On Some Triangular Summability Methods

It is to be noted that Bn (x) is defined by taking the first n + 1 terms of the series defining the function *J(x) ; the summability method is then constructed with the sequence {x.n}. As set forth in the aforementioned paper of Szasz,2 the regularity of either method (1. 1) or (1. 2) does not imply the regularity of the other method. On the other hand, if the first method is boundedness preserving, then the associated triangular methods are also boundedness preserving for all sequences {X.n}, xn > t. The conditions of regularity for the transformation (1.2 ) are