On some properties of the generalized Mittag-Leffler function

Abstract

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.

Introduction

In 1903, the Swedish mathematician Gosta Mittag-Leffler (1903) introduced the function

$$E_{\alpha }\left(z\right)=\sum _{n=0}^{\infty }\frac{z^n}{\mathrm{\Gamma }(\alpha n+1)},$$

(1.1)

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

$$E_{\alpha ,\beta }\left(z\right)=\sum _{n=0}^{\infty }\frac{z^n}{\mathrm{\Gamma }(\alpha n+\beta )},(\alpha ,\beta \in \mathbb{C},\text{Re}(\alpha )>0,\text{Re}(\beta )>0),$$

(1.2)

which is known as Wiman function.

In 1971, Prabhakar (1971) introduced the function $E_{\alpha ,\beta }^{\gamma }\left(z\right)$ in the form of

$$\begin{array}{ll}{E}_{\alpha ,\beta }^{\gamma }\left(z\right)=& \sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_n{z}^n}{\mathrm{\Gamma }(\alpha n+\beta )n!},(\alpha ,\beta ,\gamma ,\in \mathbb{C},\text{Re}(\alpha )>0,\\ \text{Re}\left(\beta \right)>0,\text{Re}\left(\gamma \right)>0),\end{array}$$

(1.3)

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

$$\begin{array}{ll}{\left(\gamma \right)}_n& =\frac{\mathrm{\Gamma }(\gamma +n)}{\mathrm{\Gamma }\left(\gamma \right)},{\left(\gamma \right)}_0=1,\\ {\left(\gamma \right)}_n& =\gamma (\gamma +1)(\gamma +2)\cdots (\gamma +n-1),n\ge 1.\end{array}$$

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

$$E_{\alpha ,\beta }^{\gamma ,q}\left(z\right)=\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_{\mathit{\text{qn}}}{z}^n}{\mathrm{\Gamma }(\alpha n+\beta )n!},$$

(1.4)

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

$$E_{\alpha ,\beta }^{\gamma ,\delta }\left(z\right)=\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_n{z}^n}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\delta \right)}_n},$$

(1.5)

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

$$E_{\alpha ,\beta ,p}^{\gamma ,\delta ,q}\left(z\right)=\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_{\mathit{\text{qn}}}{z}^n}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\delta \right)}_{\mathit{\text{pn}}}},$$

(1.6)

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

$$E_{\alpha ,\beta ,\delta }^{\gamma ,q}\left(z\right)=\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_{\mathit{\text{qn}}}{z}^n}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\delta \right)}_n},\alpha ,\beta ,\gamma ,\delta \in \mathbb{C},$$

(1.7)

where

$$min\left(\text{Re}\left(\alpha \right),\text{Re}\left(\beta \right),\text{Re}\left(\gamma \right),\text{Re}\left(\delta \right)>0\right)\text{and q}\in (0,1)\bigcup \mathbb{N},$$

(1.8)

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

$$E_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)=\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{z}^n}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}},$$

(1.9)

where,

$$\alpha ,\beta ,\gamma ,\delta ,\mu ,\nu ,\rho ,\sigma \in \mathbb{C};p,q>0\mathit{\text{and}}q\le \text{Re}\left(\alpha \right)+p,$$

(1.10)

and

$$\mathit{\text{min}}\left(\text{Re}\right(\alpha ),\text{Re}(\beta ),\text{Re}(\gamma ),\text{Re}(\delta ),\text{Re}(\mu ),\text{Re}(\nu ),\text{Re}(\rho ),\text{Re}(\sigma \left)\right)>0,$$

(1.11)

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)

Basic properties of the function$E_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)$

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

$$E_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)=\beta E_{\alpha ,\beta +1,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)+\mathrm{\alpha z}\frac{d}{\mathit{\text{dz}}}{E}_{\alpha ,\beta +1,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)$$

(2.1.1)

$$E_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)=\frac{1}{\mathrm{\Gamma }\left(\beta \right)}+\frac{{\left(\mu \right)}_{\rho }{\left(\gamma \right)}_q}{{\left(\nu \right)}_{\sigma }{\left(\delta \right)}_p}{E}_{\alpha +1,\beta ,\nu +\sigma ,\delta +p}^{\mu +\rho ,\gamma +q}\left(z\right)$$

(2.1.2)

$$E_{\alpha ,\beta -\alpha ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)-E_{\alpha ,\beta -\alpha ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma -1,q}\left(z\right)$$

$$=\frac{\mathit{\text{qz}}{\left(\mu \right)}_{\rho }}{{\left(\nu \right)}_{\sigma }}\sum _{n=0}^{\infty }\frac{(n+1){(\mu +\rho )}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}+q-1}}{\mathrm{\Gamma }(\alpha n+\beta )(\delta +\mathit{\text{pn}}){(\nu +\sigma )}_{\eta n}}\frac{z^n}{{\left(\delta \right)}_{\mathit{\text{pn}}}}$$

(2.1.3)

In particular,

$$E_{\alpha ,\beta ,\delta }^{\gamma ,q}\left(z\right)=\beta E_{\alpha ,\beta +1,\delta }^{\gamma ,q}\left(z\right)+\mathrm{\alpha z}\frac{d}{\mathit{\text{dz}}}{E}_{\alpha ,\beta +1,\delta }^{\gamma ,q}\left(z\right),$$

(2.1.4)

$$E_{\alpha ,\beta ,\delta }^{\gamma ,q}\left(z\right)=\frac{1}{\mathrm{\Gamma }\left(\beta \right)}+\frac{{\left(\gamma \right)}_q}{\delta }z{E}_{\alpha ,\alpha +\beta ,\delta +1}^{\gamma +q,q}\left(z\right)$$

(2.1.5)

$$E_{\alpha ,\beta -\alpha ,\delta }^{\gamma ,q}\left(z\right)-E_{\alpha ,\beta -\alpha ,\delta }^{\gamma -1,q}\left(z\right)=\mathit{\text{qz}}\sum _{n=0}^{\infty }\frac{(n+1){\left(\gamma \right)}_{\mathit{\text{qn}}+q-1}}{\mathrm{\Gamma }(\alpha n+\beta )(\delta +n)}\frac{z^n}{{\left(\delta \right)}_n}$$

(2.1.6)

Proof.

$$\beta E_{\alpha ,\beta +1,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)+\mathrm{\alpha z}\frac{d}{\mathit{\text{dz}}}{E}_{\alpha ,\beta +1,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)$$

$$=\beta E_{\alpha ,\beta +1,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)+\mathrm{\alpha z}\frac{d}{\mathit{\text{dz}}}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }(\alpha n+\beta +1){\left(\nu \right)}_{\eta n}}\frac{z^n}{{\left(\delta \right)}_{\mathit{\text{pn}}}}$$

$$=\beta E_{\alpha ,\beta +1,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)+\sum _{n=0}^{\infty }\frac{\alpha n{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }(\alpha n+\beta +1){\left(\nu \right)}_{\eta n}}\frac{z^n}{{\left(\delta \right)}_{\mathit{\text{pn}}}}$$

$$=\beta E_{\alpha ,\beta +1,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)+\sum _{n=0}^{\infty }\frac{(\alpha n+\beta -\beta ){\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }(\alpha n+\beta +1){\left(\nu \right)}_{\eta n}}\frac{z^n}{{\left(\delta \right)}_{\mathit{\text{pn}}}}$$

$$=E_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)$$

which is (2.1.1).

