On the stability of Taylor sections of a function \(\Sigma _{k=0}^{\infty } z^{k}/a^{k^{2}}, a>1\) (Q2378584)

In this paper the following problem is answered. Given a positive integer \(n\), which are the smallest possible values of the constants \(s_n\) such that the zeros of the polynomial \[ f_{a,n}(z)= \sum^n_{k=0} {z^k\over a^{k^2}} \] have negative real parts for \(a> s_n\)? Indeed, the authors show that, for every positive integer \(n\), there exists a constant \(s_n\) such that all zeros of \(f_{a,n}(z)\) have negative real parts if and only if \(a> s_n\). Similarly they also obtain a constant \(s_\infty\) such that all zeros of \[ f_a(z)= \sum^\infty_{k=0} {z^k\over a^{k^2}}\quad (a> 1) \] have negative real roots if and only if \(a> s_\infty\). For the proof they use the well-known Hermite-Bichler criterion.