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On some properties of the generalized Mittag-Leffler function

SpringerPlus20132:337

DOI: 10.1186/2193-1801-2-337

Received: 18 November 2012

Accepted: 24 May 2013

Published: 23 July 2013

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.

Keywords

Mittag-Leffler function Generalized hypergeometric function Fox’s H function

Introduction

In 1903, the Swedish mathematician Gosta Mittag-Leffler (1903) introduced the function
${E}_{\alpha }\left(z\right)=\sum _{n=0}^{\infty }\frac{{z}^{n}}{\mathrm{\Gamma }\left(\alpha n+1\right)},$
(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
$\phantom{\rule{-14.0pt}{0ex}}{E}_{\alpha ,\beta }\left(z\right)=\sum _{n=0}^{\infty }\frac{{z}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right)},\phantom{\rule{1em}{0ex}}\left(\alpha ,\beta \in \mathbb{C},\text{Re}\left(\alpha \right)>0,\text{Re}\left(\beta \right)>0\right),$
(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
$\begin{array}{ll}\phantom{\rule{1em}{0ex}}{E}_{\alpha ,\beta }^{\gamma }\left(z\right)=& \sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_{n}{z}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right)n!},\phantom{\rule{1em}{0ex}}\left(\alpha ,\beta ,\gamma ,\in \mathbb{C},\text{Re}\left(\alpha \right)>0,\\ \phantom{\rule{2.77626pt}{0ex}}\text{Re}\left(\beta \right)>0,\text{Re}\left(\gamma \right)>0\right),\end{array}$
(1.3)
where (γ) n is the Pochhammer symbol (Rainville (1960))
$\begin{array}{ll}\phantom{\rule{1em}{0ex}}{\left(\gamma \right)}_{n}& =\frac{\mathrm{\Gamma }\left(\gamma +n\right)}{\mathrm{\Gamma }\left(\gamma \right)},\phantom{\rule{1em}{0ex}}{\left(\gamma \right)}_{0}=1\phantom{\rule{1em}{0ex}},\\ \phantom{\rule{1em}{0ex}}{\left(\gamma \right)}_{n}& =\gamma \left(\gamma +1\right)\left(\gamma +2\right)\cdots \left(\gamma +n-1\right),n\ge 1.\end{array}$
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 $q\in \left(0,1\right)\bigcup \mathbb{N}$ as
${E}_{\alpha ,\beta }^{\gamma ,q}\left(z\right)=\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_{\mathit{\text{qn}}}{z}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right)n!},$
(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
${E}_{\alpha ,\beta }^{\gamma ,\delta }\left(z\right)=\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_{n}{z}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\delta \right)}_{n}},$
(1.5)
In 2012, a new generalization of Mittag-Leffler function was defined by Salim (2012) as
${E}_{\alpha ,\beta ,p}^{\gamma ,\delta ,q}\left(z\right)=\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_{\mathit{\text{qn}}}{z}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\delta \right)}_{\mathit{\text{pn}}}},$
(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
${E}_{\alpha ,\beta ,\delta }^{\gamma ,q}\left(z\right)=\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_{\mathit{\text{qn}}}{z}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\delta \right)}_{n}},\alpha ,\beta ,\gamma ,\delta \in \mathbb{C},$
(1.7)
where
$\phantom{\rule{-14.0pt}{0ex}}min\left(\text{Re}\left(\alpha \right),\text{Re}\left(\beta \right),\text{Re}\left(\gamma \right),\text{Re}\left(\delta \right)>0\right)\phantom{\rule{0.3em}{0ex}}\text{and q}\phantom{\rule{0.3em}{0ex}}\in \left(0,1\right)\bigcup \mathbb{N},$
(1.8)
Further the generalization of definition (1.7) is investigated and defined as follows
${E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)=\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{z}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}},$
(1.9)
where,
$\alpha ,\beta ,\gamma ,\delta ,\mu ,\nu ,\rho ,\sigma \in \mathbb{C};\phantom{\rule{1em}{0ex}}p,q>0\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}q\le \text{Re}\left(\alpha \right)+p,$
(1.10)
and
$\phantom{\rule{-15.0pt}{0ex}}\mathit{\text{min}}\left(\text{Re}\left(\alpha \right),\text{Re}\left(\beta \right),\text{Re}\left(\gamma \right),\text{Re}\left(\delta \right),\text{Re}\left(\mu \right),\text{Re}\left(\nu \right),\text{Re}\left(\rho \right),\phantom{\rule{0.3em}{0ex}}\text{Re}\left(\sigma \right)\right)\phantom{\rule{0.3em}{0ex}}>\phantom{\rule{0.3em}{0ex}}0,$
(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
$B\left\{\phantom{\rule{1em}{0ex}}f\left(z\right);a,b\right\}={\int }_{0}^{1}{z}^{a-1}{\left(1-z\right)}^{b-1}f\left(z\right)\mathit{\text{dz}}$
(1.12)
• Laplace transforms (Sneddon (1979)) of the function f(z) is defined as
$L\left\{\phantom{\rule{1em}{0ex}}f\left(z\right);s\right\}={\int }_{0}^{\infty }{e}^{-\mathit{\text{sz}}}f\left(z\right)\mathit{\text{dz}}$
(1.13)
• Mellin- transforms of the function f(z) is defined as
$M\left\{\phantom{\rule{1em}{0ex}}f\left(z\right);s\right\}={f}^{\ast }\left(s\right)={\int }_{0}^{\infty }{z}^{s-1}f\left(z\right)\mathit{\text{dz}},\phantom{\rule{1em}{0ex}}\mathrm{Re}\left(s\right)>0,$
(1.14)
• and the inverse Mellin-transform is given by
$f\left(z\right)={M}^{-1}\left\{{f}^{\ast }\left(s\right);z\right\}=\frac{1}{2\pi i}{\int }_{L}{f}^{\ast }\left(s\right){z}^{-s}\mathit{\text{ds}}$
(1.15)
• Whittaker transform (Whittaker and Watson (1962))
$\phantom{\rule{-2.0pt}{0ex}}{\int }_{0}^{\infty }{e}^{\frac{-t}{2}}{t}^{\nu -1}{W}_{\lambda ,\mu }\left(t\right)\mathit{\text{dt}}=\frac{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\nu \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu +\nu \right)}{\mathrm{\Gamma }\left(1-\lambda +\nu \right)},$
(1.16)