The proof of (2.1.2) can easily be followed from the definition (1.9). Now

$$E_{\alpha ,\beta -\alpha ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)-E_{\alpha ,\beta -\alpha ,\nu ,\sigma ,\mathrm{\delta p}}^{\mu ,\rho ,\gamma -1,q}\left(z\right)$$

$$\begin{array}{l}=\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }\left(\alpha \right(n-1)+\beta ){\left(\nu \right)}_{\eta n}}\frac{z^n}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\\ -\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{(\gamma -1)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }\left(\alpha \right(n-1)+\beta ){\left(\nu \right)}_{\eta n}}\frac{z^n}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\end{array}$$

$$=q\sum _{n=1}^{\infty }\frac{n{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}-1}}{\mathrm{\Gamma }\left(\alpha \right(n-1)+\beta )}\frac{z^n}{{\left(\delta \right)}_{\mathit{\text{pn}}}}$$

$$=\frac{{\left(\mu \right)}_{\rho }\mathit{\text{qz}}}{{\left(\nu \right)}_{\sigma }}\sum _{n=0}^{\infty }\frac{(n+1){(\mu +\rho )}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}+q-1}}{\mathrm{\Gamma }(\alpha n+\beta ){(\nu +\sigma )}_{\eta n}(\delta +\mathit{\text{pn}})}\frac{z^n}{{\left(\delta \right)}_{\mathit{\text{pn}}}}$$

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 $m\in \mathbb{N}$

$$\begin{array}{l}{\left(\frac{d}{\mathit{\text{dz}}}\right)}^m{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)\\ =\frac{{\left(1\right)}_m{\left(\mu \right)}_{\mathrm{\rho m}}{\left(\gamma \right)}_{\mathit{\text{qm}}}}{{\left(\nu \right)}_{\mathrm{\sigma m}}{\left(\delta \right)}_{\mathit{\text{pm}}}}\sum _{n=0}^{\infty }\\ \times \frac{{(\mu +\mathrm{\rho m})}_{\rho n}{(\gamma +\mathit{\text{qm}})}_{\mathit{\text{qn}}}{(1+m)}_n}{\mathrm{\Gamma }(\alpha n+\beta +\alpha ){(\nu +\mathrm{\sigma m})}_{\eta n}{(\delta +\mathit{\text{pm}})}_{\mathit{\text{pn}}}}\frac{z^n}{n!},\end{array}$$

(2.2.1)

$$\begin{array}{l}{\left(\frac{d}{\mathit{\text{dz}}}\right)}^m\left[z^{\beta -1}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\right(w{z}^{\alpha }\left)\right]=z^{\beta -m-1}{E}_{\alpha ,\beta -m,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(w{z}^{\alpha }\right),\\ \times \text{Re}(\beta -m)>0,\end{array}$$

(2.2.2)

In particular,

$${\left(\frac{d}{\mathit{\text{dz}}}\right)}^m{E}_{\alpha ,\beta ,\delta }^{\gamma ,q}\left(z\right)=\frac{{\left(1\right)}_m{\left(\gamma \right)}_{\mathit{\text{qm}}}}{{\left(\delta \right)}_m}\sum _{n=0}^{\infty }\frac{{(\gamma +\mathit{\text{qm}})}_{\mathit{\text{qn}}}{(1+m)}_n}{\mathrm{\Gamma }(\alpha n+\beta +\alpha {(\delta +m)}_n}\frac{z^n}{n!},$$

(2.2.3)

$$\begin{array}{l}{\left(\frac{d}{\mathit{\text{dz}}}\right)}^m\left[z^{\beta -1}{E}_{\alpha ,\beta ,\delta }^{\gamma ,q}\right(w{z}^{\alpha }\left)\right]=z^{\beta -m-1}{E}_{\alpha ,\beta -m,p}^{\gamma ,q}\left(w{z}^{\alpha }\right),\\ \times \text{Re}(\beta -m)>0,\end{array}$$

(2.2.4)

Proof.

From (1.9),

$$\begin{array}{l}{\left(\frac{d}{\mathit{\text{dz}}}\right)}^m\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\nu \right)}_{\eta n}}\frac{z^n}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\\ =\sum _{n=m}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}n!}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}\frac{z^{n-m}}{(n-m)!}\\ =\frac{{\left(1\right)}_m{\left(\mu \right)}_{\mathrm{\rho m}}{\left(\gamma \right)}_{\mathit{\text{qm}}}}{{\left(\nu \right)}_{\mathrm{\sigma m}}{\left(\delta \right)}_{\mathit{\text{pm}}}}\sum _{n=0}^{\infty }\\ \times \frac{{(\mu +\rho )}_{\mathrm{\rho m}}{(\gamma +\mathit{\text{qm}})}_{\mathit{\text{qn}}}{(1+m)}_n}{\mathrm{\Gamma }(\alpha n+\mathrm{\alpha m}+\beta ){(\nu +\sigma )}_{\mathrm{\sigma m}}{(\delta +\mathit{\text{pm}})}_{\mathit{\text{pn}}}}\frac{z^n}{n!}\end{array}$$

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

$$\begin{array}{l}{\left(\frac{d}{\mathit{\text{dz}}}\right)}^m\left[z^{\beta -1}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\right(w{z}^{\alpha }\left)\right]\\ ={\left(\frac{d}{\mathit{\text{dz}}}\right)}^m\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{w}^n}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\nu \right)}_{\eta n}}\frac{z^{\alpha n+\beta -1}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\\ =\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}(\alpha n+\beta -1)!w^n}{\mathrm{\Gamma }(\alpha n+\beta )(\alpha n+\beta -1-m)!{\left(\nu \right)}_{\eta n}}\frac{z^{\alpha n+\beta -m-1}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\\ =z^{\beta -m-1}{E}_{\alpha ,\beta -m,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(w{z}^{\alpha }\right)\end{array}$$

