Symmetric polynomials, Pascal matrices, and Stirling matrices (Q2469538)

For any formal power series \(a(t)=\sum_{k=0}^{\infty}\alpha_{i}t^{i}\) we let \(M_{a}\) denote the infinite Toeplitz matrix with \((i,j)\)th entry equal to \(\alpha_{i-j}\) for \(i\geq j\) and \(0\) otherwise. Furthermore for any such infinite matrix \(U\) and \(k\geq1\) we let \(U^{(k)}\) be the matrix obtained by replacing the first \(k\) rows of \(U\) by the first \(k\) rows of the infinite identity matrix, and let \(U^{[k]}\) be the matrix obtained by replacing the first \(k\) columns of \(U\) by the first \(k\) columns of the identity matrix. Let \(a_{0},a_{1},\dots \) be an infinite sequence of formal power series and \(c_{m}:=a_{0}a_{1}\dots a_{m}\). Then the following theorems are easily proved. Theorem 1: If the \(a_{i}\) are all of degree at most \(1\), and \(V\) is the infinite product \(\dots M_{a_{n}}^{(n)}\dots M_{a_{1}}^{(1)}M_{a_{0}}\) then the \(i\)th row of \(V\) is equal to the \(i\)th row of \(M_{c_{i}}\). Theorem 2: If \(W\) is the infinite product \(M_{a_{0}}M_{a_{1}}^{[1]}\dots M_{a_{n}}^{[n]}\dots \) then the \(j\)th column of \(W\) is equal to the \(j\)th column of \(M_{c_{j}}\). The authors then show that many of the results from the papers \textit{R. Brawer} and \textit{M. Pirovino} [Linear Algebra Appl. 174, 13--23 (1992; Zbl 0755.15012)], \textit{G. S. Call} and \textit{D. J. Velleman} [Am. Math. Monthly 100, No. 4, 372--376 (1993; Zbl 0788.05011)], \textit{G.-S. Cheon} and \textit{J.-S. Kim} [Linear Algebra Appl. 329, No. 1--3, 49--59 (2001; Zbl 0988.05009)], \textit{S.-L. Yang} and \textit{Z.-K. Qiao} [Int. J. Appl. Math. 14, No. 2, 145--157 (2003; Zbl 1056.15020)] and \textit{Z.-Z. Zhang} and \textit{M. Liu} [Linear Algebra Appl. 271, 169--177 (1998; Zbl 0892.15018)] can be easily derived from these theorems.