Questions of signs in power series

This paper deals with the signs of the (real) coefficients of a power series k(x) = Z,,=o k,x, which is the quotient of the two power series q(x)= Z,=0 q^xv and p(x) pvx. Questions of this type have been previously treated by Kaluza (1928) and Szego (1926) in the special case of q(x) = 1. Their significance for inclusion theorems for Norlund means was shown by Hardy (1949), ?4.5. By consideration of a general power series q(x) instead of q(x) = 1 it is possible to include the known results2 of this subject in a single theorem (?1.1). Moreover the method of proof used allows us to establish a second type of theorem using simpler conditions (?1.2). For this second type of theorem the introduction of a general q(x) is essential for applications, because the results are still trivial for q(x) = 1. There are two different principles to generalize. Thereby the second kind of theorem becomes also applicable in the case of q(x) = 1 (??2.1, 2.2, 2.4). The statements now obtained can even be used conversely for a second proof of the theorems first obtained (?2.3). The results of this paper are, as in the book of Hardy, fundamentally for application to inclusion of Norlund means and related questions. These applications will be discussed in a following paper Some new inclusion theorems for Ndrlund means. Also some concrete examples for the general theorems on distribution of signs will be found there.