1 Introduction

Mittag-Leffler functions play a vital role in determining the solutions of fractional differential and integral equations which are associated with an extensive variety of problems in diverse areas of mathematics and mathematical physics. In addition, from exponential behavior, the deviations of physical phenomena could also be represented by means of Mittag-Leffler functions. Therefore, the uses of Mittag-Leffler functions are constantly rising, especially in physics. For more details about the recent research in the field of dynamical systems theory, stochastic systems, non-equilibrium statistical mechanics and quantum mechanics, the readers may refer to the recent work of the researchers [1–5] and the references cited therein.

Now, we begin with the Mittag-Leffler functions \(E_{\alpha}(z)\) and \(E_{\alpha,\beta}(z)\) defined in the form of the following series:

$$ E_{\alpha}(z)=\sum_{n=0}^{\infty} \frac{z^{n}}{\Gamma(\alpha n+1)}, \quad z\in\mathbb{C}; \Re(\alpha)>0 $$
(1)

and

$$ E_{\alpha,\beta}(z)=\sum_{n=0}^{\infty} \frac{z^{n}}{\Gamma(\alpha n+\beta)}, \quad z, \beta\in\mathbb{C}; \Re(\alpha)>0, $$
(2)

respectively. For further study of Mittag-Leffler functions like generalizations and applications, the readers may refer to the work of researchers [6–10], Kilbas et al. [11], and Saigo and Kilbas [12]. In recent years, the Mittag-Leffler function (1) and some of its generalizations have been numerically established in the complex plane [13, 14]. A new generalization of the Mittag-Leffler functions \(E_{\alpha, \beta}(z)\) in (2) has been defined by Prabhakar [15] as given below:

$$ E_{\alpha,\beta}^{\gamma}(z)=\sum _{n=0}^{\infty} \frac{(\gamma )_{n}}{\Gamma(\alpha n+\beta)}\frac{z^{n}}{n!}, \quad z, \beta\in\mathbb{C}; \Re (\alpha)>0, $$
(3)

where \((\gamma)_{n}\) denotes the well-known Pochhammer’s symbol which is defined by

$$ (\gamma)_{n}=\left \{ \textstyle\begin{array}{l@{\quad}l}1& (n=0, \gamma\in\mathbb{C}),\\ \gamma(\gamma+1)\cdots(\gamma+n-1)& (n\in\mathbb{N}, \gamma\in\mathbb {C}). \end{array}\displaystyle \right . $$

In fact, the following special cases are satisfied:

$$ E_{\alpha,\beta}^{1}(z)=E_{\alpha, \beta}(z)=E_{\alpha, 1}^{1}(z)=E_{\alpha}(z). $$
(4)

Many researchers have established the significance and great consideration of Mittag-Leffler functions in the theory of special functions to explore the generalizations and some applications. Various extensions for these functions are found in [16–20]. Srivastava and Tomovski [21] defined the further generalization of the Mittag-Leffler function \(E_{\alpha, \beta}^{\gamma }(z)\) as given in (3), which is defined as

$$ E_{\alpha, \beta}^{\gamma,\kappa}(z)=\sum _{n=0}^{\infty} \frac {(\gamma)_{n\kappa}}{\Gamma(\alpha n+\beta)}\frac{z^{n}}{n!}, $$
(5)

where \(z,\beta,\gamma\in\mathbb{C}\); \(\Re(\alpha)>\max\{0, \Re(\kappa )-1\}\); \(\Re(\kappa)>0\).

Özarslan and Yilmaz [22] investigated an extended Mittag-Leffler function \(E_{\alpha,\beta}^{\gamma;c}(z;p)\), which is defined as

$$ E_{\alpha,\beta}^{\gamma;c}(z;p)=\sum _{n=0}^{\infty} \frac {B_{p}(\gamma+n, c-\gamma)}{B(\gamma, c-\gamma)}\frac{(c)_{n}}{\Gamma (\alpha n+\beta)} \frac{z^{n}}{n!}, $$
(6)

where \(p\geq0\), \(\Re(c)>\Re(\gamma)>0\) and \(B_{p}(x,y)\) is an extended beta function defined in [23, 24] as follows:

$$ B_{p}(x,y)= \int _{0}^{1}t^{x-1}(1-t)^{y-1}e^{-\frac{p}{t(1-t)}} \,dt, $$
(7)

where \(\Re(p)>0\), \(\Re(x)>0\) and \(\Re(y)>0\). If \(p=0\), then the function \(B_{p}(x,y)\) reduces to the following beta function:

$$ B(x,y)= \int _{0}^{1}t^{x-1}(1-t)^{y-1} \,dt. $$
(8)

