The Pascal matrix in the multivariate Riordan group
For \(n\in \mathbb{N}\), the squared matrices in \(\mathcal{M}_{(n+1)\times (n+1)}(\mathbb{N})\) \[L_n =\left(\binom{i}{j}\right)_ {0\leq i,j\leq n}, \quad U_n = L^T_n, \quad S_n= \left(\binom{i+j}{i}\right)_ {0\leq i,j\leq n} \] are the lower-triangular, upper-triangular and symmetric Pascal matrices of order \(n+1\), respectively. For the vectors \(\textbf{k} = (k_1,\ldots, k_n)\), \(\textbf{i} = (i_1,\ldots, i_n)\in Z^n_{\geq 0}\), \(\textbf{i} \leq \textbf{k}\) indicates the partial order on \(Z^n_{\geq 0}\) defined as \(0 \leq i_j \leq k_j\) for \(1 \leq j \leq n\). Denote the generalized binomials as follows: \[\binom{\textbf{k}}{\textbf{i}}=\binom{k_1}{i_1}\ldots \binom{k_n}{i_n}.\] The multivariate Riordan group is defined in terms of \(n+1\) formal power series in \(Cx_1, \dots, x_n\) as a generalization of the classical (univariate) Riordan group, which is defined in terms of two formal power series in \(Cx\). The author presents an alternative generalization of Pascal matrices. As in the classical case, there are three ways to represent Pascal matrices. Let \(R=\{\textbf{k}_1,\dots, \textbf{k}_r\}\) be a finite set of points in \(Z^n_{\geq 0}\), define the \(r \times r\) matrices \[L_R = (l_{\textbf{k}_i,\textbf{k}_j}),~~U_R = L^t_R, ~~S_R = (s_{\textbf{k}_i,\textbf{k}_j}), \] where \(l_{\textbf{k}_i,\textbf{k}_j}= \binom{\textbf{k}_i}{\textbf{k}_j}\), \(s_{\textbf{k}_i,\textbf{k}_j}= \binom{\textbf{k}_i+\textbf{k}_j}{\textbf{k}_j} \) and the rows and columns of the matrices are ordered by a total order \(<T\) on \(Z^n_{\geq 0}\). The author studies the properties of the Pascal matrices and considers their infinite versions. She also proves that the infinite Pascal matrix is an element of the multivariate Riordan group.
scientific article; zbMATH DE number 7451374
scientific article