Certain fractional calculus formulas involving extended generalized Mathieu series

Abstract

We establish fractional integral and derivative formulas by using fractional calculus operators involving the extended generalized Mathieu series. Next, we develop their composition formulas by applying the integral transforms. Finally, we discuss special cases.

Introduction and preliminaries

Fractional calculus is a very rapidly growing subject of mathematics which deals with the study of fractional order derivatives and integrals. Fractional calculus is an efficient tool to study many complex real world systems . It is demonstrated that the fractional order representation of complex processes appearing in various fields of science, engineering and finance, provides a more realistic approach with memory effects to study these problems (see e.g. ). Among the research work developing the theory of fractional calculus and presenting some applications, we point out some literature. Kumar et al.  analyzed the fractional model of a modified Kawahara equation by using a newly introduced Caputo–Fabrizio fractional derivative. One also  studied a heat transfer problem and presented a new non-integer model for convective straight fins with temperature-dependent thermal conductivity associated with Caputo–Fabrizio fractional derivative. Recently, one  presented a new fractional extension of regularized long wave equation by using an Atangana–Baleano fractional operator. In  one introduced a new numerical scheme for a fractional Fitzhugh–Nagumo equation arising in the transmission of new impulses. In  one constituted a modified numerical scheme to study fractional model of Lienard’s equations. Hajipour et al.  formulated a new scheme for a class of fractional chaotic systems. Baleanu et al.  proposed a new formulation of the fractional control problems involving a Mittag-Leffler non-singular kernel. In another work, Baleanu et al.  studied the motion of a bead sliding on a wire in a fractional analysis. Jajarmi et al.  analyzed a hyperchaotic financial system and its chaos control and synchronization by using fractional calculus.

For mathematical modeling of many complex problems appearing in various fields of science and engineering such as fluid dynamics, plasma physics, astrophysics, image processing, stochastic dynamical system, controlled thermonuclear fusion, nonlinear control theory, nonlinear biological systems, quantum physics and heat transfer problems, the fractional calculus operators involving various special functions have been used successfully. There is a rich literature available revealing the notable development in fractional order derivatives and integrals (see [1, 10, 11, 2328]). Recently, Caputo and Fabrizio  introduced a new fractional derivative which is more suitable than the classical Caputo fractional derivative for many engineering and thermodynamical processes. Atangana  used a new fractional derivative to study the nature of Fisher’s reaction diffusion equation. Riemann and Caputo fractional derivative operators both have a singular kernel which cannot exactly represent the complete memory effect of the system. To overcome these limitations of the old derivatives, very recently Atangana and Baleanu  presented a new non-integer order derivative having a non-local, non-singular and Mittag-Leffler type kernel.

In recent years, many researchers have extensively studied the properties, applications and extensions of various fractional integral and differential operators involving the various special functions (for details, see [25, 3242], etc.).

The image formulas for special functions of one or more variables are very useful in the evaluation and solution of differential and integral equations. Motivated by the above discussion, we developed new fractional calculus formulas involving extended generalized Mathieu series.

For our present study, we recall the generalized hypergeometric fractional integrals, introduced by Marichev , including the Saigo operators , and which were later on extended by Saigo and Maeda .

The generalized fractional calculus operators (the Marchichev–Saigo–Maeda operators) involving the Appell function or the Horn $$F_{3}(\cdot)$$ function in the kernel are defined thus.

Definition 1

Let $$\sigma, \sigma', \nu, \nu',\eta\in\mathbb{C}$$ and $$x>0$$, then, for $$\Re(\eta)>0$$,

\begin{aligned}[b]& \bigl(I^{\sigma, \sigma', \nu, \nu ',\eta}_{0,x}f \bigr) (x) \\ &\quad =\frac{x^{-\sigma}}{\Gamma(\eta)} \int ^{x}_{0}(x-t)^{\eta-1}t^{-\sigma'}F_{3} \biggl(\sigma, \sigma', \nu, \nu ';\eta; 1- \frac{t}{x}, 1-\frac{x}{t} \biggr)f(t)\,dt \end{aligned}
(1.1)

and

\begin{aligned}[b]& \bigl(I^{\sigma, \sigma', \nu, \nu ',\eta}_{x,\infty}f \bigr) (x) \\ &\quad =\frac{x^{-\sigma'}}{\Gamma(\eta)} \int ^{\infty}_{x}(t-x)^{\eta-1}t^{-\sigma}F_{3} \biggl(\sigma, \sigma', \nu, \nu ';\eta; 1- \frac{x}{t}, 1-\frac{t}{x} \biggr)f(t)\,dt, \end{aligned}
(1.2)

where the function $$f(t)$$ is so constrained that the integrals in (1.1) and (1.2) exist.

In (1.1) and (1.2), $$F_{3}(\cdot)$$ denotes Appell’s hypergeometric function  in two variables defined as

\begin{aligned}[b]& F_{3} \bigl(\sigma, \sigma', \nu, \nu';\eta ; x, y \bigr) \\ &\quad =\sum_{m,n=0}^{\infty}\frac{(\sigma)_{m} (\sigma')_{n}(\nu)_{m}(\nu ')_{n}}{(\eta)_{m+n}} \frac{x^{m}}{m!}\frac{x^{n}}{n!} \bigl(\max\bigl\{ |x|,|y|\bigr\} < 1 \bigr). \end{aligned}
(1.3)

The above fractional integral operators in Eqs. (1.1) and (1.2) can be written as follows:

\begin{aligned}[b]& \bigl(I^{\sigma, \sigma', \nu, \nu ',\eta}_{0,x}f \bigr) (x)= \biggl(\frac{d}{dx} \biggr)^{k} \bigl(I^{\sigma, \sigma', \nu+k, \nu', \eta+k}_{0,x}f \bigr) (x) \\ &\quad \bigl(\Re(\eta)\leq0; k= \bigl[-\Re(\eta)+1 \bigr] \bigr) \end{aligned}
(1.4)

and

\begin{aligned}[b]& \bigl(I^{\sigma, \sigma', \nu, \nu ',\eta}_{x,\infty}f \bigr) (x)= \biggl(-\frac{d}{dx} \biggr)^{k} \bigl(I^{\sigma, \sigma', \nu, \nu'+k,\eta+k}_{x,\infty}f \bigr) (x) \\ & \quad\bigl(\Re (\eta)\leq0; k= \bigl[-\Re(\eta)+1 \bigr] \bigr). \end{aligned}
(1.5)

Remark 1

The Appell function defined in Eq. (1.3) reduces to the Gauss hypergeometric function $${}_{2}F_{1}$$ as given in following relations:

\begin{aligned}[b]& F_{3}(\sigma, \eta-\sigma, \nu, \eta - \nu; \eta; x, y)={}_{2}F_{1}(\sigma, \nu;\eta; x+y-xy); \end{aligned}
(1.6)

also we have

\begin{aligned}[b]& F_{3} \bigl(\sigma, 0, \nu, \nu', \eta; x, y \bigr)={}_{2}F_{1}(\sigma, \nu; \eta; x) \end{aligned}
(1.7)

and

\begin{aligned}[b]& F_{3} \bigl(0, \sigma', \nu, \nu', \eta; x, y \bigr)={}_{2}F_{1} \bigl( \sigma', \nu';\eta; y \bigr). \end{aligned}
(1.8)

The corresponding Saigo–Maeda fractional differential operators are given as follows.