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

• The generalized hypergeometric function (Rainville (1960)) is defined as
$\phantom{\rule{-6.0pt}{0ex}}{}_{p}{F}_{q}\left[{\left(\alpha \right)}_{1},\dots ,{\left(\alpha \right)}_{p};{\left(\beta \right)}_{1},\dots ,{\left(\beta \right)}_{q};z\right)\right]=\sum _{n=0}^{\infty }\frac{\prod _{i=1}^{p}{\left({\alpha }_{i}\right)}_{n}}{\prod _{j=1}^{q}{\left(\beta \right)}_{j}{\right)}_{n}}\frac{{z}^{n}}{n!}$
(1.17)
• Wright generalized hypergeometric function (Srivastava and Manocha (1984)) is defined as
$\begin{array}{l}{}_{p}{\mathrm{\Psi }}_{q}\left[\left({a}_{1},{A}_{1}\right),\dots ,\left({a}_{p},{A}_{p}\right);\left({b}_{1},{B}_{1}\right),\mathit{\text{ldots}},\left({b}_{q},{B}_{q}\right);z\right]\\ \phantom{\rule{2em}{0ex}}=\sum _{n=0}^{\infty }\frac{\prod _{i=1}^{p}\mathrm{\Gamma }{\left({a}_{i}+{A}_{i}n\right)}_{n}}{\prod _{j=1}^{q}\mathrm{\Gamma }\left({b}_{j}+{B}_{j}n\right)}\frac{{z}^{n}}{n!}\end{array}$
(1.18)
• Fox’s H-function (Saigo and Kilbas (1998)) is given as
$\begin{array}{l}\phantom{\rule{2em}{0ex}}{H}_{p,q}^{m,n}\left[z|\left({a}_{1},{\alpha }_{1}\right),\dots ,\left({a}_{p},{\alpha }_{p}\right);\left({b}_{1},{\beta }_{1}\right),\dots ,\left({b}_{q},{\beta }_{q}\right)\right]\\ \phantom{\rule{-7.0pt}{0ex}}=\frac{1}{2\pi i}{\int }_{L}\frac{\prod _{j=1}^{m}\mathrm{\Gamma }\left({b}_{i}+{\beta }_{i}s\right)\prod _{j=1}^{n}\mathrm{\Gamma }\left(1-{a}_{i}-{\alpha }_{i}s\right)}{\prod _{j=m+1}^{q}\mathrm{\Gamma }\left(1-{b}_{j}-{\beta }_{j}s\right)\prod _{j=n+1}^{p}\mathrm{\Gamma }\left({a}_{j}+{\alpha }_{j}s\right)}{z}^{-s}\mathit{\text{ds}}\end{array}$
(1.19)
• Generalized Laguerre polynomials (Rainville (1960)). These are also known as Sonine polynomials and are defined as
${L}_{n}^{\left(\alpha \right)}\left(x\right)=\frac{{\left(1+\alpha \right)}_{n}}{n!}{}_{1}{F}_{1}\left[-n;1+\alpha ;x\right]$
(1.20)
• Incomplete Gamma function (Rainville (1960)). This is denoted by γ(α, z) and is defined by
$\gamma \left(\alpha ,z\right)={\int }_{0}^{z}{e}^{-t}{t}^{\alpha -1}\mathit{\text{dt}},\phantom{\rule{1em}{0ex}}\text{Re}\left(\alpha \right)>0,$
(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
$\phantom{\rule{-5.0pt}{0ex}}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)=\beta {E}_{\alpha ,\beta +1,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)+\mathrm{\alpha z}\frac{d}{\mathit{\text{dz}}}{E}_{\alpha ,\beta +1,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)$
(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)
${E}_{\alpha ,\beta -\alpha ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)-{E}_{\alpha ,\beta -\alpha ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma -1,q}\left(z\right)$
$=\frac{\mathit{\text{qz}}{\left(\mu \right)}_{\rho }}{{\left(\nu \right)}_{\sigma }}\sum _{n=0}^{\infty }\frac{\left(n+1\right){\left(\mu +\rho \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}+q-1}}{\mathrm{\Gamma }\left(\alpha n+\beta \right)\left(\delta +\mathit{\text{pn}}\right){\left(\nu +\sigma \right)}_{\eta n}}\frac{{z}^{n}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}$
(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)
$\phantom{\rule{-9.0pt}{0ex}}{E}_{\alpha ,\beta -\alpha ,\delta }^{\gamma ,q}\left(z\right)-{E}_{\alpha ,\beta -\alpha ,\delta }^{\gamma -1,q}\left(z\right)=\mathit{\text{qz}}\sum _{n=0}^{\infty }\frac{\left(n+1\right){\left(\gamma \right)}_{\mathit{\text{qn}}+q-1}}{\mathrm{\Gamma }\left(\alpha n+\beta \right)\left(\delta +n\right)}\frac{{z}^{n}}{{\left(\delta \right)}_{n}}$
(2.1.6)

Proof.

$\phantom{\rule{-7.0pt}{0ex}}\beta {E}_{\alpha ,\beta +1,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)+\mathrm{\alpha z}\frac{d}{\mathit{\text{dz}}}{E}_{\alpha ,\beta +1,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)$
$=\beta {E}_{\alpha ,\beta +1,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)+\mathrm{\alpha z}\frac{d}{\mathit{\text{dz}}}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }\left(\alpha n+\beta +1\right){\left(\nu \right)}_{\eta n}}\frac{{z}^{n}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}$
$=\beta {E}_{\alpha ,\beta +1,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)+\sum _{n=0}^{\infty }\frac{\alpha n{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }\left(\alpha n+\beta +1\right){\left(\nu \right)}_{\eta n}}\frac{{z}^{n}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}$
$=\beta {E}_{\alpha ,\beta +1,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)+\sum _{n=0}^{\infty }\frac{\left(\alpha n+\beta -\beta \right){\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }\left(\alpha n+\beta +1\right){\left(\nu \right)}_{\eta n}}\frac{{z}^{n}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}$
$={E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)$

which is (2.1.1).

The proof of (2.1.2) can easily be followed from the definition (1.9). Now
$\phantom{\rule{-7.0pt}{0ex}}{E}_{\alpha ,\beta -\alpha ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)-{E}_{\alpha ,\beta -\alpha ,\nu ,\sigma ,\mathrm{\delta p}}^{\mu ,\rho ,\gamma -1,q}\left(z\right)$
$\begin{array}{l}=\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }\left(\alpha \left(n-1\right)+\beta \right){\left(\nu \right)}_{\eta n}}\frac{{z}^{n}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\\ \phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}-\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma -1\right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }\left(\alpha \left(n-1\right)+\beta \right){\left(\nu \right)}_{\eta n}}\frac{{z}^{n}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\end{array}$
$\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{0.3em}{0ex}}=q\sum _{n=1}^{\infty }\frac{n{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}-1}}{\mathrm{\Gamma }\left(\alpha \left(n-1\right)+\beta \right)}\frac{{z}^{n}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}$
$\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{0.3em}{0ex}}=\frac{{\left(\mu \right)}_{\rho }\mathit{\text{qz}}}{{\left(\nu \right)}_{\sigma }}\sum _{n=0}^{\infty }\frac{\left(n+1\right){\left(\mu +\rho \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}+q-1}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\nu +\sigma \right)}_{\eta n}\left(\delta +\mathit{\text{pn}}\right)}\frac{{z}^{n}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}$

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 $m\in \mathbb{N}$
$\begin{array}{l}{\left(\frac{d}{\mathit{\text{dz}}}\right)}^{m}{E}_{\phantom{\rule{0.3em}{0ex}}\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)\\ \phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}=\frac{{\left(1\right)}_{m}{\left(\mu \right)}_{\mathrm{\rho m}}{\left(\gamma \right)}_{\mathit{\text{qm}}}}{{\left(\nu \right)}_{\mathrm{\sigma m}}{\left(\delta \right)}_{\mathit{\text{pm}}}}\sum _{n=0}^{\infty }\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{0.3em}{0ex}}×\frac{{\left(\mu +\mathrm{\rho m}\right)}_{\rho n}{\left(\gamma +\mathit{\text{qm}}\right)}_{\mathit{\text{qn}}}{\left(1+m\right)}_{n}}{\mathrm{\Gamma }\left(\alpha n+\beta +\alpha \right){\left(\nu +\mathrm{\sigma m}\right)}_{\eta n}{\left(\delta +\mathit{\text{pm}}\right)}_{\mathit{\text{pn}}}}\frac{{z}^{n}}{n!},\end{array}$
(2.2.1)
$\begin{array}{l}{\left(\frac{d}{\mathit{\text{dz}}}\right)}^{m}\left[{z}^{\beta -1}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(w{z}^{\alpha }\right)\right]={z}^{\beta -m-1}{E}_{\alpha ,\beta -m,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(w{z}^{\alpha }\right),\\ \phantom{\rule{2em}{0ex}}×\text{Re}\left(\beta -m\right)>0,\end{array}$
(2.2.2)
In particular,
$\phantom{\rule{-15.0pt}{0ex}}{\left(\frac{d}{\mathit{\text{dz}}}\right)}^{m}{E}_{\alpha ,\beta ,\delta }^{\gamma ,q}\left(z\right)=\frac{{\left(1\right)}_{m}{\left(\gamma \right)}_{\mathit{\text{qm}}}}{{\left(\delta \right)}_{m}}\sum _{n=0}^{\infty }\frac{{\left(\gamma +\mathit{\text{qm}}\right)}_{\mathit{\text{qn}}}{\left(1+m\right)}_{n}}{\mathrm{\Gamma }\left(\alpha n+\beta +\alpha {\left(\delta +m\right)}_{n}}\frac{{z}^{n}}{n!},$
(2.2.3)
$\begin{array}{l}{\left(\frac{d}{\mathit{\text{dz}}}\right)}^{m}\left[\phantom{\rule{0.3em}{0ex}}{z}^{\beta -1}{E}_{\alpha ,\beta ,\delta }^{\gamma ,q}\left(w{z}^{\alpha }\right)\right]={z}^{\beta -m-1}{E}_{\alpha ,\beta -m,p}^{\gamma ,q}\left(w{z}^{\alpha }\right),\\ \phantom{\rule{2em}{0ex}}×\text{Re}\left(\beta -m\right)>0,\end{array}$
(2.2.4)

Proof.

