Generalized Leibniz functional matrices and factorizations of some well-known matrices
Let \(f(t)\) be a function such that the \(n\)th derivative \(f^{(n)}(t)\) exists. Then the \((n+1)\times(n+1)\) Leibniz functional matrix \(L_n[f(t)]\) has entries \((L_n[f(t)])_{ij}=\frac{f^{(i-j)}(t)}{(i-j)!}\) if \(i\geq j\) and \((L_n[f(t)])_{ij}= 0\) otherwise. This paper introduces generalized Leibniz functional matrices as follows. Let \(f_0(t), \dots, f_n(t)\) and \(g_0(t), \dots, g_n(t)\) be functions such that their \(n\)th derivatives exist. Then the entries of the generalized Leibniz functional matrix with varying rows and columns are \[ (L_{n}^{V}[f_0(t),\dots,f_n(t);g_0(t), \dots, g_n(t)])_{ij}=\frac{(f_i(t)g_j(t))^{(i-j)}}{(i-j)!} \] if \(i\geq j\) and \(0\) otherwise. If \(g_0(t)=\cdots=g_n(t)=1\), then one has generalized Leibniz functional matrix with varying rows, denoted by \(L_{n}^{R}[f_0(t),\dots,f_n(t)]\). Similarly, \(L_{n}^{C}[g_0(t), \dots, g_n(t)] =L_{n}^{V}[1,\dots,1;g_0(t), \dots, g_n(t)]\) are generalized Leibniz functional matrices with varying columns. Some properties of generalized Leibniz functional matrices are explored. For instance, it is proved that \[ L_{n}^{R}[f_0(t),\dots,f_n(t)] L_{n}^{C}[g_0(t), \dots, g_n(t)]=L_{n}^{V}[f_0(t),\dots,f_n(t);g_0(t), \dots, g_n(t)]. \] This and similar factorization properties of generalized Leibniz functional matrices are applied in factorization of some well-known matrices: complete symmetric polynomial matrix, elementary symmetric polynomial matrix, Pascal matrix, Stirling matrices etc.
scientific article; zbMATH DE number 5376919