The gamma function is defined by

$$ \Gamma(z)= \int _{0}^{\infty}t^{z-1}e^{-t}\,dt; \quad \Re(z)>0. $$
(9)

By inspection, we conclude the following relation:

$$ \Gamma(z+1)=z\Gamma(z). $$
(10)

Mittal et al. [25] defined an extended generalized Mittag-Leffler function as

$$ E_{\alpha, \beta}^{\gamma, q; c}(z;p)=\sum _{n=0}^{\infty} \frac {B_{p}(\gamma+nq, c-\gamma)}{B(\gamma, c-\gamma)}\frac{(c)_{nq}}{\Gamma (\alpha n+\beta)} \frac{z^{n}}{n!}, $$
(11)

where \(\alpha, \beta, \gamma\in\mathbb{C}\), \(\Re(\alpha)>0\), \(\Re(\beta )>0\), \(\Re(\gamma)>0\), \(q>0\) and \(B_{p}(x,y)\) is an extended beta function defined in (7).

The fractional calculus is a field of applied mathematics that deals with the fractional derivatives and fractional integrals of arbitrary orders. During the last few decades, many researchers have applied fractional calculus to all fields of science such as engineering and mathematics. The researchers have developed a significant contributions in the field of fractional calculus such as fractional derivatives of constant and variable orders, global existence solution of differential equations; an alternative method for solving generalized differential equations of fractional order, a new type of fractional derivative formula containing the normalized sine function without singular kernel. For the recent development in the field of fractional calculus, the readers are referred to the work of [26–31] and [32–35].

Recently, Nair [36] introduced a pathway fractional integral operator by using the idea of Mathai [37], and Mathai and Haubold [38, 39], which is defined as

$$ \bigl( P_{0+}^{\mu, \lambda}f \bigr) (x)=x^{\mu} \int _{0}^{[\frac{x}{\alpha(1-\lambda)}]}\biggl[ 1- \frac{\alpha(1-\lambda)\tau}{x} \biggr]^{\frac{\mu}{(1-\lambda)}}f(\tau)\,d\tau, $$
(12)

where \(f\in L(a,b)\) (\(L(a,b)\) is a Lebesgue measurable real or complex-valued function), \({\mu\in\mathbb{C}}\), \(\Re(\mu)>0\), \(\alpha>0\) and \(\lambda<1\) (λ is a pathway parameter) (cf. [40]). For a given scalar λ and scalar random variables, the pathway model is defined by the following probability density function:

$$ f(x)=c|x|^{\nu-1} \bigl[ 1- \alpha(1- \lambda)|x|^{\eta} \bigr]^{\frac{\mu}{(1-\lambda)}}, $$
(13)

where \(x\in(-\infty, \infty)\); \(\eta>0\); \(\mu>0\); \([1-\alpha(1-\lambda)|x|^{\eta}]^{\frac{\mu}{(1-\lambda)}}>0\); \(\nu>0\) and where c and λ denote the normalizing constant and pathway parameter, respectively.

Moreover, the normalizing constants for \(\lambda\in\mathbb{R}\) are defined by the following:

$$ c=\left \{ \textstyle\begin{array}{l@{\quad}l}\frac{1}{2}\frac{\eta[\alpha(1-\lambda)]^{\frac{\nu}{\eta}}\Gamma(\frac {\nu}{\eta}+\frac{\mu}{\lambda-1}+1)}{\Gamma(\frac{\nu}{\eta}) \Gamma(\frac{\mu}{1-\lambda}+1)}& (\lambda< 1),\\ \frac{1}{2}\frac{\eta[\alpha(1-\lambda)]^{\frac{\nu}{\eta}}\Gamma(\frac {\mu}{\lambda-1})}{\Gamma(\frac{\nu}{\eta}) \Gamma(\frac{\mu}{\lambda-1}-\frac{\nu}{\eta})}& (\frac{1}{\lambda -1}-\frac{\nu}{\eta}>0, \lambda>1),\\ \frac{1}{2}\frac{(\alpha\mu)^{\frac{\nu}{\eta}}}{\Gamma(\frac{\nu}{\eta })}& (\lambda\rightarrow1). \end{array}\displaystyle \right . $$

It is noted that if \(\lambda<1\), we have \([ 1-\alpha(1-\lambda)|x|^{\eta} ]^{\frac{\mu}{(1-\lambda)}}>0\), and (13) can be considered a member of the extended generalized type-1 beta family. Also the extended type-1 beta density, the triangular density, the uniform density and many other probability density functions are specific cases of the pathway density function defined in (13) for \({\lambda<1}\).