From (1.9),
$\begin{array}{l}\phantom{\rule{-13.0pt}{0ex}}{\left(\frac{d}{\mathit{\text{dz}}}\right)}^{m}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\nu \right)}_{\eta n}}\frac{{z}^{n}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}=\sum _{n=m}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}n!}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}\frac{{z}^{n-m}}{\left(n-m\right)!}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}=\frac{{\left(1\right)}_{m}{\left(\mu \right)}_{\mathrm{\rho m}}{\left(\gamma \right)}_{\mathit{\text{qm}}}}{{\left(\nu \right)}_{\mathrm{\sigma m}}{\left(\delta \right)}_{\mathit{\text{pm}}}}\sum _{n=0}^{\infty }\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}×\frac{{\left(\mu +\rho \right)}_{\mathrm{\rho m}}{\left(\gamma +\mathit{\text{qm}}\right)}_{\mathit{\text{qn}}}{\left(1+m\right)}_{n}}{\mathrm{\Gamma }\left(\alpha n+\mathrm{\alpha m}+\beta \right){\left(\nu +\sigma \right)}_{\mathrm{\sigma m}}{\left(\delta +\mathit{\text{pm}}\right)}_{\mathit{\text{pn}}}}\frac{{z}^{n}}{n!}\phantom{\rule{2em}{0ex}}\end{array}$

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
$\begin{array}{l}\phantom{\rule{-16.0pt}{0ex}}{\left(\frac{d}{\mathit{\text{dz}}}\right)}^{m}\left[\phantom{\rule{0.3em}{0ex}}{z}^{\beta -1}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(w{z}^{\alpha }\right)\right]\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{-2.0pt}{0ex}}\phantom{\rule{2.56804pt}{0ex}}\phantom{\rule{2.56804pt}{0ex}}\phantom{\rule{2.56804pt}{0ex}}={\left(\frac{d}{\mathit{\text{dz}}}\right)}^{m}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{w}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\nu \right)}_{\eta n}}\frac{{z}^{\alpha n+\beta -1}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{-2.0pt}{0ex}}\phantom{\rule{2.56804pt}{0ex}}\phantom{\rule{2.56804pt}{0ex}}\phantom{\rule{2.56804pt}{0ex}}=\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}\left(\alpha n+\beta -1\right)!{w}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right)\left(\alpha n+\beta -1-m\right)!{\left(\nu \right)}_{\eta n}}\frac{{z}^{\alpha n+\beta -m-1}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{-2.0pt}{0ex}}\phantom{\rule{2.56804pt}{0ex}}\phantom{\rule{2.56804pt}{0ex}}\phantom{\rule{2.56804pt}{0ex}}={z}^{\beta -m-1}{E}_{\alpha ,\beta -m,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(w{z}^{\alpha }\right)\phantom{\rule{2em}{0ex}}\end{array}$

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
$\begin{array}{l}\frac{{d}^{m}}{d{z}^{m}}{E}_{\frac{m}{r},\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left({z}^{\frac{p}{r}}\right)=\frac{\mathrm{\Gamma }\left(\nu \right)}{\mathrm{\Gamma }\left(\mu \right)}\sum _{n=1}^{\infty }\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}×\frac{\mathrm{\Gamma }\left(\frac{\mathit{\text{nm}}}{r}+1\right)\mathrm{\Gamma }\left(\mu +\rho n\right){\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }\left(\frac{\mathit{\text{nm}}}{r}+\beta \right)\mathrm{\Gamma }\left(\frac{\mathit{\text{nm}}}{r}-m+1\right)\mathrm{\Gamma }\left(\nu +\eta n\right)}\frac{{z}^{\left(\frac{n}{r}-1\right)m}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\end{array}$
(2.3.1)
$\begin{array}{l}\frac{{d}^{m}}{d{z}^{m}}{E}_{\frac{m}{r},\beta ,\delta }^{\gamma ,q}\left({z}^{\frac{m}{r}}\right)=\mathrm{\Gamma }\left(\delta \right)\sum _{n=1}^{\infty }\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}×\frac{{\left(\gamma \right)}_{\mathit{\text{qn}}}\mathrm{\Gamma }\left(n+1\right)\mathrm{\Gamma }\left(\frac{\mathit{\text{nm}}}{r}+1\right)}{\mathrm{\Gamma }\left(\delta +n\right)\mathrm{\Gamma }\left(\frac{\mathit{\text{nm}}}{r}+\beta \right)\mathrm{\Gamma }\left(\frac{\mathit{\text{np}}}{r}-m+1\right)}\frac{{z}^{\left(\frac{n}{r}-1\right)m}}{n!}\end{array}$
(2.3.2)

Proof.

$\begin{array}{l}\phantom{\rule{-15.0pt}{0ex}}\frac{{d}^{m}}{d{z}^{m}}{E}_{\frac{m}{r},\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left({z}^{\frac{p}{r}}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{-1.8pt}{0ex}}=\frac{{d}^{m}}{d{z}^{m}}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }\left(\frac{\mathit{\text{nm}}}{r}+\beta \right){\left(\nu \right)}_{\eta n}}\frac{{z}^{\left(\frac{m}{r}\right)n}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{-1.8pt}{0ex}}=\sum _{n=1}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}\left(\frac{\mathit{\text{nm}}}{r}\right)!}{\mathrm{\Gamma }\left(\frac{\mathit{\text{nm}}}{r}+\beta \right){\left(\nu \right)}_{\eta n}\left(\frac{\mathit{\text{nm}}}{r}-m\right)!}\frac{{z}^{\left(\frac{n}{r}-1\right)m}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{-1.8pt}{0ex}}=\frac{\mathrm{\Gamma }\left(\nu \right)}{\mathrm{\Gamma }\left(\mu \right)}\sum _{n=1}^{\infty }\frac{\mathrm{\Gamma }\left(\frac{\mathit{\text{nm}}}{r}+1\right)\mathrm{\Gamma }\left(\mu +\rho n\right){\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }\left(\frac{\mathit{\text{nm}}}{r}+\beta \right)\mathrm{\Gamma }\left(\frac{\mathit{\text{nm}}}{r}-m+1\right)\mathrm{\Gamma }\left(\nu +\eta n\right)}\frac{{z}^{\left(\frac{n}{r}-1\right)m}}{{\left(\delta \right)}_{\mathit{\text{pn}}}}\phantom{\rule{2em}{0ex}}\end{array}$

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
$\phantom{\rule{-13.0pt}{0ex}}\frac{{d}^{m}}{d{z}^{m}}{E}_{\frac{m}{r},\beta }^{\gamma }\left({z}^{\frac{m}{r}}\right)=\sum _{n=1}^{\infty }\frac{{\left(\gamma \right)}_{n}\mathrm{\Gamma }\left(\frac{\mathit{\text{np}}}{r}+1\right)}{\mathrm{\Gamma }\left(\frac{\mathit{\text{nm}}}{r}+\beta \right)\mathrm{\Gamma }\left(\frac{\mathit{\text{np}}}{r}-m+1\right)}\frac{{z}^{\left(\frac{n}{r}-1\right)m}}{n!}$
(2.3.3)
For β = γ = δ = q = 1 in (2.3.2), we have
$\frac{{d}^{m}}{d{z}^{m}}{E}_{\frac{m}{r}}\left({z}^{\frac{m}{r}}\right)=\sum _{n=1}^{\infty }\frac{{z}^{\left(\frac{n}{r}-1\right)m}}{\mathrm{\Gamma }\left(\frac{\mathit{\text{nm}}}{r}-m+1\right)}$
(2.3.4)

Theorem 2.4.