Definition 2

Let $$\sigma, \sigma',\nu, \nu',\eta\in\mathbb{C}$$ and $$x>0$$, then

\begin{aligned}[b]\bigl(D^{\sigma, \sigma', \nu, \nu ', \eta}_{0,x}f \bigr) (x)& = \bigl(I^{-\sigma', -\sigma, -\nu', -\nu, -\eta }_{0,x}f \bigr) (x) \\ & = \biggl(\frac{d}{dx} \biggr)^{k} \bigl(I^{-\sigma', -\sigma, -\nu'+k, -\nu, -\eta+k}_{0,x}f \bigr) (x) \quad\bigl(\Re(\eta)>0; k= \bigl[\Re(\eta) \bigr]+1 \bigr) \\ & = \frac{1}{\Gamma(k-\eta)} \biggl(\frac{d}{dx} \biggr)^{k}(x)^{\sigma'} \int^{x}_{0}(x-t)^{k-\eta-1}t^{\sigma } \\ &\quad{} \times F_{3} \biggl(-\sigma', -\sigma, k- \nu', -\nu;k-\eta; 1-\frac {t}{x}, 1-\frac{x}{t} \biggr)f(t)\,dt \end{aligned}
(1.9)

and

\begin{aligned}[b] \bigl(D^{\sigma, \sigma', \nu, \nu ', \eta}_{x, \infty}f \bigr) (x)&= \bigl(I^{-\sigma', -\sigma, -\nu', -\nu , -\eta}_{x, \infty}f \bigr) (x) \\ & = \biggl(-\frac{d}{dx} \biggr)^{k} \bigl(I^{-\sigma', -\sigma, -\nu', -\nu+k, -\eta+k}_{x, \infty}f \bigr) (x) \quad\bigl(\Re(\eta)>0; k= \bigl[\Re(\eta) \bigr]+1 \bigr) \\ & = \frac{1}{\Gamma(k-\eta)} \biggl(-\frac{d}{dx} \biggr)^{k}(x)^{\sigma} \int^{\infty}_{x}(t-x)^{k-\eta -1}t^{\sigma'} \\ & \quad{}\times F_{3} \biggl(-\sigma', -\sigma, - \nu', k-\nu;k-\eta ; 1-\frac{x}{t}, 1-\frac{t}{x} \biggr)f(t)\,dt. \end{aligned}
(1.10)

In view of the above reduction formula as given in Eq. (1.7), the general fractional calculus operators reduce to the Saigo operators  defined as follows.

Definition 3

For $$x>0$$, $$\sigma, \nu, \eta\in\mathbb{C}$$ and $$\Re(\sigma)>0$$

\begin{aligned}[b]& \bigl(I^{\sigma, \nu, \eta }_{0,x}f \bigr) (x)= \frac{x^{-\sigma-\nu}}{\Gamma(\sigma)} \int ^{x}_{0}(x-t)^{\sigma-1}{}_{2}F_{1} \biggl(\sigma+\nu,-\eta; \sigma;1-\frac {t}{x} \biggr)f(t)\,dt \end{aligned}
(1.11)

and

\begin{aligned}[b]& \bigl(I^{\sigma, \nu, \eta }_{x,\infty}f \bigr) (x)= \frac{1}{\Gamma(\sigma)} \int^{\infty}_{x}(t-x)^{\sigma-1}t^{-\sigma-\nu}{}_{2}F_{1} \biggl(\sigma+\nu,-\eta ; \sigma;1-\frac{x}{t} \biggr)f(t)\,dt, \end{aligned}
(1.12)

where $${}_{2}F_{1}(\cdot)$$, a special case of the generalized hypergeomteric function, is the Gauss hypergeometric function and the function $$f(t)$$ is so constrained that the integrals in Eqs. (1.11) and (1.12) converge.

Remark 2

The Saigo fractional integral operators, given in Eqs. (1.11) and (1.12) can also be written as:

For $$x>0$$, $$\sigma, \nu, \eta\in\mathbb{C}$$

\begin{aligned}[b]& \bigl(I^{\sigma, \nu, \eta }_{0,x}f \bigr) (x)= \biggl(\frac{d}{dx} \biggr)^{k} \bigl(I^{\sigma+k, \nu-k, \eta-k}_{0,x}f \bigr) (x) \\ &\quad \bigl(\Re(\sigma)\leq0; k= \bigl[\Re(-\sigma) \bigr]+1 \bigr) \end{aligned}
(1.13)

and

\begin{aligned}[b]& \bigl(I^{\sigma, \nu, \eta }_{x,\infty}f \bigr) (x)= \biggl(-\frac{d}{dx} \biggr)^{k} \bigl(I^{\sigma-k, \nu-k, \eta}_{x,\infty}f \bigr) (x) \\ &\quad \bigl(\Re(\sigma)\leq0; k= \bigl[\Re(-\sigma ) \bigr]+1 \bigr). \end{aligned}
(1.14)

And the corresponding Saigo fractional differential operators are defined as:

Definition 4

Let $$\sigma, \nu,\eta\in\mathbb{C}$$ and $$x>0$$, then

\begin{aligned}[b] \bigl(D^{\sigma, \nu, \eta }_{0,x}f \bigr) (x)&= \bigl(I^{-\sigma, -\nu, \sigma+\eta}_{0,x}f \bigr) (x) \\ & = \biggl(\frac{d}{dx} \biggr)^{k} \bigl(I^{-\sigma+k, -\nu-k, \sigma+\eta -k}_{0,x}f \bigr) (x) \quad \bigl(\Re(\sigma)>0; k= \bigl[\Re(\sigma) \bigr]+1 \bigr) \end{aligned}
(1.15)

and

\begin{aligned}[b] \bigl(D^{\sigma, \nu, \eta}_{x, \infty}f \bigr) (x)&= \bigl(I^{-\sigma, -\nu, \sigma+\eta}_{x, \infty }f \bigr) (x) \\ & = \biggl(-\frac{d}{dx} \biggr)^{k} \bigl(I^{-\sigma+k, -\nu-k, \sigma+\eta }_{x, \infty}f \bigr) (x) \quad \bigl(\Re(\sigma)>0; k= \bigl[\Re(\sigma) \bigr]+1 \bigr), \end{aligned}
(1.16)

where $$[x]$$ denotes the greatest integer function.

If we take $$\nu=0$$ in Eqs. (1.11), (1.12), (1.15) and (1.16) we get the so-called Erdélyi–Kober fractional integral and derivative operators defined as follows [45, 46].