For instance, if \(\lambda>1\), and by putting \((1-\lambda)=-(\lambda -1)\) in (12), then it yields

$$ \bigl( P_{0+}^{\mu, \lambda}f \bigr) (x)=x^{\mu} \int _{0}^{[\frac{x}{-\alpha(1-\lambda)}]}\biggl[ 1+ \frac{\alpha(\lambda-1)\tau}{x} \biggr]^{\frac{\mu}{-(\lambda-1)}}f(\tau)\,d\tau $$
(14)

and

$$ f(x)=c|x|^{\nu-1} \bigl[ 1+ \alpha( \lambda-1)|x|^{\eta} \bigr]^{-\frac{\mu}{(\lambda-1)}}, $$
(15)

where \(x\in(-\infty, \infty)\); \(\eta>0\); \(\mu>0\); and \(\lambda>1\) represents the extended generalized type-2 beta model for real x. The type-2 beta density function, the F density function, the student t density function and many other density functions are particular cases of the density function defined in (15). Furthermore, if \(\lambda \rightarrow1_{-}\), then (12) reduces to the Laplace integral transform. Similarly, if \(\lambda=0\), \(\alpha=1\) and μ is replaced by \(\mu-1\), then (12) reduces to the well-known Riemann-Liouville fractional integral operator.

The pathway fractional integral operator (12) leads to several other interesting examples such as fractional calculus related to several probability density functions and their applications in statistical theory. Nisar et al. [41] presented the pathway fractional integral formulae associated with Struve function of the first kind. The results given in [40] are a slight generalization of the result provided by Agarwal and Purohit [42] and Nair [36]. Recently, Nisar et al. [43] provided the composition of pathway integral operator associated with generalized k-Mittag-Leffler functions. The main aim of this study is to obtain pathway fractional integral operators associated with extended Mittag-Leffler functions.

2 Pathway fractional integration of an extended Mittag-Leffler function

In this section, we derive the pathway integration formulae involving the extended Mittag-Leffler functions from (6).

Theorem 1

Suppose that \(\rho; \beta; \gamma; c; \mu\in\mathbb {C}\); \(\{\Re(\rho); \Re(\beta); \Re(\mu); \Re(c)\}>0\); \(\Re(\frac{\mu }{1-\lambda})>-1\); \(\lambda<1\); \(p\geq0\) and \(\omega\in\mathbb{R}\). Then the following formula holds true:

$$ P_{0+}^{\mu, \lambda}\bigl[ \tau^{\beta-1}E_{\rho, \beta}^{\gamma; c} \bigl[ \bigl(\omega \tau^{\rho}\bigr); p \bigr] \bigr](x)= \frac{\Gamma(1+\frac{\mu}{1-\lambda})x^{\mu+\beta}}{[\alpha (1-\lambda)]^{\beta}} E_{\rho, \beta+(1+\frac{\mu}{1-\lambda})}^{\gamma; c}\biggl[ \omega \biggl(\frac{x}{\alpha(1-\lambda)} \biggr)^{\rho}; p \biggr]. $$
(16)

Proof

By using (6) and (12), we have

$$ P_{0+}^{\mu, \lambda}\bigl[ \tau^{\beta-1}E_{\rho, \beta}^{\gamma; c} \bigl[ \bigl(\omega \tau^{\rho}\bigr);p \bigr] \bigr](x)=x^{\mu}\int _{0}^{[\frac{x}{\alpha(1-\lambda)}]}\tau ^{\beta-1} \biggl[1- \frac{\alpha(1-\lambda)\tau}{x} \biggr]^{\frac{\mu}{1-\lambda }}E_{\rho, \beta}^{\gamma; c} \bigl( \omega\tau^{\rho}; p \bigr)\,d\tau. $$

We denote the right-hand side of the above equation by \(I_{1}\), and after interchanging the order of integration and summation, we have