If $\alpha ,\beta ,\mu ,\rho ,\nu ,\gamma ,\lambda ,\sigma ,\delta ,\eta \in \mathbb{C};\text{Re}\left(\alpha \right)>0,\text{Re}\left(\beta \right)>0,\text{Re}\left(\gamma \right)>0,\text{Re}\left(\mu \right)>0,\text{Re}\left(\rho \right)>0,\text{Re}\left(\nu \right)>0,\text{Re}\left(\delta \right)>0,\text{Re}\left(\sigma \right)>0,\text{Re}\left(\lambda \right)>0\phantom{\rule{2.77626pt}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}q<\text{Re}\left(\alpha \right)+\mathrm{p}$ then
$\begin{array}{l}\frac{1}{\mathrm{\Gamma }\left(\eta \right)}{\int }_{0}^{1}{u}^{\beta -1}{\left(1-u\right)}^{\eta -1}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z{u}^{\alpha }\right)\mathit{\text{du}}\\ \phantom{\rule{2em}{0ex}}={E}_{\alpha ,\beta +\eta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)\mathit{\text{du}},\end{array}$
(2.4.1)
$\begin{array}{l}\frac{1}{\mathrm{\Gamma }\left(\eta \right)}{\int }_{t}^{x}{\left(x-s\right)}^{\eta -1}{\left(s-t\right)}^{\beta -1}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left[\phantom{\rule{0.3em}{0ex}}\lambda {\left(s-t\right)}^{\alpha }\right]\mathit{\text{du}}\\ \phantom{\rule{2em}{0ex}}={\left(x-t\right)}^{\eta +\beta -1}{E}_{\alpha ,\beta +\eta ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right),\end{array}$
(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,
$\phantom{\rule{-5.0pt}{0ex}}\frac{1}{\mathrm{\Gamma }\left(\eta \right)}{\int }_{0}^{1}{u}^{\beta -1}{\left(1-u\right)}^{\sigma -1}{E}_{\alpha ,\beta ,\delta }^{\gamma ,q}\left(z{u}^{\alpha }\right)\mathit{\text{du}}={E}_{\alpha ,\beta +\eta ,\delta }^{\gamma ,q}\left(z\right)\mathit{\text{du}},$
(2.4.4)
$\begin{array}{l}\phantom{\rule{-13.0pt}{0ex}}\frac{1}{\mathrm{\Gamma }\left(\eta \right)}{\int }_{t}^{x}{\left(x-s\right)}^{\eta -1}{\left(s-t\right)}^{\beta -1}{E}_{\alpha ,\beta ,\delta }^{\gamma ,q}\left[\phantom{\rule{0.3em}{0ex}}\lambda {\left(s-t\right)}^{\alpha }\right]\mathit{\text{du}}\\ \phantom{\rule{1em}{0ex}}={\left(x-t\right)}^{\eta +\beta -1}{E}_{\alpha ,\beta +\eta ,\delta }^{\gamma ,q}\left(z\right),\end{array}$
(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)
$\frac{1}{\mathrm{\Gamma }\left(\eta \right)}{\int }_{0}^{1}{u}^{\beta -1}{\left(1-u\right)}^{\sigma -1}{E}_{\alpha ,\beta }^{\gamma }\left(z{u}^{\alpha }\right)\mathit{\text{du}}={E}_{\alpha ,\beta +\eta }^{\gamma }\left(z\right)\mathit{\text{du}},$
(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.

$\begin{array}{l}\phantom{\rule{-13.0pt}{0ex}}\frac{1}{\mathrm{\Gamma }\left(\eta \right)}{\int }_{0}^{1}{u}^{\beta -1}{\left(1-u\right)}^{\eta -1}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z{u}^{\alpha }\right)\mathit{\text{du}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{-8.0pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}=\frac{1}{\mathrm{\Gamma }\left(\eta \right)}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{z}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}{\int }_{0}^{1}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}×{u}^{\alpha n+\beta -1}{\left(1-u\right)}^{\eta -1}\mathit{\text{du}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{-8.0pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}=\frac{1}{\mathrm{\Gamma }\left(\eta \right)}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{z}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}B\left(\alpha n+\beta ,\eta \right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{-8.0pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}={E}_{\alpha ,\beta +\eta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)\phantom{\rule{2em}{0ex}}\end{array}$

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
$\phantom{\rule{-12.0pt}{0ex}}\frac{1}{\mathrm{\Gamma }\left(\eta \right)}{\int }_{0}^{1}{\left(x-t\right)}^{\eta +\beta -1}{\left(1-u\right)}^{\eta -1}{u}^{\beta -1}$
$\phantom{\rule{2em}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}×\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{z}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}{\left(\lambda \right)}^{n}{\left(x-t\right)}^{\alpha n}{u}^{\alpha n}\mathit{\text{du}}$
$\begin{array}{l}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}=\frac{{\left(x-t\right)}^{\eta +\beta -1}}{\mathrm{\Gamma }\left(\eta \right)}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{\left[\phantom{\rule{0.3em}{0ex}}\lambda {\left(x-t\right)}^{\alpha }\right]}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}{\int }_{0}^{1}\\ \phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{0.3em}{0ex}}×{u}^{\alpha n+\beta -1}{\left(1-u\right)}^{\eta -1}\mathit{\text{du}}\end{array}$
$\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}=\frac{{\left(x-t\right)}^{\eta +\beta -1}}{\mathrm{\Gamma }\left(\eta \right)}\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_{\mathit{\text{qn}}}{\left[\phantom{\rule{0.3em}{0ex}}\lambda {\left(x-t\right)}^{\alpha }\right]}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\delta \right)}_{\mathit{\text{pn}}}}B\left(\alpha n+\beta ,\eta \right)$
$\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}={\left(x-t\right)}^{\eta +\beta -1}{E}_{\alpha ,\beta +\eta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}{\left[\phantom{\rule{0.3em}{0ex}}\lambda {\left(x-t\right)}^{\alpha }\right]}^{n}$

which proves (2.4.2).

Now
$\begin{array}{l}{\int }_{0}^{z}{t}^{\beta -1}{E}_{\alpha ,\beta ,\nu ,\sigma ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(w{t}^{\alpha }\right)\mathit{\text{dt}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}={\int }_{0}^{z}{t}^{\beta -1}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{w}^{n}{t}^{\alpha n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}\mathit{\text{dt}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}=\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{w}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}{\int }_{0}^{z}{t}^{\alpha n+\beta -1}\mathit{\text{dt}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}={z}^{\beta }\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{\left(w{z}^{\alpha }\right)}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta +1\right){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}={z}^{\beta }{E}_{\alpha ,\beta +1,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(w{z}^{\alpha }\right)\phantom{\rule{2em}{0ex}}\end{array}$

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
$\begin{array}{ll}\phantom{\rule{-14.0pt}{0ex}}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)& =\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\mathit{\text{ln}}}{\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }\left(\mathit{\text{kn}}+\beta \right){\left(\nu \right)}_{\mathit{\text{mn}}}{\left(\delta \right)}_{\mathit{\text{pn}}}}{z}^{n}\phantom{\rule{2em}{0ex}}\\ =\frac{1}{\mathrm{\Gamma }\left(\beta \right)}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\mathit{\text{ln}}}{\left(\gamma \right)}_{\mathit{\text{qn}}}{\left(1\right)}_{n}}{{\left(\beta \right)}_{\mathit{\text{kn}}}{\left(\nu \right)}_{\mathit{\text{mn}}}{\left(\delta \right)}_{\mathit{\text{pn}}}}\frac{{z}^{n}}{n!}\phantom{\rule{2em}{0ex}}\end{array}$
$\begin{array}{l}\phantom{\rule{-4.5pt}{0ex}}=\frac{1}{\mathrm{\Gamma }\left(\beta \right)}\sum _{n=0}^{\infty }\frac{\prod _{i=1}^{l}{\left(\frac{\mu +i-1}{l}\right)}_{n}\prod _{j=1}^{q}{\left(\frac{\gamma +j-1}{q}\right)}_{n}{\left(1\right)}_{n}}{\prod _{r=1}^{k}{\left(\frac{\beta +r-1}{k}\right)}_{n}\prod _{s=1}^{p}{\left(\frac{\delta +s-1}{p}\right)}_{n}\prod _{t=1}^{m}{\left(\frac{\nu +t-1}{m}\right)}_{n}n!}\\ \phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{0.3em}{0ex}}×{\left(\frac{{l}^{l}{q}^{q}z}{{m}^{m}{p}^{p}{k}^{k}}\right)}^{n}\end{array}$

