On the conditions for a special entire function relative to the partial theta-function and the Euler function to belong to the Laguerre-Pólya class

Abstract

In this paper, we discuss the conditions for the function $F_a(z)={\sum }_{k=0}^{\infty }\frac{z^k}{(a+1)(a^2+1)\dots (a^k+1)},a>1,$ to belong to the Laguerre-Pólya class, or to have only real zeros.

19 pages, submitted to CMFT Journal. This article was partially supported by the Akhiezer Foundation. The author is also deeply grateful to her scientific advisor, A. Vishnyakova, for her patience, support and contribution to her research career. arXiv admin note: text overlap with arXiv:1903.09070