$$ \begin{aligned} I_{1}&=x^{\mu}\int _{0}^{[\frac{x}{\alpha(1-\lambda)}]}\tau^{\beta -1} \biggl[1- \frac{\alpha(1-\lambda)\tau}{x} \biggr]^{\frac{\mu}{1-\lambda}} \sum_{n=0}^{\infty} \frac{B_{p}(\gamma+n, c-\gamma)}{B(\gamma, c-\gamma)}\frac{(c)_{n}}{\Gamma(\rho n+\beta)}\frac{(\omega\tau^{\rho})^{n}}{n!} \,d\tau \\ &=x^{\mu}\sum_{n=0}^{\infty} \frac{B_{p}(\gamma+n, c-\gamma )}{B(\gamma, c-\gamma)}\frac{(c)_{n}}{\Gamma(\rho n+\beta)}\frac{(\omega)^{n}}{n!} \int _{0}^{[\frac{x}{\alpha(1-\lambda)}]}\tau^{\beta+\rho n-1} \biggl[1- \frac{\alpha(1-\lambda)\tau}{x} \biggr]^{\frac{\mu}{1-\lambda}}\,d\tau \\ &=x^{\mu}\sum_{n=0}^{\infty} \frac{B_{p}(\gamma+n, c-\gamma )}{B(\gamma, c-\gamma)}\frac{(c)_{n}}{\Gamma(\rho n+\beta)}\frac{(\omega)^{n}}{n!} \biggl( \frac{x}{\alpha(1-\lambda)} \biggr)^{\rho n+\beta} \frac{\Gamma(1+\frac{\mu}{1-\lambda})\Gamma(\rho n+\beta)}{\Gamma(\rho n+\beta+1+\frac{\mu}{1-\lambda})} \\ &=\frac{x^{\mu+\beta}\Gamma(1+\frac{\mu}{1-\lambda})}{[\alpha(1-\lambda )]^{\beta}}\sum_{n=0}^{\infty} \frac{B_{p}(\gamma+n, c-\gamma )}{B(\gamma, c-\gamma)}\frac{(c)_{n}}{\Gamma(\rho n+\beta+1+\frac{\mu }{1-\lambda})}\frac{(\omega( \frac{x}{\alpha(1-\lambda)} )^{\rho})^{n}}{n!} \\ &=\frac{x^{\mu+\beta}\Gamma(1+\frac{\mu}{1-\lambda})}{[\alpha(1-\lambda )]^{\beta}}E_{\rho, \beta+1+\frac{\mu}{1-\lambda}}^{\gamma; c} \biggl( \omega \biggl(\frac{x}{\alpha(1-\lambda)} \biggr)^{\rho}; p \biggr), \end{aligned} $$

which completes the required proof. □

Corollary 1

If \(p=0\), then (16) leads to the following result of Mittag-Leffler function (see [36]):

$$ P_{0+}^{\mu, \lambda}\bigl[ \tau^{\beta-1}E_{\rho, \beta}^{\gamma; c} \bigl[ \bigl(\omega \tau^{\rho}\bigr) \bigr] \bigr](x)= \frac{x^{\mu+\beta}\Gamma(1+\frac{\mu}{1-\lambda})}{[\alpha (1-\lambda)]^{\beta}}E_{\rho, \beta+1+\frac{\mu}{1-\lambda}}^{\gamma} \biggl( \omega \biggl( \frac{x}{\alpha(1-\lambda)} \biggr)^{\rho} \biggr). $$
(17)

Corollary 2

If \(\delta=1\), \(\alpha=1\), \(\lambda=0\) and \(\mu=\mu -1\), then (16) leads to the following fractional integral formula of extended Mittag-Leffler function (see [44]):

$$ I_{a+}^{\mu}\bigl[ \tau^{\beta-1}E_{\alpha, \beta}^{\gamma} \bigl(\omega( \tau)^{\rho},p \bigr) \bigr](x) =\Gamma( \mu)x^{\mu+\beta}E_{\rho, \beta+\mu}^{\gamma} \bigl(\omega(x)^{\rho},p \bigr). $$
(18)

Corollary 3

If we set \(p=0\) in (18), then it leads to the following well-known result of Mittag-Leffler function (see [7]):

$$ I_{a+}^{\mu}\bigl[ \tau^{\beta-1}E_{\alpha, \beta}^{\gamma} \bigl(\omega( \tau)^{\rho}\bigr) \bigr](x) =\Gamma( \mu)x^{\mu+\beta}E_{\rho, \beta+\mu}^{\gamma} \bigl(\omega(x)^{\rho} \bigr). $$
(19)

Now, we derive the following result by assuming the case that \(\lambda >1\) and using equation (14).

