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Generalized Mittag-Leffler function and generalized fractional calculus operators. (English) Zbl 1047.33011

The paper studies the function

E ρ,μ γ (z)= k=0 (γ) k Γ(ρk+μ)k!,

where ρ, μ, and γ are complex parameters with Re(ρ)>0. This is a generalization of the classical Mittag-Leffler function E ρ,μ (z) as well as a generalization of the Kummer confluent hypergeometric function Φ(γ,μ,z).

33E12Mittag-Leffler functions and generalizations
33C15Confluent hypergeometric functions, Whittaker functions, 1 F 1
26A33Fractional derivatives and integrals (real functions)
47B38Operators on function spaces (general)
47G10Integral operators