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

$$\begin{array}{l}\frac{d^m}{d{z}^m}{E}_{\frac{m}{r},\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z^{\frac{p}{r}}\right)=\frac{\mathrm{\Gamma }\left(\nu \right)}{\mathrm{\Gamma }\left(\mu \right)}\sum _{n=1}^{\infty }\\ \times \frac{\mathrm{\Gamma }(\frac{\mathit{\text{nm}}}{r}+1)\mathrm{\Gamma }(\mu +\rho n){\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }(\frac{\mathit{\text{nm}}}{r}+\beta )\mathrm{\Gamma }(\frac{\mathit{\text{nm}}}{r}-m+1)\mathrm{\Gamma }(\nu +\eta n)}\frac{z^{(\frac{n}{r}-1)m}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\end{array}$$

(2.3.1)

$$\begin{array}{l}\frac{d^m}{d{z}^m}{E}_{\frac{m}{r},\beta ,\delta }^{\gamma ,q}\left(z^{\frac{m}{r}}\right)=\mathrm{\Gamma }\left(\delta \right)\sum _{n=1}^{\infty }\\ \times \frac{{\left(\gamma \right)}_{\mathit{\text{qn}}}\mathrm{\Gamma }(n+1)\mathrm{\Gamma }(\frac{\mathit{\text{nm}}}{r}+1)}{\mathrm{\Gamma }(\delta +n)\mathrm{\Gamma }(\frac{\mathit{\text{nm}}}{r}+\beta )\mathrm{\Gamma }(\frac{\mathit{\text{np}}}{r}-m+1)}\frac{z^{(\frac{n}{r}-1)m}}{n!}\end{array}$$

(2.3.2)

Proof.

$$\begin{array}{l}\frac{d^m}{d{z}^m}{E}_{\frac{m}{r},\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z^{\frac{p}{r}}\right)\\ =\frac{d^m}{d{z}^m}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }(\frac{\mathit{\text{nm}}}{r}+\beta ){\left(\nu \right)}_{\eta n}}\frac{z^{\left(\frac{m}{r}\right)n}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\\ =\sum _{n=1}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}\left(\frac{\mathit{\text{nm}}}{r}\right)!}{\mathrm{\Gamma }(\frac{\mathit{\text{nm}}}{r}+\beta ){\left(\nu \right)}_{\eta n}(\frac{\mathit{\text{nm}}}{r}-m)!}\frac{z^{(\frac{n}{r}-1)m}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\\ =\frac{\mathrm{\Gamma }\left(\nu \right)}{\mathrm{\Gamma }\left(\mu \right)}\sum _{n=1}^{\infty }\frac{\mathrm{\Gamma }(\frac{\mathit{\text{nm}}}{r}+1)\mathrm{\Gamma }(\mu +\rho n){\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }(\frac{\mathit{\text{nm}}}{r}+\beta )\mathrm{\Gamma }(\frac{\mathit{\text{nm}}}{r}-m+1)\mathrm{\Gamma }(\nu +\eta n)}\frac{z^{(\frac{n}{r}-1)m}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\end{array}$$

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

$$\frac{d^m}{d{z}^m}{E}_{\frac{m}{r},\beta }^{\gamma }\left(z^{\frac{m}{r}}\right)=\sum _{n=1}^{\infty }\frac{{\left(\gamma \right)}_n\mathrm{\Gamma }(\frac{\mathit{\text{np}}}{r}+1)}{\mathrm{\Gamma }(\frac{\mathit{\text{nm}}}{r}+\beta )\mathrm{\Gamma }(\frac{\mathit{\text{np}}}{r}-m+1)}\frac{z^{(\frac{n}{r}-1)m}}{n!}$$

(2.3.3)

For β = γ = δ = q = 1 in (2.3.2), we have

$$\frac{d^m}{d{z}^m}{E}_{\frac{m}{r}}\left(z^{\frac{m}{r}}\right)=\sum _{n=1}^{\infty }\frac{z^{(\frac{n}{r}-1)m}}{\mathrm{\Gamma }(\frac{\mathit{\text{nm}}}{r}-m+1)}$$

(2.3.4)

Theorem 2.4.

If $\alpha ,\beta ,\mu ,\rho ,\nu ,\gamma ,\lambda ,\sigma ,\delta ,\eta \in \mathbb{C};\text{Re}\left(\alpha \right)>0,\text{Re}\left(\beta \right)>0,\text{Re}\left(\gamma \right)>0,\text{Re}\left(\mu \right)>0,\text{Re}\left(\rho \right)>0,\text{Re}\left(\nu \right)>0,\text{Re}\left(\delta \right)>0,\text{Re}\left(\sigma \right)>0,\text{Re}\left(\lambda \right)>0\mathit{\text{and}}q<\text{Re}\left(\alpha \right)+\mathrm{p}$ then

$$\begin{array}{l}\frac{1}{\mathrm{\Gamma }\left(\eta \right)}{\int }_0^1{u}^{\beta -1}{(1-u)}^{\eta -1}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z{u}^{\alpha }\right)\mathit{\text{du}}\\ =E_{\alpha ,\beta +\eta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)\mathit{\text{du}},\end{array}$$

(2.4.1)

$$\begin{array}{l}\frac{1}{\mathrm{\Gamma }\left(\eta \right)}{\int }_t^x{(x-s)}^{\eta -1}{(s-t)}^{\beta -1}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left[\lambda {(s-t)}^{\alpha }\right]\mathit{\text{du}}\\ ={(x-t)}^{\eta +\beta -1}{E}_{\alpha ,\beta +\eta ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right),\end{array}$$

(2.4.2)

$${\int }_0^z{t}^{\beta -1}{E}_{\alpha ,\beta ,\mu ,\rho ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(w{t}^{\alpha }\right)\mathit{\text{dt}}=z^{\beta }{E}_{\alpha ,\beta +1,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(w{z}^{\alpha }\right),$$

(2.4.3)

In particular,

$$\frac{1}{\mathrm{\Gamma }\left(\eta \right)}{\int }_0^1{u}^{\beta -1}{(1-u)}^{\sigma -1}{E}_{\alpha ,\beta ,\delta }^{\gamma ,q}\left(z{u}^{\alpha }\right)\mathit{\text{du}}=E_{\alpha ,\beta +\eta ,\delta }^{\gamma ,q}\left(z\right)\mathit{\text{du}},$$