Theorem 2

Suppose that \(\rho; \beta; \gamma; c; \mu\in\mathbb {C}\); \(\{\Re(\rho); \Re(\beta); \Re(\mu); \Re(c)\}>0\); \(\Re(\frac{\mu }{1-\lambda})>-1\); \(\lambda>1\); \(p\geq0\) and \(\omega\in\mathbb{R}\). Then the following formula holds true:

$$ P_{0+}^{\mu, \lambda}\bigl[ \tau^{\beta-1}E_{\rho, \beta}^{\gamma; c} \bigl[ \bigl(\omega \tau^{\rho}\bigr);p \bigr] \bigr](x)= \frac{\Gamma(1-\frac{\mu}{\lambda-1})x^{\mu+\beta}}{[-\alpha (\lambda-1)]^{\beta}} E_{\rho, \beta+(1-\frac{\mu}{\lambda-1})}^{\gamma; c}\biggl[ \omega \biggl(\frac{x}{-\alpha(\lambda-1)} \biggr)^{\rho}; p \biggr]. $$
(20)

Proof

By using (6) and (12), we have

$$\begin{gathered} P_{0+}^{\mu, \lambda}\bigl[ \tau^{\beta-1}E_{\rho, \beta}^{\gamma; c} \bigl[ \bigl(\omega \tau^{\rho}\bigr);p \bigr] \bigr](x)\\ \quad=x^{\mu}\int _{0}^{[\frac{x}{-\alpha(1-\lambda)}]}\tau ^{\beta-1} \biggl[1+ \frac{\alpha(\lambda-1)\tau}{x} \biggr]^{\frac{\mu}{-(\lambda-1)}}\bigl[ E_{\rho, \beta}^{\gamma; c} \bigl[ \bigl(\omega\tau^{\rho}\bigr);p \bigr] \bigr]\,d\tau. \end{gathered}$$

For convenience, we denote the right-hand side of the above equation by \(I_{2}\), then

$$ I_{2}=x^{\mu}\int _{0}^{[\frac{x}{-\alpha(1-\lambda)}]}\tau^{\beta -1} \biggl[1+ \frac{\alpha(\lambda-1)\tau}{x} \biggr]^{\frac{\mu}{-(\lambda-1)}} \sum_{n=0}^{\infty} \frac{B_{p}(\gamma+n, c-\gamma)}{B(\gamma, c-\gamma)}\frac{(c)_{n}}{\Gamma(\rho n+\beta)}\frac{(\omega\tau^{\rho})^{n}}{n!} \,d\tau. $$

By interchanging the order of integration and summation, we obtain

$$ I_{2}=x^{\mu}\sum_{n=0}^{\infty} \frac{B_{p}(\gamma+n, c-\gamma )}{B(\gamma, c-\gamma)}\frac{(c)_{n}}{\Gamma(\rho n+\beta)}\frac{\omega^{n}}{n!} \int _{0}^{[\frac{x}{-\alpha(1-\lambda)}]}\tau^{\beta+\rho n-1} \biggl[1+ \frac{\alpha(\lambda-1)\tau}{x} \biggr]^{\frac{\mu}{-(\lambda-1)}}\,d\tau. $$

By substituting \(\frac{-\alpha(\lambda-1)\tau}{x}=u\) and using the definitions of beta function (8) and (10) in the above equation, we get

$$ \begin{aligned} I_{2}={}&\frac{x^{\mu+\beta}}{[-\alpha(\lambda-1)]^{\beta}}\sum _{n=0}^{\infty} \frac{B_{p}(\gamma+n, c-\gamma)}{B(\gamma, c-\gamma )}\frac{(c)_{n}}{\Gamma(\rho n+\beta)} \frac{( \omega(\frac{x}{-\alpha(\lambda-1)})^{\rho})^{n}}{n!} \\ &\times\frac{\Gamma(1-\frac{\mu}{\lambda-1})\Gamma (\rho n+\beta)}{\Gamma(\rho n+\beta+1-\frac{\mu}{\lambda-1})} \\ ={}&\frac{x^{\mu+\beta}\Gamma(1-\frac{\mu}{\lambda-1})}{[-\alpha(\lambda -1)]^{\beta}}\sum_{n=0}^{\infty} \frac{B_{p}(\gamma+n, c-\gamma )}{B(\gamma, c-\gamma)}\frac{(c)_{n}}{\Gamma(\rho n+\beta+1-\frac{\mu }{\lambda-1})}\frac{( \omega(\frac{x}{-\alpha(\lambda-1)})^{\rho})^{n}}{n!} \\ ={}&\frac{x^{\mu+\beta}\Gamma(1-\frac{\mu}{\lambda-1})}{[-\alpha(\lambda -1)]^{\beta}}E_{\rho, \beta+(1-\frac{\mu}{\lambda-1})}^{\gamma; c}\biggl[ \omega \biggl(\frac{x}{-\alpha(\lambda-1)} \biggr)^{\rho}; p \biggr], \end{aligned} $$

which completes the required proof of (20). □

Corollary 4

If \(p=0\), then (20) leads to the following result of Mittag-Leffler function recently introduced by Nair [36]:

$$ P_{0+}^{\mu, \lambda}\bigl[ \tau^{\beta-1}E_{\rho, \beta}^{\gamma} \bigl[ \bigl(\omega \tau^{\rho}\bigr) \bigr] \bigr](x)= \frac{\Gamma(1-\frac{\mu}{\lambda-1})x^{\mu+\beta}}{[-\alpha (\lambda-1)]^{\beta}} E_{\rho, \beta+(1-\frac{\mu}{\lambda-1})}^{\gamma}\biggl[ \omega \biggl( \frac{x}{-\alpha(\lambda-1)} \biggr)^{\rho}\biggr]. $$
(21)