Definition 5

For $$x>0$$, $$\sigma, \eta\in\mathbb{C}$$ with $$\Re (\sigma)> 0$$ [11, 26]

\begin{aligned}[b]& \bigl(I^{\sigma,\eta}_{0,x}f \bigr) (x)= \frac{x^{-\sigma-\eta}}{\Gamma(\sigma)} \int^{x}_{0}(x-t)^{\sigma -1}t^{\eta}f(t) \,dt \end{aligned}
(1.17)

and

\begin{aligned}[b]& \bigl(I^{\sigma, \eta}_{x,\infty }f \bigr) (x)= \frac{x^{\eta}}{\Gamma(\sigma)} \int^{\infty }_{x}(t-x)^{\sigma-1}t^{-\sigma-\eta}f(t) \,dt, \end{aligned}
(1.18)

provided that the integrals in (1.17) and (1.18) converge.

The corresponding derivative operators are defined as follows.

Definition 6

For $$x>0$$, $$\sigma, \eta\in\mathbb{C}$$ with $$\Re (\sigma)> 0$$ (see [11, 26])

\begin{aligned}[b] \bigl(D^{\sigma, \eta }_{0,x}f \bigr) (x)&=x^{-\eta} \biggl(\frac{d}{dx} \biggr)^{k} \frac{1}{\Gamma (k-\sigma)} \int^{x}_{0}t^{\sigma+\eta}(x-t)^{k-\sigma-1}f(t) \,dt \\ & = \biggl(\frac{d}{dx} \biggr)^{k} \bigl(I^{-\sigma+k, -\sigma, \sigma+\eta -k}_{0,x}f \bigr) (x)\quad \bigl(k= \bigl[\Re(\sigma) \bigr]+1 \bigr) \end{aligned}
(1.19)

and

\begin{aligned}[b] \bigl(D^{\sigma, \eta}_{x,\infty }f \bigr) (x)&=x^{\eta+\sigma} \biggl(\frac{d}{dx} \biggr)^{k} \frac{1}{\Gamma (k-\sigma)} \int^{\infty}_{x}t^{-\eta}(t-x)^{k-\sigma-1}f(t) \,dt \\ & =(-1)^{k} \biggl(\frac{d}{dx} \biggr)^{k} \bigl(I^{-\sigma+k, -\sigma, \sigma +\eta}_{x,\infty}f \bigr) (x)\quad \bigl(k= \bigl[\Re(\sigma) \bigr]+1 \bigr). \end{aligned}
(1.20)

When $$\nu=-\sigma$$, the operators in Eqs. (1.11), (1.12), (1.15) and (1.16) give the Riemann–Liouville and the Weyl fractional integral operators (see [45, 47]) are defined as follows.

Definition 7

For $$x>0$$, $$\sigma\in\mathbb{C}$$ with $$\Re(\sigma )> 0$$

\begin{aligned}[b]& \bigl(I^{\sigma}_{0,x}f \bigr) (x)= \frac{1}{\Gamma(\sigma)} \int^{x}_{0}(x-t)^{\sigma-1}f(t)\,dt \end{aligned}
(1.21)

and

\begin{aligned}[b]& \bigl(I^{\sigma}_{x,\infty}f \bigr) (x)= \frac{1}{\Gamma(\sigma)} \int^{\infty}_{x}(t-x)^{\sigma-1}f(t)\,dt, \end{aligned}
(1.22)

provided both integrals converge.

The corresponding derivative operators are defined as follows.

Definition 8

For $$x>0$$, $$\sigma\in\mathbb{C}$$ with $$\Re(\sigma )> 0$$

\begin{aligned}[b] \bigl(D^{\sigma}_{0,x}f \bigr) (x)&= \biggl(\frac{d}{dx} \biggr)^{k}\frac{1}{\Gamma(k-\sigma)} \int ^{x}_{0}(x-t)^{k-\sigma-1}f(t)\,dt \\ & = \biggl(\frac{d}{dx} \biggr)^{k} \bigl(I^{k-\sigma}_{0,x}f \bigr) (x) \quad\bigl(k= \bigl[\Re(\sigma) \bigr]+1 \bigr) \end{aligned}
(1.23)

and

\begin{aligned}[b] \bigl(D^{\sigma}_{x,\infty}f \bigr) (x)&=(-1)^{k} \biggl(\frac{d}{dx} \biggr)^{k} \frac{1}{\Gamma(k-\sigma)} \int ^{\infty}_{x}(t-x)^{k-\sigma-1}f(t)\,dt \\ & =(-1)^{k} \biggl(\frac{d}{dx} \biggr)^{k} \bigl(I^{k-\sigma}_{x,\infty}f \bigr) (x) \quad\bigl(k= \bigl[\Re(\sigma) \bigr]+1 \bigr). \end{aligned}
(1.24)

For details of such operators along with their properties and applications one may refer to [11, 26, 45, 48, 49].

Power function formulas of the above discussed fractional operators are required for our present study as given in the following lemmas [37, 40, 50].

Lemma 1

Let σ, $$\sigma'$$, ν, $$\nu'$$, η and $$\rho\in\mathbb{C}$$, $$x>0$$ be such that $$\Re(\eta)>0$$; then the following formulas hold true:

\begin{aligned}[b]& \bigl(I^{\sigma, \sigma', \nu, \nu', \eta}_{0,x}t^{\rho-1} \bigr) (x) \\ & \quad=\frac{\Gamma(\rho)\Gamma(\rho +\eta-\sigma-\sigma'-\nu)\Gamma(\rho+\nu'-\sigma')}{\Gamma(\rho+\nu ')\Gamma(\rho+\eta-\sigma-\sigma')\Gamma(\rho+\eta-\sigma'-\nu)}x^{\rho +\eta-\sigma-\sigma'-1} \\ &\qquad \bigl(\Re(\rho)>\max \bigl\{ 0,\Re \bigl(\sigma+\sigma'+\nu -\eta \bigr),\Re \bigl(\sigma'-\nu' \bigr) \bigr\} \bigr) \end{aligned}
(1.25)

and

\begin{aligned}[b]& \bigl(I^{\sigma, \sigma', \nu, \nu', \eta}_{x,\infty}t^{\rho-1} \bigr) (x) \\ &\quad =\frac{\Gamma(1-\rho-\nu )\Gamma(1-\rho-\eta+\sigma+\sigma')\Gamma(1-\rho-\eta+\sigma+\nu ')}{\Gamma(1-\rho)\Gamma(1-\rho-\eta+\sigma+\sigma'+\nu')\Gamma(1-\rho +\sigma-\nu)}x^{\rho+\eta-\sigma-\sigma'-1} \\ & \qquad\bigl(\Re(\rho)< 1+\min \bigl\{ \Re (-\nu),\Re \bigl(\sigma+ \sigma'- \eta \bigr),\Re \bigl(\sigma+\nu'-\eta \bigr) \bigr\} \bigr). \end{aligned}
(1.26)