(2.4.4)

$$\begin{array}{l}\frac{1}{\mathrm{\Gamma }\left(\eta \right)}{\int }_t^x{(x-s)}^{\eta -1}{(s-t)}^{\beta -1}{E}_{\alpha ,\beta ,\delta }^{\gamma ,q}\left[\lambda {(s-t)}^{\alpha }\right]\mathit{\text{du}}\\ ={(x-t)}^{\eta +\beta -1}{E}_{\alpha ,\beta +\eta ,\delta }^{\gamma ,q}\left(z\right),\end{array}$$

(2.4.5)

$${\int }_0^z{t}^{\beta -1}{E}_{\alpha ,\beta ,\delta }^{\gamma ,q}\left(w{t}^{\alpha }\right)\mathit{\text{dt}}=z^{\beta }{E}_{\alpha ,\beta +1,\delta }^{\gamma ,q}\left(w{z}^{\alpha }\right),$$

(2.4.6)

$$\frac{1}{\mathrm{\Gamma }\left(\eta \right)}{\int }_0^1{u}^{\beta -1}{(1-u)}^{\sigma -1}{E}_{\alpha ,\beta }^{\gamma }\left(z{u}^{\alpha }\right)\mathit{\text{du}}=E_{\alpha ,\beta +\eta }^{\gamma }\left(z\right)\mathit{\text{du}},$$

(2.4.7)

$${\int }_0^z{t}^{\beta -1}{E}_{\alpha ,\beta }\left(w{t}^{\alpha }\right)\mathit{\text{dt}}=z^{\beta }{E}_{\alpha ,\beta +1}\left(w{z}^{\alpha }\right),$$

(2.4.8)

Proof.

$$\begin{array}{l}\frac{1}{\mathrm{\Gamma }\left(\eta \right)}{\int }_0^1{u}^{\beta -1}{(1-u)}^{\eta -1}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z{u}^{\alpha }\right)\mathit{\text{du}}\\ =\frac{1}{\mathrm{\Gamma }\left(\eta \right)}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{z}^n}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}{\int }_0^1\\ \times u^{\alpha n+\beta -1}{(1-u)}^{\eta -1}\mathit{\text{du}}\\ =\frac{1}{\mathrm{\Gamma }\left(\eta \right)}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{z}^n}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}B(\alpha n+\beta ,\eta )\\ =E_{\alpha ,\beta +\eta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)\end{array}$$

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

$$\frac{1}{\mathrm{\Gamma }\left(\eta \right)}{\int }_0^1{(x-t)}^{\eta +\beta -1}{(1-u)}^{\eta -1}{u}^{\beta -1}$$

$$\times \sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{z}^n}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}{\left(\lambda \right)}^n{(x-t)}^{\alpha n}{u}^{\alpha n}\mathit{\text{du}}$$

$$\begin{array}{l}=\frac{{(x-t)}^{\eta +\beta -1}}{\mathrm{\Gamma }\left(\eta \right)}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{\left[\lambda {(x-t)}^{\alpha }\right]}^n}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}{\int }_0^1\\ \times u^{\alpha n+\beta -1}{(1-u)}^{\eta -1}\mathit{\text{du}}\end{array}$$

$$=\frac{{(x-t)}^{\eta +\beta -1}}{\mathrm{\Gamma }\left(\eta \right)}\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_{\mathit{\text{qn}}}{\left[\lambda {(x-t)}^{\alpha }\right]}^n}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\delta \right)}_{\mathit{\text{pn}}}}B(\alpha n+\beta ,\eta )$$

$$={(x-t)}^{\eta +\beta -1}{E}_{\alpha ,\beta +\eta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}{\left[\lambda {(x-t)}^{\alpha }\right]}^n$$

which proves (2.4.2).

Now

$$\begin{array}{l}{\int }_0^z{t}^{\beta -1}{E}_{\alpha ,\beta ,\nu ,\sigma ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(w{t}^{\alpha }\right)\mathit{\text{dt}}\\ ={\int }_0^z{t}^{\beta -1}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{w}^n{t}^{\alpha n}}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}\mathit{\text{dt}}\\ =\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{w}^n}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}{\int }_0^z{t}^{\alpha n+\beta -1}\mathit{\text{dt}}\\ =z^{\beta }\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{\left(w{z}^{\alpha }\right)}^n}{\mathrm{\Gamma }(\alpha n+\beta +1){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}\\ =z^{\beta }{E}_{\alpha ,\beta +1,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(w{z}^{\alpha }\right)\end{array}$$

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 $E_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)$

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

$$\begin{array}{ll}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)& =\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\mathit{\text{ln}}}{\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }(\mathit{\text{kn}}+\beta ){\left(\nu \right)}_{\mathit{\text{mn}}}{\left(\delta \right)}_{\mathit{\text{pn}}}}{z}^n\\ =\frac{1}{\mathrm{\Gamma }\left(\beta \right)}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\mathit{\text{ln}}}{\left(\gamma \right)}_{\mathit{\text{qn}}}{\left(1\right)}_n}{{\left(\beta \right)}_{\mathit{\text{kn}}}{\left(\nu \right)}_{\mathit{\text{mn}}}{\left(\delta \right)}_{\mathit{\text{pn}}}}\frac{z^n}{n!}\end{array}$$

$$\begin{array}{l}=\frac{1}{\mathrm{\Gamma }\left(\beta \right)}\sum _{n=0}^{\infty }\frac{\prod _{i=1}^l{\left(\frac{\mu +i-1}{l}\right)}_n\prod _{j=1}^q{\left(\frac{\gamma +j-1}{q}\right)}_n{\left(1\right)}_n}{\prod _{r=1}^k{\left(\frac{\beta +r-1}{k}\right)}_n\prod _{s=1}^p{\left(\frac{\delta +s-1}{p}\right)}_n\prod _{t=1}^m{\left(\frac{\nu +t-1}{m}\right)}_{n}n!}\\ \times {\left(\frac{l^l{q}^{q}z}{m^m{p}^p{k}^k}\right)}^n\end{array}$$

$$\begin{array}{l}=\frac{1}{\mathrm{\Gamma }\left(\beta \right)}{}_{q+l+1}{F}_{k+p+m}\left[\Delta (l;\mu ),\Delta (q;\gamma ),(1,1);\Delta (k,\beta ),\right.\\ \left.\Delta (p;\delta ),\Delta (m;\nu );\frac{l^l{q}^{q}z}{m^m{p}^p{k}^k}\right]\end{array}$$

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 :

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

• (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

• (iii) 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

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

Integral transforms of $E_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)$

In this section we discuss some useful integral transforms like Euler transform, laplace transform and Whittaker transform of $E_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right).$

Theorem 4.1.

