Abstract
The present paper deals with the study of a generalized Mittag-Leffler function and associated fractional operator. The operator has been discussed in the space of Lebesgue measurable functions. The composition with Riemann–Liouville fractional integration operator has been obtained.
Highlights
The well-known Mittag-Leffler function Eα(z) named after its originator, the Swedish mathematician Gosta Mittag-Leffler (1846–1927), is defined by (Mittag-Leffler 1903) ∞ zn Eα(z) =; z is a complex variable and Re(α) ≥ 0
D dx n−1 x (x − t)β+n−r−2Eαμ,βρ+,γn,q−r−1,υ,σ,δ,p[w(x−t)α]f (t)dt a Applying same procedures as above, this led the proof of the theorem. This is easy to prove by using mathematical induction method
In this paper, we proved some properties of generalized Mittag-Leffler functions and used the fractional calculus approach to prove Theorems 4, 5, 6 and 7
Summary
The well-known Mittag-Leffler function Eα(z) named after its originator, the Swedish mathematician Gosta Mittag-Leffler (1846–1927), is defined by (Mittag-Leffler 1903). Theorem 3 Let a ∈ R+ = [0, ∞), α, β, γ , δ, μ, υ, ρ, σ ∈ C; p, q > 0 and q ≤ Re(α) + p and min(Re(α), Re(β), Re(γ ), Re(δ), Re(μ), Re(υ), Re(ρ), Re(σ )) > 0,b > a., the operator Eαμ,,βρ,,υγ,,σq,δ,p;w;a+ is bounded on L(a, b) and. Theorem 4 (Composition with Riemann–Liouville fractional integration operator) Let α, β, γ , δ, μ, υ, ρ, σ ∈ C; p, q > 0; q ≤ Re(α) + p; b > a and min(Re(α), Re(β), Re(γ ), Re(δ), Re(μ), Re(υ), Re(ρ), Re(σ )) > 0. D dx n−1 x (x − t)β+n−r−2Eαμ,,βρ+,γn,q−r−1,υ,σ ,δ,p[w(x−t)α]f (t)dt a Applying same procedures as above, this led the proof of the theorem This is easy to prove by using mathematical induction method .
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