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 :
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

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 $\alpha \in {R}_{+}=\left(0\phantom{\rule{1em}{0ex}}\infty \right);\beta ,\nu ,\sigma ,\mu ,\rho ,\gamma ,\delta \in \mathbb{C}$ 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:
$\begin{array}{l}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)=\frac{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{2\pi i\Gamma \left(\gamma \right)\mathrm{\Gamma }\left(\mu \right)}{\int }_{L}\\ \phantom{\rule{2em}{0ex}}×\frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(1-s\right)\mathrm{\Gamma }\left(\mu -\rho s\right)\mathrm{\Gamma }\left(\gamma -\mathit{\text{qs}}\right)}{\mathrm{\Gamma }\left(\beta -\alpha s\right)\mathrm{\Gamma }\left(\delta -\mathit{\text{ps}}\right)\mathrm{\Gamma }\left(\nu -\rho s\right)}{\left(-z\right)}^{-s}\mathit{\text{ds}},\end{array}$
(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 $s=-n,\forall \phantom{\rule{1em}{0ex}}n\in {\mathbb{N}}_{0}$ (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
$\frac{1}{2\pi i}{\int }_{L}\frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(1-s\right)\mathrm{\Gamma }\left(\mu -\rho s\right)\mathrm{\Gamma }\left(\gamma -\mathit{\text{qs}}\right)}{\mathrm{\Gamma }\left(\beta -\alpha s\right)\mathrm{\Gamma }\left(\nu -\sigma s\right)\mathrm{\Gamma }\left(\delta -\mathit{\text{ps}}\right)}{\left(-z\right)}^{-s}\mathit{\text{ds}}$
$=\sum _{n=0}^{\infty }{\mathit{\text{Res}}}_{s=-n}\left[\phantom{\rule{0.3em}{0ex}}\frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(1-s\right)\mathrm{\Gamma }\left(\mu -\rho s\right)\mathrm{\Gamma }\left(\gamma -\mathit{\text{qs}}\right)}{\mathrm{\Gamma }\left(\beta -\alpha s\right)\mathrm{\Gamma }\left(\delta -\mathit{\text{ps}}\right)\mathrm{\Gamma }\left(\nu -\sigma s\right)}{\left(-z\right)}^{-s}\right]$
$\begin{array}{l}\phantom{\rule{-3.5pt}{0ex}}=\sum _{n=0}^{\infty }{\mathit{\text{Lim}}}_{s\to -n}\frac{\pi \left(s+n\right)}{sin\mathrm{\pi s}}.\frac{\mathrm{\Gamma }\left(\mu -\rho s\right)\mathrm{\Gamma }\left(\gamma -\mathit{\text{qs}}\right)}{\mathrm{\Gamma }\left(\beta -\alpha s\right)\mathrm{\Gamma }\left(\delta -\mathit{\text{ps}}\right)\mathrm{\Gamma }\left(\nu -\sigma s\right)}\\ \phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}×{\left(-z\right)}^{-s}\end{array}$
$=\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}\mathrm{\Gamma }\left(\mu +\rho n\right)\mathrm{\Gamma }\left(\gamma +\mathit{\text{qn}}\right)}{\mathrm{\Gamma }\left(\beta +\alpha n\right)\mathrm{\Gamma }\left(\delta +\mathit{\text{pn}}\right)\mathrm{\Gamma }\left(\nu +\eta n\right)}{\left(-z\right)}^{n}$
$=\frac{\mathrm{\Gamma }\left(\gamma \right)\mathrm{\Gamma }\left(\mu \right)}{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\delta \right)}_{\mathit{\text{pn}}}{\left(\nu \right)}_{\eta n}}{z}^{n}$
$=\frac{\mathrm{\Gamma }\left(\gamma \right)\mathrm{\Gamma }\left(\mu \right)}{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)$

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 ${E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)$
$\begin{array}{l}M\left\{{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(-\mathit{\text{wz}}\right);s\right\}=\frac{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{2\pi i\Gamma \left(\gamma \right)\mathrm{\Gamma }\left(\mu \right)}{\int }_{L}\\ \phantom{\rule{2em}{0ex}}×\frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(1-s\right)\mathrm{\Gamma }\left(\mu -\rho s\right)\mathrm{\Gamma }\left(1-\mathit{\text{qs}}\right)}{\mathrm{\Gamma }\left(\gamma \right)\mathrm{\Gamma }\left(\beta -\alpha s\right)\mathrm{\Gamma }\left(\nu -\sigma s\right)\mathrm{\Gamma }\left(\delta -\mathit{\text{ps}}\right)}{\left(\mathit{\text{wz}}\right)}^{-s}\mathit{\text{ds}},\end{array}$
(4.2.1)

Proof.

From Theorem 4.1, we have
$\begin{array}{l}\phantom{\rule{-13.0pt}{0ex}}{E}_{\alpha ,\beta ,\nu ,\mathrm{\sigma \sigma },\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(-\mathit{\text{wz}}\right)\\ =\frac{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{2\pi i\Gamma \left(\gamma \right)\mathrm{\Gamma }\left(\mu \right)}{\int }_{L}\frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(1-s\right)\mathrm{\Gamma }\left(\mu -\rho s\right)\mathrm{\Gamma }\left(\gamma -\mathit{\text{qs}}\right)}{\mathrm{\Gamma }\left(\beta -\alpha s\right)\mathrm{\Gamma }\left(\nu -\sigma s\right)\mathrm{\Gamma }\left(\delta -\mathit{\text{ps}}\right)}\\ \phantom{\rule{1em}{0ex}}×{\left(\mathit{\text{wz}}\right)}^{-s}\mathit{\text{ds}}\end{array}$
$\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}=\frac{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{2\pi i\Gamma \left(\gamma \right)\mathrm{\Gamma }\left(\mu \right)}{\int }_{L}{f}^{\ast }\left(s\right){\left(z\right)}^{-s}\mathit{\text{ds}}$
where
${f}^{\ast }\left(s\right)=\frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(1-s\right)\mathrm{\Gamma }\left(\mu -\rho s\right)\mathrm{\Gamma }\left(\gamma -\mathit{\text{qs}}\right)}{\mathrm{\Gamma }\left(\beta -\alpha s\right)\mathrm{\Gamma }\left(\nu -\sigma s\right)\mathrm{\Gamma }\left(\delta -\mathit{\text{ps}}\right)}{\left(w\right)}^{-s}$

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${E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)$
$\phantom{\rule{-13.0pt}{0ex}}\begin{array}{l}{\int }_{0}^{1}{z}^{a-1}{\left(1-z\right)}^{b-1}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(x{z}^{\eta }\right)\mathit{\text{dz}}\\ \phantom{\rule{2.56804pt}{0ex}}\phantom{\rule{2.56804pt}{0ex}}\phantom{\rule{2.56804pt}{0ex}}\phantom{\rule{2.56804pt}{0ex}}=\frac{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(\nu \right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}{}_{4}{\mathrm{\Psi }}_{4}\left[\begin{array}{ll}\left(\mu ,\rho \right)\left(\gamma ,q\right),\left(a,\eta \right),\left(1,1\right)& ;\\ \phantom{\rule{.5em}{0ex}}x\\ \left(\beta ,\alpha \right),\left(a+b,\eta \right),\left(\nu ,\sigma \right),\left(\delta ,p\right)& ;\end{array}\phantom{\rule{.5em}{0ex}}\right]\end{array}$
(4.3.1)

Proof.