Mellin-Barnes integral representation of $E_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right).$

Let (1.9) and (1.10) be satified and $\alpha \in R_{+}=\left(0\infty \right);\beta ,\nu ,\sigma ,\mu ,\rho ,\gamma ,\delta \in \mathbb{C}$ and q < R e(α) + p. Then the function $E_{\alpha ,\beta ,\delta }^{\gamma ,q}\left(z\right)$ is represented by Mellin-Barnes integral as:

$$\begin{array}{l}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)=\frac{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{2\pi i\Gamma \left(\gamma \right)\mathrm{\Gamma }\left(\mu \right)}{\int }_L\\ \times \frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }(1-s)\mathrm{\Gamma }(\mu -\rho s)\mathrm{\Gamma }(\gamma -\mathit{\text{qs}})}{\mathrm{\Gamma }(\beta -\alpha s)\mathrm{\Gamma }(\delta -\mathit{\text{ps}})\mathrm{\Gamma }(\nu -\rho s)}{(-z)}^{-s}\mathit{\text{ds}},\end{array}$$

(4.1.1)

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 $s=-n,\forall n\in {\mathbb{N}}_0$ (to the left) from those at $s=\frac{\gamma +n}{q},n\in {\mathbb{N}}_0$ (to the right).

Proof.

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

$$\frac{1}{2\pi i}{\int }_L\frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }(1-s)\mathrm{\Gamma }(\mu -\rho s)\mathrm{\Gamma }(\gamma -\mathit{\text{qs}})}{\mathrm{\Gamma }(\beta -\alpha s)\mathrm{\Gamma }(\nu -\sigma s)\mathrm{\Gamma }(\delta -\mathit{\text{ps}})}{(-z)}^{-s}\mathit{\text{ds}}$$

$$=\sum _{n=0}^{\infty }{\mathit{\text{Res}}}_{s=-n}\left[\frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }(1-s)\mathrm{\Gamma }(\mu -\rho s)\mathrm{\Gamma }(\gamma -\mathit{\text{qs}})}{\mathrm{\Gamma }(\beta -\alpha s)\mathrm{\Gamma }(\delta -\mathit{\text{ps}})\mathrm{\Gamma }(\nu -\sigma s)}{(-z)}^{-s}\right]$$

$$\begin{array}{l}=\sum _{n=0}^{\infty }{\mathit{\text{Lim}}}_{s\to -n}\frac{\pi (s+n)}{sin\mathrm{\pi s}}.\frac{\mathrm{\Gamma }(\mu -\rho s)\mathrm{\Gamma }(\gamma -\mathit{\text{qs}})}{\mathrm{\Gamma }(\beta -\alpha s)\mathrm{\Gamma }(\delta -\mathit{\text{ps}})\mathrm{\Gamma }(\nu -\sigma s)}\\ \times {(-z)}^{-s}\end{array}$$

$$=\sum _{n=0}^{\infty }\frac{{(-1)}^n\mathrm{\Gamma }(\mu +\rho n)\mathrm{\Gamma }(\gamma +\mathit{\text{qn}})}{\mathrm{\Gamma }(\beta +\alpha n)\mathrm{\Gamma }(\delta +\mathit{\text{pn}})\mathrm{\Gamma }(\nu +\eta n)}{(-z)}^n$$

$$=\frac{\mathrm{\Gamma }\left(\gamma \right)\mathrm{\Gamma }\left(\mu \right)}{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\delta \right)}_{\mathit{\text{pn}}}{\left(\nu \right)}_{\eta n}}{z}^n$$

$$=\frac{\mathrm{\Gamma }\left(\gamma \right)\mathrm{\Gamma }\left(\mu \right)}{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)$$

which completes the proof. □

Remark 4.1.

Setting μ = ρ, ν = σ and p = 1, we get the Melin Barne’s integral of the function $E_{\alpha ,\beta ,\delta }^{\gamma ,q}.$

Theorem 4.2.

(Mellin transform) of $E_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)$

$$\begin{array}{l}M\left\{E_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\right(-\mathit{\text{wz}});s\}=\frac{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{2\pi i\Gamma \left(\gamma \right)\mathrm{\Gamma }\left(\mu \right)}{\int }_L\\ \times \frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }(1-s)\mathrm{\Gamma }(\mu -\rho s)\mathrm{\Gamma }(1-\mathit{\text{qs}})}{\mathrm{\Gamma }\left(\gamma \right)\mathrm{\Gamma }(\beta -\alpha s)\mathrm{\Gamma }(\nu -\sigma s)\mathrm{\Gamma }(\delta -\mathit{\text{ps}})}{\left(\mathit{\text{wz}}\right)}^{-s}\mathit{\text{ds}},\end{array}$$

(4.2.1)

Proof.

From Theorem 4.1, we have

$$\begin{array}{l}{E}_{\alpha ,\beta ,\nu ,\mathrm{\sigma \sigma },\delta ,p}^{\mu ,\rho ,\gamma ,q}(-\mathit{\text{wz}})\\ =\frac{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{2\pi i\Gamma \left(\gamma \right)\mathrm{\Gamma }\left(\mu \right)}{\int }_L\frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }(1-s)\mathrm{\Gamma }(\mu -\rho s)\mathrm{\Gamma }(\gamma -\mathit{\text{qs}})}{\mathrm{\Gamma }(\beta -\alpha s)\mathrm{\Gamma }(\nu -\sigma s)\mathrm{\Gamma }(\delta -\mathit{\text{ps}})}\\ \times {\left(\mathit{\text{wz}}\right)}^{-s}\mathit{\text{ds}}\end{array}$$

$$=\frac{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{2\pi i\Gamma \left(\gamma \right)\mathrm{\Gamma }\left(\mu \right)}{\int }_L{f}^{\ast }\left(s\right){\left(z\right)}^{-s}\mathit{\text{ds}}$$

where

$$f^{\ast }\left(s\right)=\frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }(1-s)\mathrm{\Gamma }(\mu -\rho s)\mathrm{\Gamma }(\gamma -\mathit{\text{qs}})}{\mathrm{\Gamma }(\beta -\alpha s)\mathrm{\Gamma }(\nu -\sigma s)\mathrm{\Gamma }(\delta -\mathit{\text{ps}})}{\left(w\right)}^{-s}$$

is in the form of inverse Mellin-Transform (1.15). So applying the Mellin-transform (1.14) yields directly the required result. □

Theorem 4.3.

