kind of \(r\)-Stirling matrix defined by (23). By (31), \(\det L_{n}^{r}=1\) and we can guarantee that \((\bar{B}_{0}^{r},\ldots,\bar{B}_{n}^{r})^{T}\) is a basis of \(\mathbf{P}^{n}(\mathbb{R})\). Definition 3: We say that \((\bar{B}_{0}^{r},\ldots,\bar{B}_{n}^{r})\) is the \(r\)-Bell basis of the second kind of \(\mathbf{P}^{n}(\mathbb{R})\). Let us recall that the monomial basis \((m_{0},\ldots,m_{n})\), with \(m_{i}(x)=x^{i}\) for \(i=0,\ldots,n\), is a strictly totally positive basis of \(\mathbf{P}^{n}(0,+\infty)\), and so the collocation matrix  
\[V:=\Big{(}x_{i}^{j-1}\Big{)}_{1\leq i,j\leq n+1}, \tag{42}\]  
is strictly totally positive for any increasing sequence of positive values \(0<x_{1}<\ldots<x_{n+1}\) (see Section 3 of [19]). In fact, \(V\) is the Vandermonde matrix at the considered nodes. Let us recall that Vandermonde matrices have relevant applications in linear interpolation and numerical quadrature (see for example [28; 29]). As for \(BD(V)\), we have  
\[BD(V)_{i,j}:= \begin{cases}\prod_{k=1}^{j-1}\frac{x_{i}-x_{i-k}}{x_{i-1}-x_{i-k -1}},&\text{if $i>j$},\\ \prod_{k=1}^{i-1}(x_{i}-x_{k}),&\text{if $i=j$},\\ x_{i},&\text{if $i<j$},\end{cases}  \tag{43}\]and it can be easily checked that the computation of \(BD(V)\) does not require inaccurate cancellations and can be performed to high relative accuracy. Taking into account the total positivity of the Vandermonde matrices and the total positivity of \(r\)-Stirling matrices of the first and second kinds, we shall derive the total positivity of r-Bell bases, as well as factorizations providing computations to high relative accuracy when considering their collocation matrices. **Theorem 4**. : _The \(r\)-Bell basis of first and second kind are totally positive bases of \(\mathbf{P}^{n}(0,+\infty)\). Moreover, given \(0<x_{1}<\ldots<x_{n+1}\), the collocation matrices_  
\[B:=\left(B^{r}_{j-1}(x_{i})\right)_{1\leq i,j\leq n+1},\quad\bar{B}:=\left( \bar{B}^{r}_{j-1}(x_{i})\right)_{1\leq i,j\leq n+1}, \tag{44}\]  
_and their bidiagonal factorization (5) can be computed to high relative accuracy._  
Proof.