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Nonasymptotic results on distribution of zeros of the function E p (z,μ). (English) Zbl 0905.30004

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

E ρ (z,μ)= k=0 z k Γ(μ+k/ρ),ρ>0,μ

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) ρ>1, μ[1,1+1/ρ]; (ii) ρ=1, μ(0,2); (iii) 1/2<ρ<1, μ[1/ρ-1,1][1/ρ,2].

Then all zeros of E ρ (z,μ) are situated outside the angle {z:|argz|π/2ρ}.

The set W={(ρ,μ): all zeros of E ρ (z,μ) 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<ρ1/2, then all zeros of both E ρ (z,1) and E ρ (z,2) are negative and simple.

MSC:
30C15Zeros of polynomials, etc. (one complex variable)
30D15Special 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 α(x),Acta Math.,29(1905), 217–234. · Zbl 02650564 · doi:10.1007/BF02403204
[4]G. Pólya, Bemerkung über die Mittag-Lefflerschen FunktionenE α(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)).