(Euler(Beta)transform) of$E_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)$

$$\begin{array}{l}{\int }_0^1{z}^{a-1}{(1-z)}^{b-1}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(x{z}^{\eta }\right)\mathit{\text{dz}}\\ =\frac{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(\nu \right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}{}_4{\mathrm{\Psi }}_4\left[\begin{array}{ll}(\mu ,\rho )(\gamma ,q),(a,\eta ),(1,1)& ;\\ x\\ (\beta ,\alpha ),(a+b,\eta ),(\nu ,\sigma ),(\delta ,p)& ;\end{array}\right]\end{array}$$

(4.3.1)

Proof.

$${\int }_0^1{z}^{a-1}{(1-z)}^{b-1}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(x{z}^{\eta }\right)\mathit{\text{dz}}$$

$$={\int }_0^1{z}^{a-1}{(1-z)}^{b-1}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{\left(x{z}^{\eta }\right)}^n}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}$$

$$=\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{x}^n}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}{\int }_0^1{z}^{\eta n+a-1}{(1-z)}^{b-1}\mathit{\text{dz}}$$

$$=\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{x}^n}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}B(\eta n+a,b)$$

$$\begin{array}{l}=\frac{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(\nu \right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}\sum _{n=0}^{\infty }\\ \times \frac{\mathrm{\Gamma }(\mu +\rho n)\mathrm{\Gamma }(\mathit{\text{qn}}+\gamma )\mathrm{\Gamma }(a+\eta n)\mathrm{\Gamma }(n+1)}{\mathrm{\Gamma }(\alpha n+\beta )\mathrm{\Gamma }(\eta n+a+b)\mathrm{\Gamma }(\nu +\eta n)\mathrm{\Gamma }(\mathit{\text{pn}}+\delta )}\frac{x^n}{n!},\end{array}$$

from which the result follows. □

Corollary 4.3.

$$\begin{array}{l}{\int }_0^1{z}^{a-1}{(1-z)}^{b-1}{E}_{\alpha ,\beta ,\delta }^{\gamma ,q}\left(x{z}^{\eta }\right)\mathit{\text{dz}}\\ =\frac{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\gamma \right)}{}_3{\mathrm{\Psi }}_3\left[\begin{array}{ll}(\gamma ,q),(a,\eta ),(1,1)& ;\\ x\\ (\beta ,\alpha ),(a+b,\eta ),(\delta ,1)& ;\end{array}\right]\end{array}$$

(4.3.2)

Special properties:

• (i) For q = 1, (4.3.2) reduces to Tariq OSalim (2009)(4.1).

  $$\begin{array}{l}{\int }_0^1{z}^{a-1}{(1-z)}^{b-1}{E}_{\alpha ,\beta }^{\gamma ,\delta }\left(x{z}^{\sigma }\right)\mathit{\text{dz}}\\ =\frac{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\gamma \right)}{}_3{\mathrm{\Psi }}_3\left[\begin{array}{ll}(\gamma ,1),(a,\sigma ),(1,1)& ;\\ x\\ (\beta ,\alpha ),(\delta ,1)(a+b,\sigma )& ;\end{array}\right],\end{array}$$

  (4.3.3)

• (ii) For δ = q = 1 in (4.3.2), we have

  $$\begin{array}{l}{\int }_0^1{z}^{a-1}{(1-z)}^{b-1}{E}_{\alpha ,\beta }^{\gamma }\left(x{z}^{\sigma }\right)\mathit{\text{dz}}\\ =\frac{\mathrm{\Gamma }\left(b\right)}{\mathrm{\Gamma }\left(\gamma \right)}{}_2{\mathrm{\Psi }}_2\left[\begin{array}{ll}(\gamma ,1),(a,\sigma ),& ;\\ x\\ (\beta ,\alpha ),(a+b,\sigma )& ;\end{array}\right],\end{array}$$

  (4.3.4)

  If a = β, α = σ, then (4.3.2) reduces to

  $${\int }_0^1{z}^{\beta -1}{(1-z)}^{b-1}{E}_{\sigma ,\beta ,\delta }^{\gamma ,q}\left(x{z}^{\sigma }\right)\mathit{\text{dz}}=\frac{\mathrm{\Gamma }\left(b\right)}{\mathrm{\Gamma }\left(\gamma \right)}{E}_{\sigma ,\beta +b,\delta }^{\gamma ,q}\left(x\right),$$

  (4.3.5)

  Putting α = β = γ = δ = q = 1 in (4.3.2), we have

  $$\begin{array}{l}{\int }_0^1{z}^{a-1}{(1-z)}^{b-1}exp\left(x{z}^{\sigma }\right)\mathit{\text{dz}}\\ =\mathrm{\Gamma }\left(b\right){}_1{\mathrm{\Psi }}_1\left[\begin{array}{ll}(a,\sigma ),& ;\\ x\\ (a+b,\sigma )& ;\end{array}\right]\end{array}$$

  (4.3.6)

Theorem 4.4. (Laplace transform)

$$\begin{array}{l}{\int }_0^{\infty }{z}^{a-1}{e}^{-\mathit{\text{sz}}}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(x{z}^{\eta }\right)\mathit{\text{dz}}\\ =\frac{\mathrm{\Gamma }\left(\nu \right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}{s}^{-a}{}_3{\mathrm{\Psi }}_3\left[\begin{array}{ll}(\mu ,\rho )(\gamma ,q),(1,1)& ;\\ \frac{x}{s^{\eta }}\\ (\beta ,\alpha ),(\nu ,\sigma )(\delta ,p)& ;\end{array}\right],\end{array}$$

(4.4.1)

Proof.

$$\begin{array}{l}{\int }_0^{\infty }{z}^{a-1}{e}^{-\mathit{\text{sz}}}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(x{z}^{\eta }\right)\mathit{\text{dz}}\\ ={\int }_0^{\infty }{z}^{a-1}{e}^{-\mathit{\text{sz}}}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{\left(x{z}^{\sigma }\right)}^n}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}\mathit{\text{dz}}\end{array}$$

$$=\frac{\mathrm{\Gamma }\left(\nu \right)\mathrm{\Gamma }\left(\delta \right)s^{-a}}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}\sum _{n=0}^{\infty }\frac{\mathrm{\Gamma }(\mu +\rho n)\mathrm{\Gamma }(\gamma +\mathit{\text{qn}})\mathrm{\Gamma }(1+n)}{\mathrm{\Gamma }(\beta +\alpha n)\mathrm{\Gamma }(\nu +\eta n)\mathrm{\Gamma }(\delta +\mathit{\text{pn}})}\frac{{\left(\frac{x}{s^{\sigma }}\right)}^n}{n!},$$

from which the result follows. □

Corollary 4.4.