Lemma 2

Let $$\sigma, \sigma', \nu, \nu', \eta$$ and $$\rho\in\mathbb{C}, x>0$$ be such that $$\Re(\eta)>0$$, then the following formulas hold true:

\begin{aligned}[b]& \bigl(D^{\sigma, \sigma', \nu, \nu', \eta}_{0,x}t^{\rho-1} \bigr) (x) \\ &\quad =\frac{\Gamma(\rho)\Gamma(\rho -\eta+\sigma+\sigma'+\nu')\Gamma(\rho-\nu+\sigma)}{\Gamma(\rho-\nu )\Gamma(\rho-\eta+\sigma+\sigma')\Gamma(\rho-\eta+\sigma+\nu')}x^{\rho -\eta+\sigma+\sigma'-1} \\ &\qquad \bigl(\Re(\rho)>\max \bigl\{ 0,\Re \bigl(\eta-\sigma-\sigma '- \nu' \bigr),\Re(\nu-\sigma) \bigr\} \bigr) \end{aligned}
(1.27)

and

\begin{aligned}[b]& \bigl(D^{\sigma, \sigma', \nu, \nu', \eta}_{x,\infty}t^{\rho-1} \bigr) (x) \\ & \quad=\frac{\Gamma(1-\rho+\nu ')\Gamma(1-\rho+\eta-\sigma-\sigma')\Gamma(1-\rho+\eta-\sigma'-\nu )}{\Gamma(1-\rho)\Gamma(1-\rho+\eta-\sigma-\sigma'-\nu)\Gamma(1-\rho -\sigma'+\nu')}x^{\rho-\eta+\sigma+\sigma'-1} \\ & \qquad\bigl(\Re(\rho)< 1+\min \bigl\{ \Re \bigl(\nu' \bigr),\Re \bigl( \eta- \sigma-\sigma' \bigr),\Re \bigl(\eta-\sigma'-\nu \bigr) \bigr\} \bigr). \end{aligned}
(1.28)

Lemma 3

Let $$\sigma, \nu, \eta, \rho\in\mathbb{C}$$, $$x>0$$ be such that $$\Re(\sigma)>0$$, then the following formulas hold true:

\begin{aligned}[b]& \bigl(I^{\sigma,\nu,\eta }_{0,x}t^{\rho-1} \bigr) (x)=\frac{\Gamma(\rho)\Gamma(\rho+\eta-\nu )}{\Gamma(\rho-\nu)\Gamma(\rho+\eta+\sigma)}x^{\rho-\nu-1} \\ & \quad\bigl(\Re(\rho )>\max \bigl\{ 0,\Re(\nu-\eta) \bigr\} \bigr) \end{aligned}
(1.29)

and

\begin{aligned}[b]& \bigl(I^{\sigma,\nu,\eta}_{x, \infty}t^{\rho-1} \bigr) (x)=\frac{\Gamma(1-\rho+\nu)\Gamma(1-\rho+\eta )}{\Gamma(1-\rho)\Gamma(1-\rho+\eta+\sigma+\nu)}x^{\rho-\nu-1} \\ & \quad\bigl(\Re (\rho)< 1+\min \bigl\{ \Re(\nu),\Re(\eta) \bigr\} \bigr). \end{aligned}
(1.30)

Lemma 4

Let $$\sigma, \nu, \eta, \rho\in\mathbb{C}$$, $$x>0$$ be such that $$\Re(\sigma)>0$$, then the following formulas hold true:

\begin{aligned}[b]& \bigl(D^{\sigma, \nu, \eta }_{0,x}t^{\rho-1} \bigr) (x)=\frac{\Gamma(\rho)\Gamma(\rho+\eta+\sigma +\nu)}{\Gamma(\rho+\eta)\Gamma(\rho+\nu)}x^{\rho+\nu-1} \\ & \quad\bigl(\Re(\rho )>-\min \bigl\{ 0,\Re(\sigma+\nu+\eta) \bigr\} \bigr) \end{aligned}
(1.31)

and

\begin{aligned}[b]& \bigl(D^{\sigma, \nu, \eta }_{x,\infty}t^{\rho-1} \bigr) (x)=\frac{\Gamma(1-\rho-\nu)\Gamma(1-\rho +\sigma+\eta)}{\Gamma(1-\rho+\eta-\nu)\Gamma(1-\rho)}x^{\rho+\nu-1} \\ &\quad \bigl(\Re(\rho)< 1+\min \bigl\{ \Re(-\nu-n),\Re(\eta+\sigma) \bigr\} \textit{ and } n= \bigl[\Re (\sigma) \bigr]+1 \bigr). \end{aligned}
(1.32)

Lemma 5

Let $$\sigma, \eta, \rho\in\mathbb{C}$$, $$x>0$$ be such that $$\Re(\sigma)>0$$, then the following formulas hold true:

\begin{aligned}[b]& \bigl(I^{\sigma,\eta }_{0,x}t^{\rho-1} \bigr) (x)=\frac{\Gamma(\rho+\eta)}{\Gamma(\rho+\eta +\sigma)}x^{\rho-1} \\ &\quad \bigl(\Re(\rho)>-\Re(\eta) \bigr) \end{aligned}
(1.33)

and

\begin{aligned}[b]& \bigl(I^{\sigma,\eta}_{x, \infty }t^{\rho-1} \bigr) (x)=\frac{\Gamma(1-\rho+\eta)}{\Gamma(1-\rho+\eta +\sigma)}x^{\rho-1} \\ &\quad \bigl(\Re(\rho)< 1+\Re(\eta) \bigr). \end{aligned}
(1.34)

Lemma 6

Let $$\sigma, \eta, \rho\in\mathbb{C}$$, $$x>0$$ be such that $$\Re(\sigma)>0$$, then the following formulas hold true:

\begin{aligned}[b]& \bigl(D^{\sigma, \eta }_{0,x}t^{\rho-1} \bigr) (x)=\frac{\Gamma(\rho+\eta+\sigma)}{\Gamma(\rho +\eta)}x^{\rho-1} \\ & \quad\bigl(\Re(\rho)>-\Re(\eta+\sigma) \bigr) \end{aligned}
(1.35)

and

\begin{aligned}[b]& \bigl(D^{\sigma, \eta}_{x,\infty }t^{\rho-1} \bigr) (x)=\frac{\Gamma(1-\rho+\sigma+\eta)}{\Gamma(1-\rho +\eta)}x^{\rho-1} \\ & \quad\bigl(\Re(\rho)< 1+\Re(\eta+\sigma)-n \textit{ and } n= \bigl[\Re (\sigma) \bigr]+1 \bigr). \end{aligned}
(1.36)

Lemma 7

Let $$\sigma, \rho\in\mathbb{C}$$, $$x>0$$ be such that $$\Re(\sigma)>0$$, then the following formulas hold true:

\begin{aligned}[b]& \bigl(I^{\sigma}_{0,x}t^{\rho -1} \bigr) (x)=\frac{\Gamma(\rho)}{\Gamma(\rho+\sigma)}x^{\rho+\sigma -1} \\ &\quad \bigl(\Re(\rho)>0 \bigr) \end{aligned}
(1.37)

and

\begin{aligned}[b]& \bigl(I^{\sigma}_{x, \infty }t^{\rho-1} \bigr) (x)=\frac{\Gamma(1-\rho-\sigma)}{\Gamma(1-\rho )}x^{\rho+\sigma-1} \\ &\quad \bigl(0< \Re(\sigma)< 1-\Re(\rho) \bigr). \end{aligned}
(1.38)

