On the conditions for entire functions related to the partial theta function to belong to the Laguerre-Pólya class

In this paper, we study entire functions of order zero with positive Taylor coefficients and investigate sufficient or necessary conditions for such functions to have only real zeros. We answer the following question: for which values of a>1 and m≥1 an entire function φa,m(z)=∑k=0∞zkak2(k!)m and its Taylor sections belong to the Laguerre-Pólya class?