$$\begin{array}{l}{\int }_0^{\infty }{z}^{a-1}{e}^{-\mathit{\text{sz}}}{E}_{\alpha ,\beta ,\delta }^{\gamma ,q}\left(x{z}^{\eta }\right)\mathit{\text{dz}}\\ =\frac{\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\gamma \right)}{s}^{-a}{}_2{\mathrm{\Psi }}_2\left[\begin{array}{ll}(\gamma ,q),(1,1)& ;\\ \frac{x}{s^{\eta }}\\ (\beta ,\alpha ),(\delta ,1)& ;\end{array}\right],\end{array}$$

(4.4.2)

Remark 4.4.

For q = 1, (4.4.2) reduces to Tariq O Salim (2009)(4.2).

Theorem 4.5. (Whittaker transform)

$$\begin{array}{l}{\int }_0^{\infty }{e}^{\frac{-\mathrm{\varphi t}}{2}}{t}^{\xi -1}{W}_{\lambda ,\psi }\left(\mathrm{\varphi t}\right)E_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(\omega t^{\eta }\right)\mathit{\text{dt}}=\frac{\mathrm{\Gamma }\left(\nu \right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}{\varphi }^{-\xi }{}_5{\mathrm{\Psi }}_4\\ \times \left[\begin{array}{ll}(\mu ,\rho )(\gamma ,q),(\frac{1}{2}+\psi +\xi ,\eta ),(\frac{1}{2}-\psi +\xi ,\eta ),(1,1)& ;\\ \frac{\omega }{{\varphi }^{\eta }}\\ (\beta ,\alpha ),(\nu ,\sigma ),(1-\lambda +\xi ,\eta ),(\delta ,p)& ;\end{array}\right]\end{array}$$

(4.5.1)

Proof.

Substituting ϕ t = v in L.H.S. of Theorem 4.5, we have

$${\int }_0^{\infty }{\left(\frac{v}{\varphi }\right)}^{\xi -1}{e}^{\frac{-v}{2}}{W}_{\lambda ,\psi }\left(v\right)\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{\left(\omega \right)}^n}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\delta \right)}_n}{\left(\frac{v}{\varphi }\right)}^{\mathrm{n\eta }}\frac{1}{\varphi }\mathit{\text{dv}}$$

$$\begin{array}{l}=\frac{{\varphi }^{-\xi }\mathrm{\Gamma }\left(\nu \right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}\sum _{n=0}^{\infty }\frac{\mathrm{\Gamma }(\mu +\rho n)\mathrm{\Gamma }(\gamma +\mathit{\text{qn}})}{\mathrm{\Gamma }(\alpha n+\beta )\mathrm{\Gamma }(\nu +\eta n)\mathrm{\Gamma }(\delta +\mathit{\text{pn}}}\\ \times {\left(\frac{\omega }{{\varphi }^{\eta }}\right)}^n{\int }_0^{\infty }{\left(v\right)}^{\eta n+\xi -1}{e}^{\frac{-v}{2}}{W}_{\lambda ,\psi }\left(v\right)\mathit{\text{dv}}\end{array}$$

$$\begin{array}{l}=\frac{{\varphi }^{-\xi }\mathrm{\Gamma }\left(\nu \right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}\sum _{n=0}^{\infty }\frac{\mathrm{\Gamma }(\mu +\rho n)\mathrm{\Gamma }(\gamma +\mathit{\text{qn}})}{\mathrm{\Gamma }(\alpha n+\beta )\mathrm{\Gamma }(\nu +\eta n)\mathrm{\Gamma }(\delta +\mathit{\text{pn}})}\\ \times {\left(\frac{\omega }{{\varphi }^{\eta }}\right)}^n\frac{\mathrm{\Gamma }(\frac{1}{2}+\psi +\xi +\mathrm{n\eta })\mathrm{\Gamma }(\frac{1}{2}-\psi +\xi +\mathrm{n\eta })}{\mathrm{\Gamma }(1-\lambda +\xi +\mathrm{n\eta })}\end{array}$$

$$\begin{array}{l}=\frac{{\varphi }^{-\xi }\mathrm{\Gamma }\left(\nu \right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}\sum _{n=0}^{\infty }\\ \times \frac{\mathrm{\Gamma }(\mu +\rho n)\mathrm{\Gamma }(\gamma +\mathit{\text{qn}})\mathrm{\Gamma }(\frac{1}{2}\pm \psi +\xi +\mathrm{n\eta })\mathrm{\Gamma }(n+1)}{\mathrm{\Gamma }(1-\lambda +\xi +\mathrm{n\eta })}\\ \times \frac{{\left(\frac{\omega }{{\varphi }^{\eta }}\right)}^n}{n!},\end{array}$$

from which the result follows. □

Corollary 4.5.

$$\begin{array}{l}{\int }_0^{\infty }{\mathrm{e}}^{\frac{-\mathrm{\varphi t}}{2}}{t}^{\xi -1}{W}_{\lambda ,\psi }\left(\mathrm{\varphi t}\right)E_{\alpha ,\beta ,\delta }^{\gamma ,q}\left(\omega t^{\eta }\right)\mathit{\text{dt}}=\frac{\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\gamma \right)}{\varphi }^{-\xi }{}_4{\mathrm{\Psi }}_3\\ \times \left[\begin{array}{ll}(\gamma ,q),(\frac{1}{2}+\psi +\xi ,\eta ),(\frac{1}{2}-\psi +\xi ,\eta ),(1,1)& ;\\ \frac{\omega }{{\varphi }^{\eta }}\\ (\beta ,\alpha ),(1-\lambda +\xi ,\eta ),(\delta ,1)& ;\end{array}\right],\end{array}$$

(4.5.2)

Special properties :

