On the entire functions from the Laguerre-P\'olya I class with non-monotonic second quotients of Taylor coefficients

TL;DR

This paper characterizes entire functions with specific non-monotonic second quotients of Taylor coefficients that belong to the Laguerre-Pólya I class, providing necessary and sufficient conditions for this classification.

Contribution

It introduces a new class of entire functions with non-monotonic coefficient quotients and establishes criteria for their inclusion in the Laguerre-Pólya I class.

Findings

Derived necessary and sufficient conditions for functions to be in the Laguerre-Pólya I class.

Identified the impact of non-monotonic second quotients on the function's class membership.

Extended understanding of coefficient behavior in entire functions within the Laguerre-Pólya class.

Abstract

We study the entire functions $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ with non-monotonic second quotients of Taylor coefficients, namely, such that $\frac{a_{2m-1}^2}{a_{2m-2}a_{2m}} = a>1$ and $\frac{a_{2m}^2}{a_{2m-1}a_{2m+1}} = b>1$ for all $m \in \mathbb{N}.$ We obtain necessary and sufficient conditions under which such functions belong to the Laguerre-P\'olya I class.