${\int }_{0}^{1}{z}^{a-1}{\left(1-z\right)}^{b-1}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(x{z}^{\eta }\right)\mathit{\text{dz}}$
$={\int }_{0}^{1}{z}^{a-1}{\left(1-z\right)}^{b-1}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{\left(x{z}^{\eta }\right)}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}$
$=\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{x}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}{\int }_{0}^{1}{z}^{\eta n+a-1}{\left(1-z\right)}^{b-1}\mathit{\text{dz}}$
$=\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{x}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}B\left(\eta n+a,b\right)$
$\begin{array}{l}\phantom{\rule{-4.0pt}{0ex}}=\frac{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(\nu \right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}\sum _{n=0}^{\infty }\\ \phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{0.3em}{0ex}}×\frac{\mathrm{\Gamma }\left(\mu +\rho n\right)\mathrm{\Gamma }\left(\mathit{\text{qn}}+\gamma \right)\mathrm{\Gamma }\left(a+\eta n\right)\mathrm{\Gamma }\left(n+1\right)}{\mathrm{\Gamma }\left(\alpha n+\beta \right)\mathrm{\Gamma }\left(\eta n+a+b\right)\mathrm{\Gamma }\left(\nu +\eta n\right)\mathrm{\Gamma }\left(\mathit{\text{pn}}+\delta \right)}\frac{{x}^{n}}{n!},\end{array}$

from which the result follows. □

Corollary 4.3.

$\phantom{\rule{-13.0pt}{0ex}}\begin{array}{l}{\int }_{0}^{1}{z}^{a-1}{\left(1-z\right)}^{b-1}{E}_{\alpha ,\beta ,\delta }^{\gamma ,q}\left(x{z}^{\eta }\right)\mathit{\text{dz}}\\ \phantom{\rule{1em}{0ex}}=\frac{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\gamma \right)}{}_{3}{\mathrm{\Psi }}_{3}\left[\begin{array}{ll}\left(\gamma ,q\right),\left(a,\eta \right),\left(1,1\right)& ;\\ \phantom{\rule{.5em}{0ex}}x\\ \left(\beta ,\alpha \right),\left(a+b,\eta \right),\left(\delta ,1\right)& ;\end{array}\phantom{\rule{.5em}{0ex}}\right]\end{array}$
(4.3.2)
Special properties:
1. (i)
For q = 1, (4.3.2) reduces to Tariq OSalim (2009)(4.1).
$\phantom{\rule{-13.0pt}{0ex}}\begin{array}{l}{\int }_{0}^{1}{z}^{a-1}{\left(1-z\right)}^{b-1}{E}_{\alpha ,\beta }^{\gamma ,\delta }\left(x{z}^{\sigma }\right)\mathit{\text{dz}}\\ \phantom{\rule{1em}{0ex}}=\frac{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\gamma \right)}{}_{3}{\mathrm{\Psi }}_{3}\left[\begin{array}{ll}\left(\gamma ,1\right),\left(a,\sigma \right),\left(1,1\right)& ;\\ \phantom{\rule{.5em}{0ex}}x\\ \left(\beta ,\alpha \right),\left(\delta ,1\right)\left(a+b,\sigma \right)& ;\end{array}\phantom{\rule{.5em}{0ex}}\right]\phantom{\rule{.5em}{0ex}},\end{array}$
(4.3.3)

2. (ii)
For δ = q = 1 in (4.3.2), we have
$\phantom{\rule{-13.0pt}{0ex}}\begin{array}{l}{\int }_{0}^{1}{z}^{a-1}{\left(1-z\right)}^{b-1}{E}_{\alpha ,\beta }^{\gamma }\left(x{z}^{\sigma }\right)\mathit{\text{dz}}\\ \phantom{\rule{1em}{0ex}}=\frac{\mathrm{\Gamma }\left(b\right)}{\mathrm{\Gamma }\left(\gamma \right)}{}_{2}{\mathrm{\Psi }}_{2}\left[\begin{array}{ll}\left(\gamma ,1\right),\left(a,\sigma \right),& ;\\ \phantom{\rule{.5em}{0ex}}x\\ \left(\beta ,\alpha \right),\left(a+b,\sigma \right)& ;\end{array}\phantom{\rule{.5em}{0ex}}\right]\phantom{\rule{.5em}{0ex}},\end{array}$
(4.3.4)
If a = β, α = σ, then (4.3.2) reduces to
${\int }_{0}^{1}{z}^{\beta -1}{\left(1-z\right)}^{b-1}{E}_{\sigma ,\beta ,\delta }^{\gamma ,q}\left(x{z}^{\sigma }\right)\mathit{\text{dz}}=\frac{\mathrm{\Gamma }\left(b\right)}{\mathrm{\Gamma }\left(\gamma \right)}{E}_{\sigma ,\beta +b,\delta }^{\gamma ,q}\left(x\right),$
(4.3.5)
Putting α = β = γ = δ = q = 1 in (4.3.2), we have
$\phantom{\rule{-13.0pt}{0ex}}\begin{array}{l}{\int }_{0}^{1}{z}^{a-1}{\left(1-z\right)}^{b-1}exp\left(x{z}^{\sigma }\right)\mathit{\text{dz}}\\ \phantom{\rule{1em}{0ex}}=\mathrm{\Gamma }\left(b\right){}_{1}{\mathrm{\Psi }}_{1}\left[\begin{array}{ll}\left(a,\sigma \right),& ;\\ \phantom{\rule{0.5em}{0ex}}x\\ \left(a+b,\sigma \right)& ;\end{array}\phantom{\rule{0.5em}{0ex}}\right]\end{array}$
(4.3.6)

Theorem 4.4. (Laplace transform)

$\begin{array}{l}{\int }_{0}^{\infty }{z}^{a-1}{e}^{-\mathit{\text{sz}}}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(x{z}^{\eta }\right)\mathit{\text{dz}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}=\frac{\mathrm{\Gamma }\left(\nu \right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}{s}^{-a}{}_{3}{\mathrm{\Psi }}_{3}\left[\begin{array}{ll}\left(\mu ,\rho \right)\left(\gamma ,q\right),\left(1,1\right)& ;\\ \phantom{\rule{0.5em}{0ex}}\frac{x}{{s}^{\eta }}\\ \left(\beta ,\alpha \right),\left(\nu ,\sigma \right)\left(\delta ,p\right)& ;\end{array}\phantom{\rule{0.5em}{0ex}}\right]\phantom{\rule{0.5em}{0ex}},\end{array}$
(4.4.1)

Proof.

$\phantom{\rule{-14.0pt}{0ex}}\begin{array}{l}{\int }_{0}^{\infty }{z}^{a-1}{e}^{-\mathit{\text{sz}}}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(x{z}^{\eta }\right)\mathit{\text{dz}}\\ \phantom{\rule{1em}{0ex}}={\int }_{0}^{\infty }{z}^{a-1}{e}^{-\mathit{\text{sz}}}\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{\left(x{z}^{\sigma }\right)}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}\mathit{\text{dz}}\end{array}$
$=\frac{\mathrm{\Gamma }\left(\nu \right)\mathrm{\Gamma }\left(\delta \right){s}^{-a}}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}\sum _{n=0}^{\infty }\frac{\mathrm{\Gamma }\left(\mu +\rho n\right)\mathrm{\Gamma }\left(\gamma +\mathit{\text{qn}}\right)\mathrm{\Gamma }\left(1+n\right)}{\mathrm{\Gamma }\left(\beta +\alpha n\right)\mathrm{\Gamma }\left(\nu +\eta n\right)\mathrm{\Gamma }\left(\delta +\mathit{\text{pn}}\right)}\frac{{\left(\frac{x}{{s}^{\sigma }}\right)}^{n}}{n!},$

from which the result follows. □

Corollary 4.4.