Lemma 8

Let $$\sigma, \rho\in\mathbb{C}$$, $$x>0$$ be such that $$\Re(\sigma)>0$$, then the following formulas hold true:

\begin{aligned}[b]& \bigl(D^{\sigma}_{0,x}t^{\rho -1} \bigr) (x)=\frac{\Gamma(\rho)}{\Gamma(\rho-\sigma)}x^{\rho-\sigma -1} \\ & \quad\bigl(\Re(\rho)>\Re(\sigma)>0 \bigr) \end{aligned}
(1.39)

and

\begin{aligned}[b]& \bigl(D^{\sigma}_{x,\infty }t^{\rho-1} \bigr) (x)=\frac{\Gamma(1-\rho+\sigma)}{\Gamma(1-\rho )}x^{\rho-\sigma-1} \\ & \quad\bigl(\Re(\rho)< 1+\Re(\sigma)-n \textit{ and } n= \bigl[\Re (\sigma) \bigr]+1 \bigr). \end{aligned}
(1.40)

Mathieu series and its generalizations

In 1890 Mathieu introduced and investigated the infinite series of the form

\begin{aligned}[b]& S(r)=\sum_{n=1}^{\infty} \frac {2n}{(n^{2}+r^{2})^{2}} \quad\bigl(r\in\mathbb{R}^{+} \bigr), \end{aligned}
(2.1)

in his work  on elasticity of solid bodies; it is known as the Mathieu series.

Integral representations of $$S(r)$$ are given by (see [52, 53])

\begin{aligned}[b]& S(r)=\frac{1}{r} \int_{0}^{\infty }\frac{x\sin(rx)}{e^{x}-1}\,dx \quad\bigl(r\in \mathbb{R}^{+} \bigr). \end{aligned}
(2.2)

A generalized form of the Mathieu series with a fractional power is defined as

\begin{aligned}[b]& S_{\mu}(r)=\sum _{n=1}^{\infty} \frac {2n}{(n^{2}+r^{2})^{\mu}}\quad \bigl(r\in \mathbb{R}^{+}; \mu>1 \bigr), \end{aligned}
(2.3)

and it has been extensively studied by Cerone and Lenard , Diananda , Tomovski and Trencevski  and Pogány et al. .

Recently, Tomovski and Pogány  studied the several integral representations of the generalized fractional order Mathieu-type power series (see also )

\begin{aligned}[b]& S_{\mu}(r;z)=\sum _{n \geq1} \frac {2n z^{n}}{(n^{2}+r^{2})^{\mu+1}}\quad \bigl(\mu>0, r\in \mathbb{R}^{+}, |z|< 1 \bigr) \end{aligned}
(2.4)

and

\begin{aligned}[b]& S_{\mu}(r;1)=S_{\mu}(r). \end{aligned}
(2.5)

Srivastava and Tomovski in  defined a family of more generalized Mathieu series as

\begin{aligned}[b]& S_{\mu}^{(\alpha, \beta )}(r;a)=S_{\mu}^{(\alpha, \beta)} \bigl(r;\{a_{n}\}_{n=1}^{\infty} \bigr)=\sum _{n=1}^{\infty}\frac{2a_{n}^{\beta}}{(a_{n}^{\alpha}+r^{2})^{\mu}} \\ &\quad \bigl(r,\alpha,\beta,\mu\in\mathbb{R}^{+} \bigr), \end{aligned}
(2.6)

where the positive sequence

\begin{aligned}[b]& a=\{a_{n}\}_{n=1}^{\infty}= \{a_{1}, a_{2}, a_{3},\ldots\}\quad \Bigl(\lim _{n\to\infty} a_{n}=\infty \Bigr)\end{aligned}
(2.7)

is so chosen that the infinite series

$$\sum_{n=1}^{\infty}\frac{1}{a_{n}^{\mu\alpha-\beta}}$$

is convergent.

Also from Eqs. (2.1), (2.3) and (2.6), we see that

$$\begin{gathered} S_{2}(r)=S(r), \\ S_{\mu}(r)=S_{\mu}^{(2, 1)} \bigl(r;\{n \} _{n=1}^{\infty} \bigr), \end{gathered}$$

and furthermore the special cases

\begin{aligned}[b] S_{\mu}^{(2, 1)} \bigl(r;\{n \}_{n=1}^{\infty } \bigr)=S_{\mu}(r),\qquad S_{\mu}^{(2, 1)} \bigl(r; \bigl\{ n^{\gamma} \bigr\} _{n=1}^{\infty} \bigr) \quad\text{and}\quad S_{\mu}^{(\alpha, \alpha/2)} \bigl(r;\{n\} _{n=1}^{\infty} \bigr),\end{aligned}

of the Mathieu series were investigated by Cerone and Lenard , Diananda  and Tomovski . For more details one may refer to [53, 56, 57, 59, 6164].

Recently, Tomovski and Mehrez , considered a power series defined as

\begin{aligned}[b]& S_{\mu,\lambda}^{(\alpha, \beta )}(r,a;z)=S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,\{a_{n}\}_{n=1}^{\infty };z \bigr)=\sum _{n=1}^{\infty}\frac{2a_{n}^{\beta}(\lambda )_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}}\frac{z^{n}}{n!} \\ &\quad \bigl(r,\alpha,\beta,\mu \in\mathbb{R}^{+}; |z|\leq1 \bigr) \end{aligned}
(2.8)

and

\begin{aligned}& S_{\mu,\lambda}^{(\alpha, \beta )}(r,a;1)=S_{\mu,\lambda}^{(\alpha, \beta)} (r,a ), \end{aligned}
(2.9)
\begin{aligned}& S_{\mu,1}^{(\alpha, \beta )}(r,a;1)=S_{\mu}^{(\alpha, \beta)} (r,a ). \end{aligned}
(2.10)

The concept of the Hadamard product (or the convolution) of two analytic functions is very useful in our present study. It can help us to decompose a newly emerging function into two known functions. Let

\begin{aligned} f(z):=\sum_{n=0}^{\infty}a_{n}z^{n} \quad\bigl(|z|< R_{f}\bigr) \end{aligned}
(2.11)

and

\begin{aligned} g(z):=\sum_{n=0}^{\infty}b_{n}z^{n} \quad\bigl(|z|< R_{g}\bigr) \end{aligned}
(2.12)

be two power series whose radii of convergence are denoted by $$R_{f}$$ and $$R_{g}$$, respectively. Then their Hadamard product is the power series defined by

\begin{aligned} (f*g) (z):=\sum_{n=0}^{\infty}a_{n} b_{n}z^{n}=(g*f) (z) \quad\bigl(|z|< R\bigr), \end{aligned}
(2.13)

where

\begin{aligned}[b]& R=\lim_{n \to\infty} \biggl\vert \frac {a_{n} b_{n}}{a_{n+1} b_{n+1}} \biggr\vert = \biggl(\lim_{n \to\infty} \biggl\vert \frac{a_{n}}{a_{n+1}} \biggr\vert \biggr). \biggl(\lim_{n \to\infty} \biggl\vert \frac{b_{n}}{b_{n+1}} \biggr\vert \biggr)=R_{f}.R_{g}, \end{aligned}
(2.14)

therefore, in general, we have $$R\geq R_{f} . R_{g}$$ [66, 67]. For various investigations involving the Hadamard product (or the convolution), the interested reader may refer to recent papers on the subject (see, for example, [68, 69] and the references cited therein).

