# Computation of zeros of the alpha exponential function

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## 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.

### Keywords

alpha exponential function degenerate hypergeometric function asymptotic expansions complex zeros Newton’s method## Preview

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### References

- 1.A. Wünsche, “Realization of SU(1,1) by boson operators with application to phase states,” Acta Phys. Slovac.
**49**, 771–782 (1999).Google Scholar - 2.G. H. Hardy, “On the zeros of certain class of integral Taylor Series II,” Proc. London Math. Soc.
**2**(2), 401–431 (1905).MathSciNetCrossRefMATHGoogle Scholar - 3.I. V. Ostrovskii, “Hardy’s generalization of and related analogs of cosine and sine,” Comput. Methods Funct. Theory
**6**, 1–14 (2006).MathSciNetCrossRefMATHGoogle Scholar - 4.Y. L. Luke,
*Mathematical Functions and Their Approximations*(Academic, New York, 1975; Mir, Moscow, 1980).MATHGoogle Scholar - 5.
*Higher Transcendental Functions*(Bateman Manuscript Project), Ed. by A. Erdélyi (McGraw-Hill, New York, 1953; Nauka, Moscow, 1973), Vol. 1.Google Scholar - 6.A. I. Markushevich,
*Theory of Functions of a Complex Variable*(Prentice Hall, Englewood Cliffs, N.J., 1965; Nauka, Moscow, 1968), Vols. 1 and 2.MATHGoogle Scholar - 7.M. A. Lavrent’ev and B. V. Shabat,
*Methods of Complex Analysis*(Nauka, Moscow, 1973) [in Russian].MATHGoogle Scholar