On the entire functions from the Laguerre-Pólya I class with non-monotonic second quotients of Taylor coefficients

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-Pólya I class.

20 pages. arXiv admin note: text overlap with arXiv:2008.04754, arXiv:2101.11757