Let
formula here
be a formal power series. In 1913, G. Pólya
[7] proved that if, for all sufficiently large
n, the sections
formula here
have real negative zeros only, then the series (0.1) converges in the
whole complex
plane C, and its sum f(z) is an
entire function of order 0. Since then, formal power
series with restrictions on zeros of their sections have been deeply investigated
by
several mathematicians. We cannot present an exhaustive bibliography here,
and
restrict ourselves to the references [1, 2, 3],
where the reader can find detailed
information.
In this paper, we propose a different kind of generalisation
of Pólya's theorem. It
is based on the concept of multiple positivity introduced by M. Fekete
in 1912, and
it has been treated in detail by S. Karlin [4].