On necessary and sufficient conditions for entire functions to belong to the Laguerre-Pólya class in terms of their Taylor coefficients

We discuss new necessary and sufficient conditions for the entire functions with positive Taylor coefficients to belong to the Laguerre--Pólya class. It is an important class of entire functions which are locally the limit of a sequence of real polynomials having only real zeros. For an entire function $f(z) = \sum_{k=0}^{\infty} a_k z^k,$ we define the second quotients of Taylor coefficients as $q_n(f) := \frac{a_{n-1}^2}{a_{n-2} a_{n}}, n\geq 2$ and find conditions on $q_n(f)$ for $f$ to belong to the Laguerre--Pólya class. Besides, we show the relation of the conditions to the partial theta function $g_a =\sum _{k=0}^{\infty} \frac {z^k}{a^{k^2}}$, when $a>1.$