On some properties of the generalized Mittag-Leffler function

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 \frac{1}{1-z}. 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 E_{\alpha ,\beta }^{\gamma }\left(z\right) in the form of

where (γ) _n is the Pochhammer symbol (Rainville (1960))

In 2007, Shukla and Prajapati (2007) introduced the function E_{\alpha ,\beta }^{\gamma ,q}\left(z\right) which is defined for \alpha ,\beta ,\gamma \in \mathbb{C}; Re(α) > 0,Re(β) > 0,Re(γ) > 0 and q\in (0,1)\bigcup \mathbb{N} as

In 2009, Tariq O. Salim (2009) introduced the function the function E_{\alpha ,\beta }^{\gamma ,\delta }\left(z\right) which is defined for \alpha ,\beta ,\gamma ,\delta \in \mathbb{C};\text{Re}\left(\alpha \right)>0,\text{Re}\left(\beta \right)>0,\text{Re}\left(\gamma \right)>0,\text{Re}\left(\delta \right)>0 as

In 2012, a new generalization of Mittag-Leffler function was defined by Salim (2012) as

where \alpha ,\beta ,\gamma ,\delta \in \mathbb{C}; 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

where

Further the generalization of definition (1.7) is investigated and defined as follows

where,

and

The definition (1.9) is a generalization of all above functions defined by (1.1)-(1.7).

• Setting μ = ν, ρ = σ, it reduces to Eq. (1.6) defined by Salim (2012), in addition of that if p = 1, it reduces to Eq. (1.7).

• Setting μ = ν, ρ, = σ and p = q = 1, it reduces to Eq. (1.5) defined by Salim (2009).

• Setting μ = ν, ρ = σ and p = δ = 1, it reduces to Eq. (1.4) defined by Shukla and Prajapati (2007), in addition of that if q = 1, then we get Eq. (1.3) defined by Prabhakar (1971).

• Setting μ = ν, ρ = σ and p = q = δ = 1, it reduces to Eq. (1.2) defined by Wiman (1905), moreover if β = 1, Mittag-Leffler function E _α (z) will be the result.

Throughout this investigation, we need the following well-known facts to study the various properties and relation formulas of the function E_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right).

• Beta(Euler) transforms (Sneddon (1979)) of the function f(z) is defined as

  B\left\{f\right(z);a,b\}={\int }_0^1{z}^{a-1}{(1-z)}^{b-1}f\left(z\right)\mathit{\text{dz}}

  (1.12)

• Laplace transforms (Sneddon (1979)) of the function f(z) is defined as

  L\left\{f\right(z);s\}={\int }_0^{\infty }{e}^{-\mathit{\text{sz}}}f\left(z\right)\mathit{\text{dz}}

  (1.13)

• Mellin- transforms of the function f(z) is defined as

  M\left\{f\right(z);s\}=f^{\ast }\left(s\right)={\int }_0^{\infty }{z}^{s-1}f\left(z\right)\mathit{\text{dz}},\mathrm{Re}\left(s\right)>0,

  (1.14)

• and the inverse Mellin-transform is given by

  f\left(z\right)=M^{-1}\left\{f^{\ast }\right(s);z\}=\frac{1}{2\pi i}{\int }_L{f}^{\ast }\left(s\right)z^{-s}\mathit{\text{ds}}

  (1.15)

• Whittaker transform (Whittaker and Watson (1962))

  {\int }_0^{\infty }{e}^{\frac{-t}{2}}{t}^{\nu -1}{W}_{\lambda ,\mu }\left(t\right)\mathit{\text{dt}}=\frac{\mathrm{\Gamma }(\frac{1}{2}+\mu +\nu )\mathrm{\Gamma }(\frac{1}{2}-\mu +\nu )}{\mathrm{\Gamma }(1-\lambda +\nu )},

  (1.16)

  where \text{Re}(\mu \pm \nu )>\frac{-1}{2} and W _{λ, μ} (t) is the Whittaker confluent hypergeometric function.