Also we require the Fox–Wright function $${}_{p}\Psi_{q}(z)$$ ($$p,q\in \mathbb{N}_{0}$$) with p numerator and q denominator parameters defined for $$a_{1},\ldots,a_{p} \in\mathbb{C}$$ and $$b_{1},\ldots,b_{q} \in \mathbb{C} \setminus\mathbb{Z}_{0}^{-}$$ by (for details see [11, 26, 44, 45])

\begin{aligned}[b]&{}_{p} \Psi_{q}\left [ \textstyle\begin{array}{c}(a_{1},\alpha_{1}),\ldots, (a_{p},\alpha_{p});\\ (b_{1},\beta_{1}),\ldots, (b_{q},\beta_{q}); \end{array}\displaystyle z \right ]=\sum^{\infty}_{n=0} \frac{\Gamma(a_{1}+\alpha_{1} n)\cdots\Gamma (a_{p}+\alpha_{p} n)}{\Gamma(b_{1}+\beta_{1} n)\cdots\Gamma(b_{q}+\beta_{q} n)}\frac {z^{n}}{n!}, \end{aligned}
(2.15)

where the coefficients $$\alpha_{1},\ldots,\alpha_{p}, \beta_{1},\ldots,\beta _{q}\in\mathbb{R}^{+}$$ are such that

\begin{aligned}[b]& 1+\sum^{q}_{j=1} \beta_{j}-\sum^{p}_{i=1} \alpha_{i}\geq0.\end{aligned}
(2.16)

For $$\alpha_{i}=\beta_{j}=1$$ ($$i=1,\ldots,p$$; $$j=1,\ldots,q$$), Eq. (2.15) reduces immediately to the generalized hypergeometric function $${}_{p}F_{q}$$ ($$p,q\in\mathbb{N}_{0}$$) (see ):

\begin{aligned}[b]&{}_{p} F_{q}\left [ \textstyle\begin{array}{c}a_{1},\ldots,a_{p};\\ b_{1},\ldots,b_{q}; \end{array}\displaystyle z \right ]=\frac{\Gamma(b_{1})\cdots\Gamma(b_{q})}{\Gamma(a_{1})\cdots\Gamma (a_{q})}{}_{p} \Psi_{q}\left [ \textstyle\begin{array}{c}(a_{1},1),\ldots, (a_{p},1);\\ (b_{1},1),\ldots, (b_{q},1); \end{array}\displaystyle z \right ]. \end{aligned}
(2.17)

Fractional integration of extended generalized Mathieu series

In this section, we present certain fractional integral formulas involving the extended generalized Mathieu series $$S_{\mu,\lambda }^{(\alpha, \beta)}(r,a;z)$$ by using fractional integral operators.

Theorem 1

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho+\xi n)>\max\{0,\Re(\sigma+\sigma'+\nu-\eta ),\Re(\sigma'-\nu')\}$$ then the following fractional integral formula holds true:

\begin{aligned}[b]& \bigl(I^{\sigma,\sigma',\nu,\nu ',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta )} \bigl(r,a;t^{\xi} \bigr) \bigr\} \bigr) (x)\\ &\quad=x^{\rho+\eta-\sigma-\sigma'-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;x^{\xi} \bigr) \\ &\qquad{}*{}_{3}\Psi_{3} \left [ \textstyle\begin{array}{c}(\rho,\xi), (\rho+\eta-\sigma-\sigma'-\nu, \xi), (\rho +\nu'-\sigma', \xi); \\ (\rho+\nu', \xi), (\rho+\eta-\sigma-\sigma', \xi ), (\rho+\eta-\sigma'-\nu, \xi); \end{array}\displaystyle x^{\xi} \right ]. \end{aligned}
(3.1)

Proof

Using the definition (2.8) and then interchanging the order of integration and summation, we get

\begin{aligned}[b] \bigl(I^{\sigma,\sigma',\nu,\nu ',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta )} \bigl(r,a;t^{\xi} \bigr) \bigr\} \bigr) (x)={}&\sum_{n=1}^{\infty} \frac{2a_{n}^{\beta }(\lambda)_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}}\frac{1}{n!} \\ & \times \bigl(I^{\sigma,\sigma',\nu,\nu',\eta}_{0,x}t^{\rho+\xi n-1} \bigr) (x), \end{aligned}
(3.2)

applying the result (1.25), Eq. (3.2) reduces to

\begin{aligned}[b]& \bigl(I^{\sigma,\sigma',\nu,\nu ',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta )} \bigl(r,a;t^{\xi} \bigr) \bigr\} \bigr) (x)\\&\quad=\sum_{n=1}^{\infty} \frac{2a_{n}^{\beta }(\lambda)_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}}\frac{1}{n!} \\ & \qquad{}\times\frac {\Gamma(\rho+\xi n)\Gamma(\rho+\xi n+\eta-\sigma-\sigma'-\nu)\Gamma(\rho +\xi n+\nu'-\sigma')}{\Gamma(\rho+\xi n+\nu')\Gamma(\rho+\xi n+\eta -\sigma-\sigma')\Gamma(\rho+\xi n+\eta-\sigma'-\nu)} \\ &\qquad{} \times x^{\rho +\xi n+\eta-\sigma-\sigma'-1}, \end{aligned}
(3.3)

after a little simplification, Eq. (3.3) reduces to

\begin{aligned}[b]& \bigl(I^{\sigma,\sigma',\nu,\nu ',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta )} \bigl(r,a;t^{\xi} \bigr) \bigr\} \bigr) (x)\\ &\quad=x^{\rho+\eta-\sigma-\sigma'-1}\sum_{n=1}^{\infty} \frac{2a_{n}^{\beta}(\lambda)_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu }} \\ &\qquad{} \times\frac{\Gamma(\rho+\xi n)\Gamma(\rho+\eta-\sigma-\sigma'-\nu +\xi n)\Gamma(\rho+\nu'-\sigma'+\xi n)}{\Gamma(\rho+\nu'+\xi n)\Gamma (\rho+\eta-\sigma-\sigma'+\xi n)\Gamma(\rho+\eta-\sigma'-\nu+\xi n)}\frac{x^{\xi n}}{n!}. \end{aligned}
(3.4)

