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Nonasymptotic results on distribution of zeros of the function ${E}_{p}\left(z,\mu \right)$. (English) Zbl 0905.30004

The authors have obtained some nonasymptotic results on the distribution of zeros of the function

${E}_{\rho }\left(z,\mu \right)=\sum _{k=0}^{\infty }\frac{{z}^{k}}{{\Gamma }\left(\mu +k/\rho \right)},\phantom{\rule{2.em}{0ex}}\rho >0,\phantom{\rule{1.em}{0ex}}\mu \in ℂ$

which is a generalization of the classical Mittag-Leffler function. One of the main results of the article is contained in the following theorem.

Theorem. Assume that one of the following conditions is satisfied:

(i) $\rho >1$, $\mu \in \left[1,1+1/\rho \right]$; (ii) $\rho =1$, $\mu \in \left(0,2\right)$; (iii) $1/2<\rho <1$, $\mu \in \left[1/\rho -1,1\right]\cup \left[1/\rho ,2\right]$.

Then all zeros of ${E}_{\rho }\left(z,\mu \right)$ are situated outside the angle $\left\{z:|argz|\le \pi /2\rho \right\}$.

The set $W=\left\{\left(\rho ,\mu \right)$: all zeros of ${E}_{\rho }\left(z,\mu \right)$ are negative and simple} has been investigated in the article and as a corollary the generalization of Wiman’s theorem has been proved:

If $0<\rho \le 1/2$, then all zeros of both ${E}_{\rho }\left(z,1\right)$ and ${E}_{\rho }\left(z,2\right)$ are negative and simple.

##### MSC:
 30C15 Zeros of polynomials, etc. (one complex variable) 30D15 Special classes of entire functions; growth estimates
##### References:
 [1] M. M. Dzhrbashyan,Integral transforms and representations of functions in the complex domain, Nauka (Moscow, 1966) (in Russian). [2] A. M. Sedletskii, Asymptotic formulas for zeros of a function of Mittag-Leffler’s type,Analysis Math.,20(1994), 117–132 (in Russian). · Zbl 0798.30023 · doi:10.1007/BF01908643 [3] A. Wiman, Über die Nulstellen der FunktionenE $\alpha$(x),Acta Math.,29(1905), 217–234. · Zbl 02650564 · doi:10.1007/BF02403204 [4] G. Pólya, Bemerkung über die Mittag-Lefflerschen FunktionenE $\alpha$(z),Tôhoku Math. J.,19(1921), 241–248. [5] M. M. Dzhrbashyan andA. B. Nersesyan, Expansions associated with some biorthogonal systems and boundary problems for differential equations of fractional order,Trudy Moskow. Math. Obshch.,10(1961), 89–179 (in Russian). [6] M. M. Dzhrbashyan, Differential operators of fractional order and boundary value problems in the complex domain,Oper. Theory Adv. Appl.,41(1989), 153–172. [7] J. V. Linnik andI. V. Ostrovskii,Decomposition of random variables and vectors, Amer. Math. Soc. (Providence, RI, 1977); (Russian original: Nauka (Moscow, 1972)). [8] G. H. Hardy andW. W. Rogosinski,Fourier series, Univ. Press (Cambridge, 1956). [9] B. Ja. Levin,Distribution of zeros of entire functions, Amer. Math. Soc. (Providence, RI, 1980); (Russian original: Gostekhizdat (Moscow, 1956)).