• The generalized hypergeometric function (Rainville (1960)) is defined as

  {}_p{F}_q\left[{\left(\alpha \right)}_1,\dots ,{\left(\alpha \right)}_p;{\left(\beta \right)}_1,\dots ,{\left(\beta \right)}_q;z)\right]=\sum _{n=0}^{\infty }\frac{\prod _{i=1}^p{\left({\alpha }_i\right)}_n}{\prod _{j=1}^q{\left(\beta \right)}_j{)}_n}\frac{z^n}{n!}

  (1.17)

• Wright generalized hypergeometric function (Srivastava and Manocha (1984)) is defined as

  \begin{array}{l}{}_p{\mathrm{\Psi }}_q\left[(a_1,A_1),\dots ,(a_p,A_p);(b_1,B_1),\mathit{\text{ldots}},(b_q,B_q);z\right]\\ =\sum _{n=0}^{\infty }\frac{\prod _{i=1}^p\mathrm{\Gamma }{(a_i+A_{i}n)}_n}{\prod _{j=1}^q\mathrm{\Gamma }(b_j+B_{j}n)}\frac{z^n}{n!}\end{array}

  (1.18)

• Fox’s H-function (Saigo and Kilbas (1998)) is given as

  \begin{array}{l}{H}_{p,q}^{m,n}\left[z\left|\right(a_1,{\alpha }_1),\dots ,(a_p,{\alpha }_p);(b_1,{\beta }_1),\dots ,(b_q,{\beta }_q)\right]\\ =\frac{1}{2\pi i}{\int }_L\frac{\prod _{j=1}^m\mathrm{\Gamma }(b_i+{\beta }_{i}s)\prod _{j=1}^n\mathrm{\Gamma }(1-a_i-{\alpha }_{i}s)}{\prod _{j=m+1}^q\mathrm{\Gamma }(1-b_j-{\beta }_{j}s)\prod _{j=n+1}^p\mathrm{\Gamma }(a_j+{\alpha }_{j}s)}{z}^{-s}\mathit{\text{ds}}\end{array}

  (1.19)

• Generalized Laguerre polynomials (Rainville (1960)). These are also known as Sonine polynomials and are defined as

  L_n^{\left(\alpha \right)}\left(x\right)=\frac{{(1+\alpha )}_n}{n!}{}_1{F}_1\left[-n;1+\alpha ;x\right]

  (1.20)

• Incomplete Gamma function (Rainville (1960)). This is denoted by γ(α, z) and is defined by

  \gamma (\alpha ,z)={\int }_0^z{e}^{-t}{t}^{\alpha -1}\mathit{\text{dt}},\text{Re}\left(\alpha \right)>0,

  (1.21)

As a consequence of definitions (1.1)-(1.9) the following results hold:

Theorem 2.1.

If \alpha ,\beta ,\gamma ,\delta ,\mu ,\nu ,\rho ,\sigma \in \mathbb{C}, 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

In particular,

Proof.

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).

Theorem 2.2.

If \alpha ,\beta ,\gamma ,\delta ,\mu ,\nu ,\sigma ,\rho ,w\in \mathbb{C}, 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 m\in \mathbb{N}

In particular,

Proof.

From (1.9),

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 \mathbb{C}), 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).

Theorem 2.3.

If \alpha =\frac{m}{r}, with m,r\in \mathbb{N} relatively prime;\beta ,\gamma ,\delta ,\mu ,\nu ,\rho ,\sigma \in \mathbb{C} and q < Re(α + p), then

Proof.

which proves (2.3.1). □

Corollary 2.3.

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).

Remark 2.3.

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

Theorem 2.4.

If then

In particular,

Proof.

which proves (2.4.1).

Now change the variable from s to u=\frac{s-t}{x-t}. Then the L.H.S. of (2.4.2) becomes

which proves (2.4.2).

Now

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. □

Using (1.9) with \alpha =k,\rho =l,\sigma =m\in \mathbb{N} and q\in \mathbb{N}, we have

where Δ(l;μ) is a l-tupple \frac{\mu }{l},\frac{\mu +1}{l},\dots \frac{\mu +l-1}{l}; Δ(q;γ) is a q-tupple \frac{\gamma }{q},\frac{\gamma +1}{q},\dots \frac{\gamma +q-1}{q}; Δ(k, β) is a k-tupple \frac{\beta }{k},\frac{\beta +1}{k},\dots ,\frac{\beta +k-1}{k} and so on, which is the required hypergeometric representation.

Convergence criterion of generalized Mittag-leffler function _q+l+1 F _k+p+m :

1. (i)

   If q + l + 1 ≤ k + p + m, the function _q+l+1 F _k+p+m converges for all finite z.

2. (ii)

   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

3. (iii)

   If q + l + 1 > k + p + m + 1, the function _q+1+1 F _k+p+m+1 is divergent for |z| ≠ 0

4. (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

   \begin{array}{l}\text{Re}\left(\sum _{r=1}^k\sum _{s=1}^p\sum _{t=1}^m\frac{\beta +r-1}{k}.\frac{\delta +s-1}{p}\frac{\nu +t-1}{m}\right.\\ \left.-\sum _{i=1}^l\sum _{j=1}^q\frac{\mu +i-1}{l}\frac{\gamma +j-1}{q}\right)>0\end{array}