Results & Lemmas (7)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Theorem 1.
Theorem 1. Let α, β, γ, τ be complex parameters, ℜ(α) > 0, and ℜ(γ) > 0. Let also α0, β0 ∈C be fixed numbers, ℜ(α0) > 0, ℜ(β0) > 0, ℜ(αγ−α0)…
Theorem 1. Let α, β, γ, τ be complex parameters, ℜ(α) > 0, and ℜ(γ) > 0. Let also α0, β0 ∈C be fixed numbers, ℜ(α0) > 0, ℜ(β0) > 0, ℜ(αγ−α0) > 0, and the parameter λ be positive. Then the Laplace transform L of the product of the power function tβ0−1 and Prabhakar function of Le Roy type (1.5), is given by the formula L n tβ0−1F (γ) α, β; τ(λtα0) o (s) := eF (γ) α, β; τ(s) = s−β0 F γi; 2 αi, βi; τi(λs−α0). (3.1)
Corollary 2.
Corollary 2. Let α, β, γ, τ be complex parameters and λ be positive. Let also ℜ(α) > 0, ℜ(β) > 0 and ℜ(αγ −α) > 0. Then, the following…
Corollary 2. Let α, β, γ, τ be complex parameters and λ be positive. Let also ℜ(α) > 0, ℜ(β) > 0 and ℜ(αγ −α) > 0. Then, the following relation holds true: L n tβ−1F (γ) α, β; τ(λtα) o (s) = s−β F (γ−1) α, β; τ (λs−α), (3.4) Moreover, in the right half-plane ℜ(s) > 0, the function (3.4) is an analytic function.
Theorem 3.
Theorem 3. Let αi, βi, γi, τi (i = 1,..., m) be complex parameters, ℜ(αi) > 0, and ℜ(γi) > 0. Let also α0, β0 ∈C be fixed numbers, ℜ(α0) >…
Theorem 3. Let αi, βi, γi, τi (i = 1, . . . , m) be complex parameters, ℜ(αi) > 0, and ℜ(γi) > 0. Let also α0, β0 ∈C be fixed numbers, ℜ(α0) > 0, ℜ(β0) > 0, ℜ(α1γ1 + · · · + αmγm −α0) > 0, and the parameter λ be positive. Then the Laplace transform L of the product of the power function tβ0−1 and multi-MLPR function (1.9) is given by the formula L n tβ0−1Fγi; m αi,βi;τi(λtα0) o (s) := eFγi;m αi,βi;τi(s) = s−β0 Fγi; m+1 αi,βi;τi (λs−α0). (3.5)
Corollary 4.
Corollary 4. Under the conditions of Theorem 3 and if additionally 0 ≤i0 ≤m, the following relation holds true: L n tβi0−1Fγi; m…
Corollary 4. Under the conditions of Theorem 3 and if additionally 0 ≤i0 ≤m, the following relation holds true: L n tβi0−1Fγi; m αi,βi;τi(λtαi0) o (s) = s−βi0 Feγi; m αi,βi;τi(λs−αi0), (3.7) with eγi0 = γi0 −1 and eγi = γi when i ̸= i0. Moreover, in the right half-plane ℜ(s) > 0, the function (3.7) is analytic.
Theorem 3.
Theorem 3. 4. Special cases Choosing different values of the parameters, interesting special cases can be obtained. Some of them are listed…
Theorem 3. 4. Special cases Choosing different values of the parameters, interesting special cases can be obtained. Some of them are listed in this section.
Corollary 5.
Corollary 5. L n tβ0−1F (γ)m (α,β)m(λtα0) o (s) = s−β0 F (γ)m+1 (α,β)m+1(λs−α0), (4.2) where α0, β0 ∈R, λ ∈C are such that 0 < α0 < m P j=1…
Corollary 5. L n tβ0−1F (γ)m (α,β)m(λtα0) o (s) = s−β0 F (γ)m+1 (α,β)m+1(λs−α0), (4.2) where α0, β0 ∈R, λ ∈C are such that 0 < α0 < m P j=1 Re (αj · γj), β0 > 0, and (α, β)m+1 = (α0, β0; α1, β1; . . . , αm, βm) ,
Corollary 6.
Corollary 6. Under the conditions of Theorem 1, the following Laplace transform relations hold true: L n tβ−1F (2) α, β; τ(λtα) o (s) = s−β…
Corollary 6. Under the conditions of Theorem 1, the following Laplace transform relations hold true: L n tβ−1F (2) α, β; τ(λtα) o (s) = s−β Eτ α, β(λs−α), (4.15) L n F (γ)(λt) o
Related Papers