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On some properties of the generalized Mittag-Leffler function
SpringerPlusvolume 2, Article number: 337 (2013)
This paper deals with the study of a generalized function of Mittag-Leffler type. Various properties including usual differentiation and integration, Euler(Beta) transforms, Laplace transforms, Whittaker transforms, generalized hypergeometric series form with their several special cases are obtained and relationship with Wright hypergeometric function and Laguerre polynomials is also established.
2000 Mathematics Subject Classification
33C45, 47G20, 26A33.
In 1903, the Swedish mathematician Gosta Mittag-Leffler (1903) introduced the function
where z is a complex variable and Γ is a Gamma function α ≥ 0. The Mittag-Leffler function is a direct generalisation of exponential function to which it reduces for α = 1. For 0 < α < 1 it interpolates between the pure exponential and hypergeometric function Its importance is realized during the last two decades due to its involvement in the problems of physics, chemistry, biology, engineering and applied sciences. Mittag-Leffler function naturally occurs as the solution of fractional order differential or fractional order integral equation.
The generalisation of E α (z) was studied by Wiman (1905) in 1905 and he defined the function as
which is known as Wiman function.
In 1971, Prabhakar (1971) introduced the function in the form of
where (γ) n is the Pochhammer symbol (Rainville (1960))
In 2007, Shukla and Prajapati (2007) introduced the function which is defined for ; Re(α) > 0,Re(β) > 0,Re(γ) > 0 and as
In 2009, Tariq O. Salim (2009) introduced the function the function which is defined for as
In 2012, a new generalization of Mittag-Leffler function was defined by Salim (2012) as
where min (Re(α), Re(β), Re(γ), Re(δ)) > 0
In this paper a new definition of generalized Mittag-Leffler function is investigated and defined as
Further the generalization of definition (1.7) is investigated and defined as follows
The definition (1.9) is a generalization of all above functions defined by (1.1)-(1.7).
Throughout this investigation, we need the following well-known facts to study the various properties and relation formulas of the function .
Beta(Euler) transforms (Sneddon (1979)) of the function f(z) is defined as(1.12)
Laplace transforms (Sneddon (1979)) of the function f(z) is defined as(1.13)
Mellin- transforms of the function f(z) is defined as(1.14)
and the inverse Mellin-transform is given by(1.15)
Whittaker transform (Whittaker and Watson (1962))(1.16)
where and W λ, μ (t) is the Whittaker confluent hypergeometric function.
The generalized hypergeometric function (Rainville (1960)) is defined as(1.17)
Wright generalized hypergeometric function (Srivastava and Manocha (1984)) is defined as(1.18)
Fox’s H-function (Saigo and Kilbas (1998)) is given as(1.19)
Generalized Laguerre polynomials (Rainville (1960)). These are also known as Sonine polynomials and are defined as(1.20)
Incomplete Gamma function (Rainville (1960)). This is denoted by γ(α, z) and is defined by(1.21)
Basic properties of the function
As a consequence of definitions (1.1)-(1.9) the following results hold:
If , Re(α) > 0, Re(β) > 0, Re(γ) > 0, Re(δ) > 0, Re(μ) > 0, Re(ν) > 0, Re(ρ) > 0, Re(σ) > 0 and p, q > 0 a n d q ≤ Re(α) + p, then
which is (2.1.1).
The proof of (2.1.2) can easily be followed from the definition (1.9). Now
which proves (2.1.3). □
Substituting μ = ν, ρ = σ and p = 1 in (2.1.1) immediately leads to (2.1.4).
Substituting μ = ν, ρ = σ and p = 1 in (2.1.2) immediately leads to (2.1.5).
Putting μ = ν, ρ = σ and p = 1 in (2.1.3) immediately leads to (2.1.6).
If , Re(α) > 0, Re(β) > 0, Re(γ) > 0, Re(δ) > 0, Re(μ) > 0, Re(ν) > 0, Re(ρ) > 0, Re(σ) > 0, Re(w) > 0; a n d q ≤ Re(α) + p then for
which is the proof of (2.2.1).
Again using (1.9) and term by term differentiation under the sign summation(which is possible in accordance with the uniform convergence of the series (1.9) in any compact set ), we have
which is the proof of (2.2.2). □
Setting μ = ρ, ν = σ, in (2.2.1), we get (2.2.3).
