On some properties of the generalized Mittag-Leffler function
© Khan and Ahmed; licensee Springer. 2013
Received: 18 November 2012
Accepted: 24 May 2013
Published: 23 July 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.
KeywordsMittag-Leffler function Generalized hypergeometric function Fox’s H 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.
which is known as Wiman function.
where min (Re(α), Re(β), Re(γ), Re(δ)) > 0
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:
which is (2.1.1).
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).
which is the proof of (2.2.1).
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).
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).
which proves (2.4.1).
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
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.
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
- (iv)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
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).
which completes the proof. □
Setting μ = ρ, ν = σ and p = 1, we get the Melin Barne’s integral of the function
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. □
- (i)For q = 1, (4.3.2) reduces to Tariq OSalim (2009)(4.1).(4.3.3)
- (ii)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)
from which the result follows. □
- (i)Putting q = δ = 1 in (4.5.2), we have(4.5.3)
- (ii)For q = γ = δ = 1 in (4.5.2), we have(4.5.4)
- (iii)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
Relationship with Fox H-function
Relationship with generalized Laguerre polynomials
where is a generalization of (given by Shukla et al 2007).
Note that is a polynomial of degree in z k .
which is the required relationship.
The authors wish to thank the refrees for valuable suggestions and comments.
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