Computation of zeros of the alpha exponential function

Abstract

This paper deals with the function F(α; z) of complex variable z defined by the expansion $$F\left( {\alpha ;z} \right) = \sum\nolimits_{k = 0}^\infty {\frac{{{z^k}}}{{{{\left( {k!} \right)}^\alpha }}}} $$ 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 α ∈ (0,1). Note that the function F(α; z) arises in a number of modern problems in quantum mechanics and optics. For α = 1/2, 1/3,..., approximations of F(α; z) are constructed using combinations of degenerate hypergeometric functions 1 F 1(a; c; z) and their asymptotic expansions as z → ∞. These approximations to F(α; 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 α ∈ (0,1] have a complex structure. For α = 1/2 and 1/3, the first 30 complex zeros of the function are calculated to high accuracy.