Corollary 5

Again, if \(\delta=1\), \(\alpha=1\), \(\lambda =0\) and \(\mu=\mu-1\), then (21) leads to the fractional integral formula of extended Mittag-Leffler function as defined in (18).

Corollary 6

If we set \(p=0\) in Corollary 5, then it leads to the fractional integral formula of Mittag-Leffler function as defined in (19).

3 Pathway fractional integral operator of an extended generalized Mittag-Leffler function

In this section, we present the composition of pathway fractional integral operator associated with an extended form of the generalized Mittag-Leffler function as defined in (11).

Theorem 3

Suppose that \(\rho; \beta; \gamma; c; \mu\in\mathbb {C}\); \(\{\Re(\rho); \Re(\beta); \Re(\mu); \Re(c)\}>0\); \(\Re(\frac{\mu }{1-\lambda})>-1\); \(\lambda<1\); \(p\geq0\); \(q>0\) and \(\omega\in\mathbb {R}\). Then the following formula holds true:

$$ P_{0+}^{\mu, \lambda}\bigl[ \tau^{\beta-1}E_{\rho, \beta}^{\gamma,q; c} \bigl[ \bigl(\omega \tau^{\rho}\bigr); p \bigr] \bigr](x)= \frac{\Gamma(1+\frac{\mu}{1-\lambda})x^{\mu+\beta}}{[\alpha (1-\lambda)]^{\beta}} E_{\rho, \beta+(1+\frac{\mu}{1-\lambda})}^{\gamma,q; c}\biggl[ \omega \biggl(\frac{x}{\alpha(1-\lambda)} \biggr)^{\rho}; p \biggr]. $$
(22)

Proof

By using (11) and (12), we have

$$ P_{0+}^{\mu, \lambda}\bigl[ \tau^{\beta-1}E_{\rho, \beta}^{\gamma, q; c} \bigl[ \bigl(\omega \tau^{\rho}\bigr);p \bigr] \bigr](x)=x^{\mu}\int _{0}^{[\frac{x}{\alpha(1-\lambda)}]}\tau ^{\beta-1} \biggl[1- \frac{\alpha(1-\lambda)\tau}{x} \biggr]^{\frac{\mu}{1-\lambda }}E_{\rho, \beta}^{\gamma, q; c} \bigl( \omega\tau^{\rho}; p \bigr)\,d\tau. $$

For simplicity, denote the right-hand side of the above equation by \(I_{3}\), we have

$$ I_{3}=x^{\mu}\int _{0}^{[\frac{x}{\alpha(1-\lambda)}]}\tau^{\beta -1} \biggl[1- \frac{\alpha(1-\lambda)\tau}{x} \biggr]^{\frac{\mu}{1-\lambda}} \sum_{n=0}^{\infty} \frac{B_{p}(\gamma+nq, c-\gamma)}{B(\gamma, c-\gamma)}\frac{(c)_{nq}}{\Gamma(\rho n+\beta)}\frac{(\omega\tau^{\rho})^{n}}{n!} \,d\tau. $$

By interchanging the order of integration and summation, we obtain the following:

$$ \begin{aligned} I_{3}&=x^{\mu}\sum _{n=0}^{\infty} \frac{B_{p}(\gamma+nq, c-\gamma )}{B(\gamma, c-\gamma)}\frac{(c)_{nq}}{\Gamma(\rho n+\beta)} \frac {(\omega)^{n}}{n!} \int _{0}^{[\frac{x}{\alpha(1-\lambda)}]}\tau^{\beta+\rho n-1} \biggl[1- \frac{\alpha(1-\lambda)\tau}{x} \biggr]^{\frac{\mu}{1-\lambda}}\,d\tau \\ &=x^{\mu}\sum_{n=0}^{\infty} \frac{B_{p}(\gamma+nq, c-\gamma )}{B(\gamma, c-\gamma)}\frac{(c)_{nq}}{\Gamma(\rho n+\beta)}\frac {(\omega)^{n}}{n!} \biggl( \frac{x}{\alpha(1-\lambda)} \biggr)^{\rho n+\beta} \frac{\Gamma(1+\frac{\mu}{1-\lambda})\Gamma(\rho n+\beta)}{\Gamma(\rho n+\beta+1+\frac{\mu}{1-\lambda})} \\ &=\frac{x^{\mu+\beta}\Gamma(1+\frac{\mu}{1-\lambda})}{[\alpha(1-\lambda )]^{\beta}}\sum_{n=0}^{\infty} \frac{B_{p}(\gamma+nq, c-\gamma)}{B(\gamma, c-\gamma)}\frac {(c)_{nq}}{\Gamma(\rho n+\beta+1+\frac{\mu}{1-\lambda})}\frac{(\omega ( \frac{x}{\alpha(1-\lambda)} )^{\rho})^{n}}{n!} \\ &=\frac{x^{\mu+\beta}\Gamma(1+\frac{\mu}{1-\lambda})}{[\alpha(1-\lambda )]^{\beta}}E_{\rho, \beta+1+\frac{\mu}{1-\lambda}}^{\gamma, q; c} \biggl( \omega \biggl(\frac{x}{\alpha(1-\lambda)} \biggr)^{\rho}; p \biggr), \end{aligned} $$

which completes the required proof. □

Corollary 7

If \(p=0\), and \(q=1\), then (22) leads to the following result of Mittag-Leffler function (see [36]):

$$ \begin{aligned}[b] &P_{0+}^{\mu, \lambda}\bigl[ \tau^{\beta-1}E_{\rho, \beta}^{\gamma} \bigl[ \bigl(\omega \tau^{\rho}\bigr) \bigr] \bigr](x) \\ &\quad=\frac{x^{\mu+\beta}\Gamma(1+\frac{\mu}{1-\lambda})}{[\alpha(1-\lambda )]^{\beta}}E_{\rho, \beta+1+\frac{\mu}{1-\lambda}}^{\gamma} \biggl( \omega \biggl(\frac{x}{\alpha(1-\lambda)} \biggr)^{\rho} \biggr). \end{aligned} $$
(23)

Corollary 8

If \(\alpha=1\), \(\lambda=0\) and \(\mu=\mu -1\), then (22) leads to the following result of extended Mittag-Leffler function (see [45]):

$$ P_{0+}^{\mu-1, 0}\bigl[ \tau^{\beta-1}E_{\rho, \beta}^{\gamma, q} \bigl[ \bigl(\omega \tau^{\rho}\bigr);p \bigr] \bigr](x)= \Gamma( \mu)x^{\mu+\beta}E_{\rho, \beta+\mu}^{\gamma, q} \bigl( \omega(x)^{\rho};p \bigr). $$
(24)

Remark 1

If we set \(p=0\) in Corollary 8, then we get the fractional integrals of Mittag-Leffler function earlier proved in [21].

Now, we derive the following result by assuming the case that \(\lambda >1\).

Theorem 4

Suppose that \(\rho; \beta; \gamma; c; \mu\in\mathbb {C}\); \(\{\Re(\rho); \Re(\beta); \Re(\mu); \Re(c)\}>0\); \(\Re(\frac{\mu }{1-\lambda})>-1\); \(\lambda>1\); \(p\geq0\); \(q>0\) and \(\omega\in\mathbb {R}\). Then the following formula holds true:

$$ P_{0+}^{\mu, \lambda}\bigl[ \tau^{\beta-1}E_{\rho, \beta}^{\gamma, q; c} \bigl[ \bigl(\omega \tau^{\rho}\bigr);p \bigr] \bigr](x)= \frac{\Gamma(1-\frac{\mu}{\lambda-1})x^{\mu+\beta}}{[-\alpha (\lambda-1)]^{\beta}} E_{\rho, \beta+(1-\frac{\mu}{\lambda-1})}^{\gamma, q; c}\biggl[ \omega \biggl(\frac{x}{-\alpha(\lambda-1)} \biggr)^{\rho}; p \biggr]. $$
(25)

Proof

By using (11) and (14), we obtain

$$\begin{gathered} P_{0+}^{\mu, \lambda}\bigl[ \tau^{\beta-1}E_{\rho, \beta}^{\gamma, q; c} \bigl[ \bigl(\omega \tau^{\rho}\bigr);p \bigr] \bigr](x)\\\quad=x^{\mu}\int _{0}^{[\frac{x}{-\alpha(\lambda-1)}]}\tau ^{\beta-1} \biggl[1+ \frac{\alpha(\lambda-1)\tau}{x} \biggr]^{\frac{\mu}{-(\lambda-1)}}\bigl[ E_{\rho, \beta}^{\gamma, q; c} \bigl[ \bigl(\omega\tau^{\rho}\bigr);p \bigr] \bigr]\,d\tau. \end{gathered}$$

