Zeros of polynomials related to Jensen polynomial associated with Riemann xi function $\xi(x)$

Just two pointers for this problem (sorry, no solution).

1. Hurwitz used a version of Sturm's theorem on numerators of continued fractions to study the zeros of the Bessel functions, and Watson's treatise on Bessel functions (1944) has this in section 9.7. The series in question can be written as $$\sum_{k=0}^\infty (-1)^k \frac{x^{2k}}{(2k)!} \prod_{j=1}^k \frac{2j-1}{2j+2p},\quad(p>-1).$$ (Hurwitz 1889 Ueber die Nullstellen der Bessel'schen Function)

2. There is a converse to Newton's inequalities for real polynomials with positive coefficients: if $a_i>0$ and $a_i^2 > 4 a_{i-1} a_{i+1}$, then all roots are real and distinct: A Sufficient Condition for All the Roots of a Polynomial To Be Real David C. Kurtz, The American Mathematical Monthly Vol. 99, No. 3 (Mar., 1992), pp. 259-263

J. Gélinas

Just wanted to mention the new paper of Griffin, Ono, Rolen and Zagier https://www.pnas.org/content/early/2019/05/20/1902572116 which deals with exactly this kind of problem (if I understand you and them correctly).