On some conjectures by Morris et al. about zeros of an entire function (Q1268733)

Suppose \(y'(t)=-y(qt)\) where \(q\in(0,1)\) and \(y(0)=1\). The only differentiable solution is \(y(t)=\sum^\infty_{n=0} {(-1)^nq^{n(n-1)/2} \over n!}t^n\). The author shows the following: \(t_{n+1}>q^{-1}t_n\) for all \(n\geq 0\) and \(t_{n+1}\leq q^{-2}t_n+1\) for all \(n\geq m\), where \(m\) is the minimal nonnegative integer that obeys \(t_n\geq q^3/(1-q)\), and \(t_0,t_1,\dots\) are the zeros in monotonic increasing order.