By applying the Hadamard product (2.13) in Eq. (3.4), which, in view of (2.8) and (2.15), gives the required result (3.1). □

Theorem 2

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta, \mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho-\xi n)<1+\min\{\Re(-\nu),\Re(\sigma+\sigma '-\eta),\Re(\sigma+\nu'-\eta)\}$$ then the following fractional integral formula holds true:

\begin{aligned}[b]& \biggl(I^{\sigma,\sigma',\nu,\nu ',\eta}_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta )} \biggl(r,a;\frac{1}{t^{\xi}} \biggr) \biggr\} \biggr) (x)\\&\quad=x^{\rho+\eta -\sigma-\sigma'-1} S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a;\frac {1}{x^{\xi}} \biggr) \\ & \qquad{}*{}_{3}\Psi_{3} \left [ \textstyle\begin{array}{c}(1-\rho-\nu,\xi), (1-\rho-\eta+\sigma+\sigma', \xi), (1-\rho-\eta+\sigma+\nu', \xi); \\ (1-\rho, \xi), (1-\rho-\eta+\sigma +\sigma'+\nu', \xi), (1-\rho+\sigma-\nu, \xi); \end{array}\displaystyle \frac{1}{x^{\xi}} \right ]. \end{aligned}
(3.5)

Proof

The proof of Theorem 2 is similar to that of Theorem 1. □

Special cases

Here we present some special cases by choosing suitable values of the parameters σ, $$\sigma'$$, ν, $$\nu'$$ and η. If we put $$\sigma=\sigma+\nu$$, $$\sigma'=\nu'=0$$, $$\nu=-\eta$$, $$\eta=\sigma$$ in Theorems 1 and 2, we get certain interesting results concerning the Saigo fractional integral operators given by the following corollaries.

Corollary 1

Let $$x>0$$, $$\sigma, \nu, \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho+\xi n)>\max\{0,\Re(\nu-\eta)\}$$ then the following fractional integral formula holds true:

\begin{aligned}[b]& \bigl(I^{\sigma,\nu,\eta }_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;t^{\xi } \bigr) \bigr\} \bigr) (x) \\ &\quad =x^{\rho-\nu-1}S_{\mu,\lambda}^{(\alpha, \beta )} \bigl(r,a;x^{\xi} \bigr)*{}_{2}\Psi_{2} \left [ \textstyle\begin{array}{c}(\rho, \xi), (\rho+\eta-\nu, \xi); \\ (\rho-\nu, \xi), (\rho+\eta+\sigma, \xi); \end{array}\displaystyle x^{\xi} \right ]. \end{aligned}
(3.6)

Corollary 2

Let $$x>0$$, $$\sigma, \nu, \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho-\xi n)<1+\min\{\Re(\nu),\Re(\eta)\}$$ then the following fractional integral formula holds true:

\begin{aligned}[b]& \biggl(I^{\sigma,\nu,\eta }_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a;\frac{1}{t^{\xi}} \biggr) \biggr\} \biggr) (x) \\ & \quad=x^{\rho-\nu-1}S_{\mu ,\lambda}^{(\alpha, \beta)} \biggl(r,a; \frac{1}{x^{\xi}} \biggr)*{}_{2}\Psi _{2} \left [ \textstyle\begin{array}{c}(1-\rho+\nu,\xi), (1-\rho+\eta, \xi); \\ (1-\rho, \xi ), (1-\rho+\sigma+\nu+\eta, \xi); \end{array}\displaystyle \frac{1}{x^{\xi}} \right ]. \end{aligned}
(3.7)

Further, if we put $$\nu=0$$ in (3.6) and (3.7) then these Saigo fractional integrals reduce to the following Erdélyi–Kober type fractional integral operators.

Corollary 3

Let $$x>0$$, $$\sigma, \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho+\xi n)>-\Re(\eta)$$ then the following fractional integral formula holds true:

\begin{aligned}[b]& \bigl(I^{\sigma, \eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;t^{\xi} \bigr) \bigr\} \bigr) (x) \\ &\quad =x^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;x^{\xi } \bigr)*{}_{1}\Psi_{1} \left [ \textstyle\begin{array}{c}(\rho+\eta, \xi); \\ (\rho+\sigma+\eta, \xi); \end{array}\displaystyle x^{\xi} \right ]. \end{aligned}
(3.8)

Corollary 4

Let $$x>0$$, $$\sigma, \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho-\xi n)<1+\Re(\eta)$$ then the following fractional integral formula holds true:

\begin{aligned}[b]& \biggl(I^{\sigma,\eta}_{x,\infty } \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a;\frac {1}{t^{\xi}} \biggr) \biggr\} \biggr) (x) \\ & \quad=x^{\rho-1}S_{\mu,\lambda }^{(\alpha, \beta)} \biggl(r,a; \frac{1}{x^{\xi}} \biggr)*{}_{1}\Psi_{1} \left [ \textstyle\begin{array}{c}(1-\rho+\eta, \xi); \\ (1-\rho+\sigma+\eta, \xi); \end{array}\displaystyle \frac{1}{x^{\xi}} \right ]. \end{aligned}
(3.9)

Further, if we put $$\nu=-\sigma$$ in (3.6) and (3.7), then these Saigo fractional integrals reduce to the Riemann–Liouville and the Weyl type fractional integral operators as given in the following results.

Corollary 5

Let $$x>0$$, $$\sigma, \rho\in\mathbb{C}$$ and $$r,\alpha ,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho+\xi n)>0$$ then the following fractional integral formula holds true:

\begin{aligned}[b]& \bigl(I^{\sigma}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;t^{\xi} \bigr) \bigr\} \bigr) (x) \\ &\quad =x^{\rho+\sigma-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;x^{\xi } \bigr)*{}_{1}\Psi_{1} \left [ \textstyle\begin{array}{c}(\rho, \xi); \\ (\rho+\sigma, \xi); \end{array}\displaystyle x^{\xi} \right ]. \end{aligned}
(3.10)

Corollary 6

Let $$x>0$$, $$\sigma, \rho\in\mathbb{C}$$ and $$r,\alpha ,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$1-\Re(\rho-\xi n)>\Re(\sigma)>0$$ then the following fractional integral formula holds true:

\begin{aligned}[b]& \biggl(I^{\sigma}_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a;\frac{1}{t^{\xi }} \biggr) \biggr\} \biggr) (x) \\ & \quad=x^{\rho+\sigma-1}S_{\mu,\lambda }^{(\alpha, \beta)} \biggl(r,a; \frac{1}{x^{\xi}} \biggr)*{}_{1}\Psi_{1} \left [ \textstyle\begin{array}{c}(1-\rho-\sigma, \xi); \\ (1-\rho, \xi); \end{array}\displaystyle \frac{1}{x^{\xi}} \right ]. \end{aligned}
(3.11)

If we put $$\xi=1$$ in (3.1), (3.5), (3.6), (3.7), (3.8), (3.9), (3.10) and (3.11) then we get the following results.