Setting μ = ρ, ν = σ, in (2.2.2), we get (2.2.4).
If , with relatively prime; and q < Re(α + p), then
which proves (2.3.1). □
For μ = ν, ρ = σ, δ = p = 1, (2.3.1) reduces to the known result of Shukla and Prajapati Shukla and Prajapati (2007) (2.3.1).
Setting μ = ν, ρ = σ and p = 1 in (2.3.1), we get (2.3.2).
Special Properties: Setting putting μ = ν, ρ = σ and p = q = δ = 1 in (2.3.1), we have
For β = γ = δ = q = 1 in (2.3.2), we have
which proves (2.4.1).
Now change the variable from s to Then the L.H.S. of (2.4.2) becomes
which proves (2.4.2).
which proves (2.4.3).
Putting q = δ = 1 and γ = q = δ = 1 in (2.4.1) and (2.4.3) yields (2.4.4) and (2.4.5) respectively. □
Generalized hypergeometric function representation of
Using (1.9) with and , we have
where Δ(l;μ) is a l-tupple ; Δ(q;γ) is a q-tupple ; Δ(k, β) is a k-tupple and so on, which is the required hypergeometric representation.
Convergence criterion of generalized Mittag-leffler function q+l+1 F k+p+m :
If q + l + 1 ≤ k + p + m, the function q+l+1 F k+p+m converges for all finite z.
If q + l + 1 = k + p + m + 1, the function q+l+1 F k+p+m converges for |z| < 1 and diverges for |z| > 1
If q + l + 1 > k + p + m + 1, the function q+1+1 F k+p+m+1 is divergent for |z| ≠ 0
If q + l + 1 = k + p + m + 1, the function q+l+1 F k+p+m+1 is absolutely convergent on the circle for |z| = 1, if
Integral transforms of
In this section we discuss some useful integral transforms like Euler transform, laplace transform and Whittaker transform of
Mellin-Barnes integral representation of
Let (1.9) and (1.10) be satified and and q < R e(α) + p. Then the function is represented by Mellin-Barnes integral as:
where | arg(z)| < 1; the contour of integration beginning at −i ∞ and ending at +i ∞, and indented to separate the poles of the integrand at (to the left) from those at (to the right).
We shall evaluate the integral on R.H.S. of (4.1.1) as the sum of the residues at the poles s = 0, − 1, − 2, …, we have
which completes the proof. □
Setting μ = ρ, ν = σ and p = 1, we get the Melin Barne’s integral of the function
(Mellin transform) of
From Theorem 4.1, we have
is in the form of inverse Mellin-Transform (1.15). So applying the Mellin-transform (1.14) yields directly the required result. □
from which the result follows. □
For q = 1, (4.3.2) reduces to Tariq OSalim (2009)(4.1).(4.3.3)
For δ = q = 1 in (4.3.2), we have(4.3.4)
If a = β, α = σ, then (4.3.2) reduces to(4.3.5)
Putting α = β = γ = δ = q = 1 in (4.3.2), we have(4.3.6)
Theorem 4.4. (Laplace transform)
from which the result follows. □
For q = 1, (4.4.2) reduces to Tariq O Salim (2009)(4.2).
Theorem 4.5. (Whittaker transform)
Substituting ϕ t = v in L.H.S. of Theorem 4.5, we have
from which the result follows. □
Special properties :
Putting q = δ = 1 in (4.5.2), we have(4.5.3)
For q = γ = δ = 1 in (4.5.2), we have(4.5.4)
Now putting q = β = α = γ = δ = 1 in (4.5.2), we have(4.5.5)
Relationship with some known special functions
Relationship with Wright hypergeometric function
If the condition (1.10) be satisfied, then (1.9) can be written as
Relationship with Fox H-function
Using (4.1.1), we have from
Relationship with generalized Laguerre polynomials
Putting α = k, β = μ + 1, γ = − m, q ∈ N with q|m and replacing z by z k in (1.6), we get
where is a generalization of (given by Shukla et al 2007).
Note that is a polynomial of degree in z k.
Further for , where is a generalized Laguerre polynomial. So that
which is the required relationship.
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The authors wish to thank the refrees for valuable suggestions and comments.
Authors declare that they have no competing interests.
Both the authors, viz. MAK and SA with the consultation of each other, carried out this work and drafted the manuscript together. Both the authors read and approved the final manuscript.