$\phantom{\rule{-13.0pt}{0ex}}\begin{array}{l}{\int }_{0}^{\infty }{z}^{a-1}{e}^{-\mathit{\text{sz}}}{E}_{\alpha ,\beta ,\delta }^{\gamma ,q}\left(x{z}^{\eta }\right)\mathit{\text{dz}}\\ \phantom{\rule{1em}{0ex}}=\frac{\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\gamma \right)}{s}^{-a}{}_{2}{\mathrm{\Psi }}_{2}\left[\begin{array}{ll}\left(\gamma ,q\right),\left(1,1\right)& ;\\ \phantom{\rule{0.5em}{0ex}}\frac{x}{{s}^{\eta }}\\ \left(\beta ,\alpha \right),\left(\delta ,1\right)& ;\end{array}\phantom{\rule{0.5em}{0ex}}\right]\phantom{\rule{0.5em}{0ex}},\end{array}$
(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)

$\phantom{\rule{-16.0pt}{0ex}}\begin{array}{l}{\int }_{0}^{\infty }{e}^{\frac{-\mathrm{\varphi t}}{2}}{t}^{\xi -1}{W}_{\lambda ,\psi }\left(\mathrm{\varphi t}\right){E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(\omega {t}^{\eta }\right)\mathit{\text{dt}}=\frac{\mathrm{\Gamma }\left(\nu \right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}{\varphi }^{-\xi }{}_{5}{\mathrm{\Psi }}_{4}\\ \phantom{\rule{2.56804pt}{0ex}}\phantom{\rule{2.56804pt}{0ex}}\phantom{\rule{0.3em}{0ex}}×\left[\begin{array}{ll}\left(\mu ,\rho \right)\left(\gamma ,q\right),\left(\frac{1}{2}+\psi +\xi ,\eta \right),\left(\frac{1}{2}-\psi +\xi ,\eta \right),\left(1,1\right)& ;\\ \phantom{\rule{0.5em}{0ex}}\frac{\omega }{{\varphi }^{\eta }}\\ \left(\beta ,\alpha \right),\left(\nu ,\sigma \right),\left(1-\lambda +\xi ,\eta \right),\left(\delta ,p\right)& ;\end{array}\phantom{\rule{0.5em}{0ex}}\right]\end{array}$
(4.5.1)

Proof.

Substituting ϕ t = v in L.H.S. of Theorem 4.5, we have
$\phantom{\rule{-14.0pt}{0ex}}{\int }_{0}^{\infty }{\left(\frac{v}{\varphi }\right)}^{\xi -1}{e}^{\frac{-v}{2}}{W}_{\lambda ,\psi }\left(v\right)\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{\left(\omega \right)}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\delta \right)}_{n}}{\left(\frac{v}{\varphi }\right)}^{\mathrm{n\eta }}\frac{1}{\varphi }\mathit{\text{dv}}$
$\begin{array}{l}=\frac{{\varphi }^{-\xi }\mathrm{\Gamma }\left(\nu \right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}\sum _{n=0}^{\infty }\frac{\mathrm{\Gamma }\left(\mu +\rho n\right)\mathrm{\Gamma }\left(\gamma +\mathit{\text{qn}}\right)}{\mathrm{\Gamma }\left(\alpha n+\beta \right)\mathrm{\Gamma }\left(\nu +\eta n\right)\mathrm{\Gamma }\left(\delta +\mathit{\text{pn}}}\\ \phantom{\rule{1em}{0ex}}×{\left(\frac{\omega }{{\varphi }^{\eta }}\right)}^{n}{\int }_{0}^{\infty }{\left(v\right)}^{\eta n+\xi -1}{e}^{\frac{-v}{2}}{W}_{\lambda ,\psi }\left(v\right)\mathit{\text{dv}}\end{array}$
$\begin{array}{l}=\frac{{\varphi }^{-\xi }\mathrm{\Gamma }\left(\nu \right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}\sum _{n=0}^{\infty }\frac{\mathrm{\Gamma }\left(\mu +\rho n\right)\mathrm{\Gamma }\left(\gamma +\mathit{\text{qn}}\right)}{\mathrm{\Gamma }\left(\alpha n+\beta \right)\mathrm{\Gamma }\left(\nu +\eta n\right)\mathrm{\Gamma }\left(\delta +\mathit{\text{pn}}\right)}\\ \phantom{\rule{1em}{0ex}}×{\left(\frac{\omega }{{\varphi }^{\eta }}\right)}^{n}\frac{\mathrm{\Gamma }\left(\frac{1}{2}+\psi +\xi +\mathrm{n\eta }\right)\mathrm{\Gamma }\left(\frac{1}{2}-\psi +\xi +\mathrm{n\eta }\right)}{\mathrm{\Gamma }\left(1-\lambda +\xi +\mathrm{n\eta }\right)}\end{array}$
$\begin{array}{l}=\frac{{\varphi }^{-\xi }\mathrm{\Gamma }\left(\nu \right)\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}\sum _{n=0}^{\infty }\\ \phantom{\rule{1em}{0ex}}×\frac{\mathrm{\Gamma }\left(\mu +\rho n\right)\mathrm{\Gamma }\left(\gamma +\mathit{\text{qn}}\right)\mathrm{\Gamma }\left(\frac{1}{2}±\psi +\xi +\mathrm{n\eta }\right)\mathrm{\Gamma }\left(n+1\right)}{\mathrm{\Gamma }\left(1-\lambda +\xi +\mathrm{n\eta }\right)}\\ \phantom{\rule{1em}{0ex}}×\frac{{\left(\frac{\omega }{{\varphi }^{\eta }}\right)}^{n}}{n!},\end{array}$

from which the result follows. □

Corollary 4.5.

$\phantom{\rule{-17.0pt}{0ex}}\begin{array}{l}{\int }_{0}^{\infty }{\mathrm{e}}^{\frac{-\mathrm{\varphi t}}{2}}{t}^{\xi -1}{W}_{\lambda ,\psi }\left(\mathrm{\varphi t}\right){E}_{\alpha ,\beta ,\delta }^{\gamma ,q}\left(\omega {t}^{\eta }\right)\mathit{\text{dt}}=\frac{\mathrm{\Gamma }\left(\delta \right)}{\mathrm{\Gamma }\left(\gamma \right)}{\varphi }^{-\xi }{}_{4}{\mathrm{\Psi }}_{3}\\ \phantom{\rule{1em}{0ex}}×\left[\begin{array}{ll}\left(\gamma ,q\right),\left(\frac{1}{2}+\psi +\xi ,\eta \right),\left(\frac{1}{2}-\psi +\xi ,\eta \right),\left(1,1\right)& ;\\ \phantom{\rule{0.5em}{0ex}}\frac{\omega }{{\varphi }^{\eta }}\\ \left(\beta ,\alpha \right),\left(1-\lambda +\xi ,\eta \right),\left(\delta ,1\right)& ;\end{array}\phantom{\rule{0.5em}{0ex}}\right]\phantom{\rule{0.5em}{0ex}},\end{array}$
(4.5.2)
Special properties :
1. (i)
Putting q = δ = 1 in (4.5.2), we have
$\phantom{\rule{-13.0pt}{0ex}}{\int }_{0}^{\infty }\stackrel{\frac{-\mathrm{\varphi t}}{2}}{exp}{t}^{\xi -1}{W}_{\lambda ,\psi }\left(\mathrm{\varphi t}\right){E}_{\alpha ,\beta }^{\gamma }\left(\omega {t}^{\eta }\right)\mathit{\text{dt}}$
$=\frac{{\varphi }^{-\xi }}{\mathrm{\Gamma }\left(\gamma \right)}{}_{3}{\mathrm{\Psi }}_{2}\left[\begin{array}{ll}\left(\gamma ,q\right),\left(\frac{1}{2}±\varphi +\xi ,\eta \right)& ;\\ \phantom{\rule{0.5em}{0ex}}\frac{\omega }{{\varphi }^{\eta }}\\ \left(\beta ,\alpha \right),\left(1-\lambda +\xi ,\eta \right)& ;\end{array}\phantom{\rule{0.5em}{0ex}}\right]\phantom{\rule{0.5em}{0ex}},$
(4.5.3)