• (i) Putting q = δ = 1 in (4.5.2), we have

  $${\int }_0^{\infty }\stackrel{\frac{-\mathrm{\varphi t}}{2}}{exp}{t}^{\xi -1}{W}_{\lambda ,\psi }\left(\mathrm{\varphi t}\right)E_{\alpha ,\beta }^{\gamma }\left(\omega t^{\eta }\right)\mathit{\text{dt}}$$
  $$=\frac{{\varphi }^{-\xi }}{\mathrm{\Gamma }\left(\gamma \right)}{}_3{\mathrm{\Psi }}_2\left[\begin{array}{ll}(\gamma ,q),(\frac{1}{2}\pm \varphi +\xi ,\eta )& ;\\ \frac{\omega }{{\varphi }^{\eta }}\\ (\beta ,\alpha ),(1-\lambda +\xi ,\eta )& ;\end{array}\right],$$

  (4.5.3)

• (ii) For q = γ = δ = 1 in (4.5.2), we have

  $${\int }_0^{\infty }\stackrel{\frac{-\mathrm{\varphi t}}{2}}{exp}{t}^{\xi -1}{W}_{\lambda ,\psi }\left(\mathrm{\varphi t}\right)E_{\alpha ,\beta }\left(\omega t^{\eta }\right)\mathit{\text{dt}}$$
  $$={\varphi }^{-\xi }{}_3{\mathrm{\Psi }}_2\left[\begin{array}{ll}(\frac{1}{2}\pm \varphi +\xi ,\eta ),(1,1),& ;\\ \frac{w}{{\varphi }^{\eta }}\\ (\beta ,\alpha ),(1-\lambda +\xi ,\sigma )& ;\end{array}\right]$$

  (4.5.4)

• (iii) Now putting q = β = α = γ = δ = 1 in (4.5.2), we have

  $${\int }_0^{\infty }{\mathrm{e}}^{\frac{-t}{2}}{t}^{\rho -1}{W}_{\lambda ,\psi }\left(\mathrm{\varphi t}\right)exp\left(\omega t^{\eta }\right)\mathit{\text{dt}}$$
  $$={\varphi }^{-\xi }{}_2{\mathrm{\Psi }}_1\left[\begin{array}{ll}(\frac{1}{2}\pm \varphi +\xi ,\eta )& ;\\ \frac{w}{{\varphi }^{\eta }}\\ (1-\lambda +\xi ,\sigma )& ;\end{array}\right]$$

  (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

$$E_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)=\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{z}^n}{\mathrm{\Gamma }(\alpha n+\beta ){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}$$

$$=\frac{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}\sum _{n=0}^{\infty }\frac{\mathrm{\Gamma }(\mu +\rho n)\mathrm{\Gamma }(\gamma +\mathit{\text{qn}})\mathrm{\Gamma }(n+1)}{\mathrm{\Gamma }(\alpha n+\beta )\mathrm{\Gamma }(\nu +\eta n)\mathrm{\Gamma }(\delta +\mathit{\text{pn}})}\frac{z^n}{n!}$$

$$=\frac{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}{}_3{\mathrm{\Psi }}_3\left[\begin{array}{ll}(\mu ,\rho )(\gamma ,q),(1,1),& ;\\ z\\ (\beta ,\alpha ),(\nu ,\sigma ),(\delta ,p)& ;\end{array}\right],$$

(5.1.1)

Relationship with Fox H-function

Using (4.1.1), we have from

$$\begin{array}{l}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)\\ =\frac{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{2\pi i\Gamma \left(\gamma \right)}{\int }_L\frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }(1-s)\mathrm{\Gamma }(\mu -\rho s)\mathrm{\Gamma }(\gamma -\mathit{\text{qs}})}{\mathrm{\Gamma }(\beta -\alpha s)\mathrm{\Gamma }(\nu -\sigma s)\mathrm{\Gamma }(\delta -\mathit{\text{ps}})}{(-z)}^{-s}\mathit{\text{ds}}\end{array}$$

$$\begin{array}{l}=\frac{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{2\pi i\Gamma \left(\gamma \right)\mathrm{\Gamma }\left(\mu \right)}{H}_{3,3}^{1,3}\\ \times \left[\begin{array}{ll}(0,1),(1-\mu ,\rho ),(1-\gamma ,q)& ;\\ z\\ (0,1)(1-\beta ,\alpha ),(1-\nu ,\sigma ),(1-\delta ,p)& ;\end{array}\right],\end{array}$$

(5.2.1)

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

$$E_{k,\mu +1,\delta }^{-m,q}\left(z^k\right)=\sum _{n=0}^{\left[\frac{m}{q}\right]}\frac{{(-m)}_{\mathit{\text{qn}}}{z}^{\mathit{\text{kn}}}}{\mathrm{\Gamma }(\mathit{\text{kn}}+\mu +1){\left(\delta \right)}_n}$$

$$=\sum _{n=0}^{\left[\frac{m}{q}\right]}\frac{{(-1)}^{\mathit{\text{qn}}}m!}{(m-\mathit{\text{qn}})!\mathrm{\Gamma }(\mathit{\text{kn}}+\mu +1)}\frac{z^{\mathit{\text{kn}}}}{{\left(\delta \right)}_n}$$

$$=\frac{\mathrm{\Gamma }(m+1)}{\mathrm{\Gamma }(\mathit{\text{km}}+\mu +1)}\sum _{n=0}^{\left[\frac{m}{q}\right]}\frac{{(-1)}^{\mathit{\text{qn}}}\mathrm{\Gamma }(\mathit{\text{km}}+\mu +1)}{(m-\mathit{\text{qn}})!\mathrm{\Gamma }(\mathit{\text{kn}}+\mu +1)}\frac{z^{\mathit{\text{kn}}}}{{\left(\delta \right)}_n}$$

$$=\frac{\mathrm{\Gamma }(m+1)}{\mathrm{\Gamma }(\mathit{\text{km}}+\mu +1)}{Z}_{\left[\frac{m}{q}\right],\delta }^{\mu }(z,k)$$

where $Z_{\left[\frac{m}{q}\right],\delta }^{\mu }(z,k)$ is a generalization of $Z_{\left[\frac{m}{q}\right]}^{\mu }(z,k)$(given by Shukla et al 2007).

Note that $Z_{\left[\frac{m}{q}\right]}^{\mu }(z,k)$ is a polynomial of degree $\left[\frac{m}{q}\right]$ in z ^k.

Further for $q=k=1,Z_{m,1}^{\mu }(z,1)=L_m^{\mu }\left(z\right)$, where $L_m^{\mu }\left(z\right)$ is a generalized Laguerre polynomial. So that

$$E_{k,\mu +1,1}^{-m,q}\left(z\right)=\frac{\mathrm{\Gamma }(m+1)}{\mathrm{\Gamma }(\mathit{\text{km}}+\mu +1)}{L}_m^{\mu }\left(z\right)$$

which is the required relationship.