For convenience, we denote the right-hand side of the above equation by \(I_{2}\), then

$$ I_{4}=x^{\mu}\int _{0}^{[\frac{x}{-\alpha(\lambda-1)}]}\tau^{\beta-1} \biggl[1+ \frac{\alpha(\lambda-1)\tau}{x} \biggr]^{\frac{\mu}{-(\lambda-1)}} \sum_{n=0}^{\infty} \frac{B_{p}(\gamma+nq, c-\gamma)}{B(\gamma, c-\gamma)}\frac{(c)_{nq}}{\Gamma(\rho n+\beta)}\frac{(\omega\tau^{\rho})^{n}}{n!} \,d\tau. $$

By interchanging the order of integration and summation, we obtain

$$ I_{4}=x^{\mu}\sum_{n=0}^{\infty} \frac{B_{p}(\gamma+nq, c-\gamma )}{B(\gamma, c-\gamma)}\frac{(c)_{nq}}{\Gamma(\rho n+\beta)}\frac{\omega^{n}}{n!} \int _{0}^{[\frac{x}{-\alpha(\lambda-1)}]}\tau^{\beta+\rho n-1} \biggl[1+ \frac{\alpha(\lambda-1)\tau}{x} \biggr]^{\frac{\mu}{-(\lambda-1)}}\,d\tau. $$

By substituting \(\frac{-\alpha(\lambda-1)\tau}{x}=u\) and using the definitions of beta function (8) and (10) in the above equation, we obtain

$$ \begin{aligned} I_{4}={}&\frac{x^{\mu+\beta}}{[-\alpha(\lambda-1)]^{\beta}}\sum _{n=0}^{\infty} \frac{B_{p}(\gamma+nq, c-\gamma)}{B(\gamma, c-\gamma )}\frac{(c)_{nq}}{\Gamma(\rho n+\beta)} \frac{( \omega(\frac{x}{-\alpha(\lambda-1)})^{\rho})^{n}}{n!} \\ &\times\frac{\Gamma(1-\frac{\mu}{\lambda-1})\Gamma (\rho n+\beta)}{\Gamma(\rho n+\beta+1-\frac{\mu}{\lambda-1})} \\ ={}&\frac{x^{\mu+\beta}\Gamma(1-\frac{\mu}{\lambda-1})}{[-\alpha(\lambda -1)]^{\beta}}\sum_{n=0}^{\infty} \frac{B_{p}(\gamma+nq, c-\gamma)}{B(\gamma, c-\gamma)}\frac {(c)_{nq}}{\Gamma(\rho n+\beta+1-\frac{\mu}{\lambda-1})}\frac{( \omega(\frac{x}{-\alpha(\lambda-1)})^{\rho})^{n}}{n!} \\ ={}&\frac{x^{\mu+\beta}\Gamma(1-\frac{\mu}{\lambda-1})}{[-\alpha(\lambda -1)]^{\beta}}E_{\rho, \beta+(1-\frac{\mu}{\lambda-1})}^{\gamma, q; c}\biggl[ \omega \biggl(\frac{x}{-\alpha(\lambda-1)} \biggr)^{\rho}; p \biggr], \end{aligned} $$

which completes the required proof of (25). □

Corollary 9

If \(p=0\) and \(q=1\), then (25) leads to the following result of Mittag-Leffler function recently introduced by Nair [36]:

$$ P_{0+}^{\mu, \lambda}\bigl[ \tau^{\beta-1}E_{\rho, \beta}^{\gamma} \bigl[ \bigl(\omega \tau^{\rho}\bigr); p \bigr] \bigr](x)= \frac{\Gamma(1-\frac{\mu}{\lambda-1})x^{\mu+\beta}}{[-\alpha (\lambda-1)]^{\beta}} E_{\rho, \beta+(1-\frac{\mu}{\lambda-1})}^{\gamma}\biggl[ \omega \biggl( \frac{x}{-\alpha(\lambda-1)} \biggr)^{\rho}\biggr]. $$
(26)

Corollary 10

If \(\alpha=1\), \(\lambda=0\) and \(\mu=\mu-1\), then (25) leads to the result of fractional integral containing an extended Mittag-Leffler function in its kernel defined in Corollary 8.

4 Conclusion

In this paper, we have presented two pathway fractional integration formulae associated with extended Mittag-Leffler functions. The obtained result provided extended forms of the known results earlier proved by Nair [36].