2. (ii)
For q = γ = δ = 1 in (4.5.2), we have
$\phantom{\rule{-13.0pt}{0ex}}{\int }_{0}^{\infty }\stackrel{\frac{-\mathrm{\varphi t}}{2}}{exp}{t}^{\xi -1}{W}_{\lambda ,\psi }\left(\mathrm{\varphi t}\right){E}_{\alpha ,\beta }\left(\omega {t}^{\eta }\right)\mathit{\text{dt}}$
$={\varphi }^{-\xi }{}_{3}{\mathrm{\Psi }}_{2}\left[\begin{array}{ll}\left(\frac{1}{2}±\varphi +\xi ,\eta \right),\left(1,1\right),& ;\\ \phantom{\rule{0.5em}{0ex}}\frac{w}{{\varphi }^{\eta }}\\ \left(\beta ,\alpha \right),\left(1-\lambda +\xi ,\sigma \right)& ;\end{array}\phantom{\rule{0.5em}{0ex}}\right]$
(4.5.4)

3. (iii)
Now putting q = β = α = γ = δ = 1 in (4.5.2), we have
$\phantom{\rule{-13.0pt}{0ex}}{\int }_{0}^{\infty }{\mathrm{e}}^{\frac{-t}{2}}{t}^{\rho -1}{W}_{\lambda ,\psi }\left(\mathrm{\varphi t}\right)exp\left(\omega {t}^{\eta }\right)\mathit{\text{dt}}$
$={\varphi }^{-\xi }{}_{2}{\mathrm{\Psi }}_{1}\left[\begin{array}{ll}\left(\frac{1}{2}±\varphi +\xi ,\eta \right)& ;\\ \phantom{\rule{0.5em}{0ex}}\frac{w}{{\varphi }^{\eta }}\\ \left(1-\lambda +\xi ,\sigma \right)& ;\end{array}\phantom{\rule{0.5em}{0ex}}\right]$
(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
$\phantom{\rule{-14.0pt}{0ex}}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)=\sum _{n=0}^{\infty }\frac{{\left(\mu \right)}_{\rho n}{\left(\gamma \right)}_{\mathit{\text{qn}}}{z}^{n}}{\mathrm{\Gamma }\left(\alpha n+\beta \right){\left(\nu \right)}_{\eta n}{\left(\delta \right)}_{\mathit{\text{pn}}}}$
$=\frac{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}\sum _{n=0}^{\infty }\frac{\mathrm{\Gamma }\left(\mu +\rho n\right)\mathrm{\Gamma }\left(\gamma +\mathit{\text{qn}}\right)\mathrm{\Gamma }\left(n+1\right)}{\mathrm{\Gamma }\left(\alpha n+\beta \right)\mathrm{\Gamma }\left(\nu +\eta n\right)\mathrm{\Gamma }\left(\delta +\mathit{\text{pn}}\right)}\frac{{z}^{n}}{n!}$
$=\frac{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\gamma \right)}{}_{3}{\mathrm{\Psi }}_{3}\left[\phantom{\rule{0.3em}{0ex}}\begin{array}{ll}\left(\mu ,\rho \right)\left(\gamma ,q\right),\left(1,1\right),& ;\\ \phantom{\rule{0.5em}{0ex}}z\\ \left(\beta ,\alpha \right),\left(\nu ,\sigma \right),\left(\delta ,p\right)& ;\end{array}\phantom{\rule{0.3em}{0ex}}\right]\phantom{\rule{0.5em}{0ex}},$
(5.1.1)

Relationship with Fox H-function

Using (4.1.1), we have from
$\phantom{\rule{-13.0pt}{0ex}}\begin{array}{l}{E}_{\alpha ,\beta ,\nu ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q}\left(z\right)\\ =\frac{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{2\pi i\Gamma \left(\gamma \right)}{\int }_{L}\frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(1-s\right)\mathrm{\Gamma }\left(\mu -\rho s\right)\mathrm{\Gamma }\left(\gamma -\mathit{\text{qs}}\right)}{\mathrm{\Gamma }\left(\beta -\alpha s\right)\mathrm{\Gamma }\left(\nu -\sigma s\right)\mathrm{\Gamma }\left(\delta -\mathit{\text{ps}}\right)}{\left(-z\right)}^{-s}\mathit{\text{ds}}\end{array}$
$\phantom{\rule{-13.0pt}{0ex}}\begin{array}{l}=\frac{\mathrm{\Gamma }\left(\delta \right)\mathrm{\Gamma }\left(\nu \right)}{2\pi i\Gamma \left(\gamma \right)\mathrm{\Gamma }\left(\mu \right)}{H}_{3,3}^{1,3}\\ \phantom{\rule{1em}{0ex}}×\left[\begin{array}{ll}\left(0,1\right),\left(1-\mu ,\rho \right),\left(1-\gamma ,q\right)& ;\\ \phantom{\rule{0.5em}{0ex}}z\\ \left(0,1\right)\left(1-\beta ,\alpha \right),\left(1-\nu ,\sigma \right),\left(1-\delta ,p\right)& ;\end{array}\phantom{\rule{0.5em}{0ex}}\right]\phantom{\rule{0.5em}{0ex}},\end{array}$
(5.2.1)

Relationship with generalized Laguerre polynomials

Putting α = k, β = μ + 1, γ = − m, qN with q|m and replacing z by z k in (1.6), we get
$\phantom{\rule{-14.0pt}{0ex}}{E}_{k,\mu +1,\delta }^{-m,q}\left({z}^{k}\right)=\sum _{n=0}^{\left[\frac{m}{q}\right]}\frac{{\left(-m\right)}_{\mathit{\text{qn}}}{z}^{\mathit{\text{kn}}}}{\mathrm{\Gamma }\left(\mathit{\text{kn}}+\mu +1\right){\left(\delta \right)}_{n}}$
$=\sum _{n=0}^{\left[\frac{m}{q}\right]}\frac{{\left(-1\right)}^{\mathit{\text{qn}}}m!}{\left(m-\mathit{\text{qn}}\right)!\mathrm{\Gamma }\left(\mathit{\text{kn}}+\mu +1\right)}\frac{{z}^{\mathit{\text{kn}}}}{{\left(\delta \right)}_{n}}$
$=\frac{\mathrm{\Gamma }\left(m+1\right)}{\mathrm{\Gamma }\left(\mathit{\text{km}}+\mu +1\right)}\sum _{n=0}^{\left[\frac{m}{q}\right]}\frac{{\left(-1\right)}^{\mathit{\text{qn}}}\mathrm{\Gamma }\left(\mathit{\text{km}}+\mu +1\right)}{\left(m-\mathit{\text{qn}}\right)!\mathrm{\Gamma }\left(\mathit{\text{kn}}+\mu +1\right)}\frac{{z}^{\mathit{\text{kn}}}}{{\left(\delta \right)}_{n}}$
$=\frac{\mathrm{\Gamma }\left(m+1\right)}{\mathrm{\Gamma }\left(\mathit{\text{km}}+\mu +1\right)}{Z}_{\left[\frac{m}{q}\right],\delta }^{\mu }\left(z,k\right)$

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

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

Further for $q=k=1,{Z}_{m,1}^{\mu }\left(z,1\right)={L}_{m}^{\mu }\left(z\right)$, where ${L}_{m}^{\mu }\left(z\right)$ is a generalized Laguerre polynomial. So that
${E}_{k,\mu +1,1}^{-m,q}\left(z\right)=\frac{\mathrm{\Gamma }\left(m+1\right)}{\mathrm{\Gamma }\left(\mathit{\text{km}}+\mu +1\right)}{L}_{m}^{\mu }\left(z\right)$

which is the required relationship.

Declarations

Acknowledgements

The authors wish to thank the refrees for valuable suggestions and comments.

Authors’ Affiliations

(1)
Department of Applied Mathematics, Faculty of Engineering, Aligarh Muslim University

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