P. N. Shivakumar, Yang Zhang, and Ashish Gupta
On Zeros of Polynomials and Infinite Series With Some Bounds
Dynamic Systems and Applications 32 (2023) 157-163

https://doi.org/10.46719/dsa2023.32.09

ABSTRACT.
In this paper, we consider a given infinite series in $x$ of the form $y(x) = \sum^{\infty}{k=0} b_k x^k$ expressed formally also by an infinite product as $y(x) = \Pi^{\infty}{k=1} (1- \frac{x}{a_k})$ into real positive zeros $a_i, i = 1, 2, \dots, \infty$ forming a strictly increasing sequence. For consideration of polynomials of degree $n$, we replace suitably $\infty$ by $n$.

Using the known formal solution of a second linear differential $y” = f(x) y, y(0) = y_0, y'(0) = y’0$ in the form $y(x)= \sum^{\infty}{k=0} d_k x^k$, we demonstrate that the above infinite product form of $y(x)$ yields the set of infinite equations of the form for a suitable $f(x) $.

$\sum^{\infty}_{k=1} (a_k)^{-p} = c_p$, $ p = 1, 2, \dots, \infty$ with $c_k’$s depending on $f(x)$, its derivarives at $x = 0$ and $b_k’$s.
Recognizing the infinite matrix as the infinite Vandermonde matrix, some bounds for the zeros are given.

AMS (MOS) Subject Classication. 15A99, 34A30.

Key Words and Phrases. Formal implicit series solution, differential equations, Vandermonde system.