Browsing by Subject "Power series"

Now showing 1 - 4 of 4

Open Access
On power serieshaving sectionswith multiply positive coefficients
(Taylor & Francis, 2002-03-15) Zheltukhina, N. A.
Polya’s theorem of 1913 says that if all sections of power series have real negative zeros only, then the series converges in the whole complex plane and its sum satisfies a certain growth condition. Here we show that the assertion of Po´ lya’s theorem remains valid for a much larger class of formal power series and, moreover, a better growth estimate holds.
Open Access
On the zero distributionof remainders of entire power series
(Taylor & Francis, 2001) Ostrovskll, I. V.
It has been shown by the author that, if all remainders of the power series of an entire function f have only real positive zeros, then log M(r, f) = O((1og r)'), r -+ ca. The main results of the paper are the following: (i) if at least two different remainders have only real positive zeros, then logM(r,f) = O(fi, r+ ca; (ii) this estimate cannot be improved even in the case if one replaces two by any given finite number of remainders.
Open Access
On the zeros of tails of power series
(Springer, 2000) Ostrovskii, Iossif Vladimirovich; Havin, V. P.; Nikolski, N. K.
Let f(z)=∑k=0∞akzkf(z)=∑k=0∞akzk (1.1) be a power series with a positive radius of convergence. Let sn(z)=∑k=0nakzk,tn(z)=∑k=n+1∞akzksn(z)=∑k=0nakzk,tn(z)=∑k=n+1∞akzk be its nth section and nth tail, respectively.
Open Access
Strongly clean matrices over power series
(Kyungpook National University, 2016) Chen, H.; Kose, H.; Kurtulmaz, Y.
An n × n matrix A over a commutative ring is strongly clean provided that it can be written as the sum of an idempotent matrix and an invertible matrix that commute. Let R be an arbitrary commutative ring, and let A(x) ∈ Mn ( R[[x]]) . We prove, in this note, that A(x) ∈ Mn ( R[[x]]) is strongly clean if and only if A(0) ∈ Mn(R) is strongly clean. Strongly clean matrices over quotient rings of power series are also determined.