Some properties of D-matrices (Q1851547)

The author studies the properties of D-matrices [see \textit{W. Darevsky}, Mat. Sborn. 7(49), 549-590 (1940; Zbl 0024.02801)]. It is proved that D-matrices that are normal and regular are translative from the left and \(\text{ D}\in R^{(0)} (|E|,D)\), where \(R^{(0)}(|E|,D)\) is a class of methods \((r_{n,k})\) such that for any series \(\sum\limits_{n=0}^\infty u_n\), \(\sum\limits_{n=0}^\infty v_n\), one of which converges absolutely and the other is summable by the matrix method D, the relation \( \sum\limits_{n=0}^\infty w_n=UV(r_{n,k})\) holds, where \( w_n = \sum\limits_{k=0}^n u_kv_{n-k}\), and \(U\) and \(V\) are the corresponding sums of the series under consideration.