Computation of zeros of the alpha exponential function

This paper deals with the function $F(\alpha; z)$ of complex variable $z$ defined by the expansion $F(\alpha; z)=\sum_{k=0}^\infty\frac{z^k}{(k!)^\alpha}$ which is a natural generalization of the exponential function (hence the name). Primary attention is given to finding relations concerning the locations of its zeros for $\alpha\in(0, 1)$. Note that the function $F(\alpha; z)$ arises in a number of modern problems in quantum mechanics and optics. For $\alpha=1/2,~1/3,~\dots$, approximations of $F(\alpha; z)$ are constructed using combinations of degenerate hypergeometric functions $_1F_1(a; c; z)$ and their asymptotic expansions as $z\to\infty$. These approximations to $F(\alpha; z)$ are used to approximate the countable set of complex zeros of this function in explicit form, and the resulting approximations are improved by applying Newton’s high-order accurate iterative method. A detailed numerical study reveals that the trajectories of the zeros under a varying parameter $\alpha\in(0, 1]$ have a complex structure. For $\alpha = 1/2$ and $1/3$, the first $30$ complex zeros of the function are calculated to high accuracy.