On the real zeros of functions of Mittag-Leffler type. (English. Russian original) Zbl 1095.33010

Pskhu, A. V.

Math. Notes 77, No. 4, 546-552 (2005); translation from Mat. Zametki 77, No. 4, 592-599 (2005).

Summary: In the present paper, we prove an assertion allowing us to extend results related to the presence or absence of real zeros of functions of Mittag-Leffler type \[ E_{1/\alpha}(z;\mu)= \sum_{k = 0}^\infty \frac{z^k}{\Gamma (\alpha k + \mu)} \] for certain values of \(\alpha\) and \(\mu\) to more extensive ranges of these parameters. We give a geometric description of the sets of pairs \((\alpha,\mu)\) for which the function \(E_{1/\alpha}(z;\mu)\) has and does not have real zeros.

MSC:

33E12	Mittag-Leffler functions and generalizations
30C15	Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)

Keywords:

functions of Mittag-Leffler type; real zeros of Mittag-Leffler functions; Wright function; Stankovich transformation

References:

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