Corollary 7

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta, \rho\in \mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho+n)>\max\{0,\Re(\sigma+\sigma'+\nu-\eta),\Re (\sigma'-\nu')\}$$ then the following fractional integral formula holds true:

\begin{aligned}[b]& \bigl(I^{\sigma,\sigma',\nu,\nu ',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta )}(r,a;t) \bigr\} \bigr) (x) \\ &\quad =x^{\rho+\eta-\sigma-\sigma'-1}\frac {\Gamma(\rho)\Gamma(\rho+\eta-\sigma-\sigma'-\nu)\Gamma(\rho+\nu'-\sigma ')}{\Gamma(\rho+\nu')\Gamma(\rho+\eta-\sigma-\sigma')\Gamma(\rho+\eta -\sigma'-\nu)}S_{\mu,\lambda}^{(\alpha, \beta )}(r,a;x) \\ &\qquad{}*{}_{3}F_{3} \left [ \textstyle\begin{array}{c}\rho, \rho+\eta-\sigma-\sigma'-\nu, \rho+\nu'-\sigma'; \\ \rho+\nu', \rho+\eta-\sigma-\sigma', \rho+\eta-\sigma'-\nu; \end{array}\displaystyle x^{\xi} \right ]. \end{aligned}
(3.12)

Corollary 8

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta, \rho\in \mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho-n)<1+\min\{\Re(-\nu),\Re(\sigma+\sigma'-\eta ),\Re(\sigma+\nu'-\eta)\}$$ then the following fractional integral formula holds true:

\begin{aligned}[b]& \biggl(I^{\sigma,\sigma',\nu,\nu ',\eta}_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta )} \biggl(r,a;\frac{1}{t} \biggr) \biggr\} \biggr) (x) \\ & \quad=x^{\rho+\eta-\sigma -\sigma'-1}\frac{\Gamma(1-\rho-\nu)\Gamma(1-\rho-\eta+\sigma+\sigma ')\Gamma(1-\rho-\eta+\sigma+\nu')}{\Gamma(1-\rho)\Gamma(1-\rho-\eta +\sigma+\sigma'+\nu')\Gamma(1-\rho+\sigma-\nu)}S_{\mu,\lambda }^{(\alpha, \beta)} \biggl(r,a;\frac{1}{x} \biggr) \\ &\qquad{}*{}_{3}F_{3} \left [ \textstyle\begin{array}{c}1-\rho-\nu, 1-\rho-\eta+\sigma+\sigma', 1-\rho-\eta +\sigma+\nu'; \\ 1-\rho, 1-\rho-\eta+\sigma+\sigma'+\nu', 1-\rho+\sigma -\nu; \end{array}\displaystyle \frac{1}{x} \right ]. \end{aligned}
(3.13)

Corollary 9

Let $$x>0$$, $$\sigma, \nu, \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho+n)>\max\{0,\Re(\nu-\eta)\}$$ then the following fractional integral formula holds true:

\begin{aligned}[b]& \bigl(I^{\sigma,\nu,\eta }_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)}(r,a;t) \bigr\} \bigr) (x)\\&\quad=x^{\rho-\nu-1}\frac{\Gamma(\rho)\Gamma(\rho+\eta-\nu)}{\Gamma (\rho-\nu)\Gamma(\rho+\eta+\sigma)} S_{\mu,\lambda}^{(\alpha, \beta)}(r,a;x)*{}_{2}F_{2} \left [ \textstyle\begin{array}{c}\rho, \rho+\eta-\nu; \\ \rho-\nu, \rho+\eta+\sigma; \end{array}\displaystyle x \right ]. \end{aligned}
(3.14)

Corollary 10

Let $$x>0$$, $$\sigma, \nu, \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho-n)<1+\min\{\Re(\nu),\Re(\eta)\}$$ then the following fractional integral formula holds true:

\begin{aligned}[b]& \biggl(I^{\sigma,\nu,\eta }_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a;\frac{1}{t} \biggr) \biggr\} \biggr) (x)\\&\quad=x^{\rho-\nu-1}\frac{\Gamma (1-\rho+\nu)\Gamma(1-\rho+\eta)}{\Gamma(1-\rho)\Gamma(1-\rho+\sigma+\nu +\eta)} S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a;\frac {1}{x} \biggr)\\ &\qquad{}*{}_{2}F_{2} \left [ \textstyle\begin{array}{c}1-\rho+\nu, 1-\rho+\eta; \\ 1-\rho, 1-\rho+\sigma+\nu +\eta; \end{array}\displaystyle \frac{1}{x} \right ]. \end{aligned}
(3.15)

Corollary 11

Let $$x>0$$, $$\sigma, \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho+n)>-\Re(\eta)$$ then the following fractional integral formula holds true:

\begin{aligned}[b]& \bigl(I^{\sigma, \eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)}(r,a;t) \bigr\} \bigr) (x) \\ & \quad=x^{\rho-1}\frac{\Gamma(\rho+\eta)}{\Gamma(\rho+\sigma+\eta)} S_{\mu ,\lambda}^{(\alpha, \beta)}(r,a;x)*{}_{1}F_{1} \left [ \textstyle\begin{array}{c}\rho+\eta; \\ \rho+\sigma+\eta; \end{array}\displaystyle x \right ]. \end{aligned}
(3.16)

Corollary 12

Let $$x>0$$, $$\sigma, \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho-n)<1+\Re(\eta)$$ then the following fractional integral formula holds true:

\begin{aligned}[b]& \biggl(I^{\sigma,\eta}_{x,\infty } \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a;\frac {1}{t} \biggr) \biggr\} \biggr) (x) \\ &\quad =x^{\rho-1}\frac{\Gamma(1-\rho+\eta )}{\Gamma(1-\rho+\sigma+\eta)}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a; \frac{1}{x} \biggr)*{}_{1}F_{1} \left [ \textstyle\begin{array}{c}1-\rho+\eta; \\ 1-\rho+\sigma+\eta; \end{array}\displaystyle \frac{1}{x} \right ]. \end{aligned}
(3.17)

Corollary 13

Let $$x>0$$, $$\sigma, \rho\in\mathbb{C}$$ and $$r,\alpha ,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho+n)>0$$ then the following fractional integral formula holds true:

\begin{aligned}[b]& \bigl(I^{\sigma}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)}(r,a;t) \bigr\} \bigr) (x) \\ &\quad =x^{\rho+\sigma-1}\frac{\Gamma(\rho)}{\Gamma(\rho+\sigma)}S_{\mu,\lambda }^{(\alpha, \beta)}(r,a;x)*{}_{1}F_{1} \left [ \textstyle\begin{array}{c}\rho; \\ \rho+\sigma; \end{array}\displaystyle x \right ]. \end{aligned}
(3.18)

Corollary 14

Let $$x>0$$, $$\sigma, \rho\in\mathbb{C}$$ and $$r,\alpha ,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$1-\Re(\rho-n)>\Re(\sigma)>0$$ then the following fractional integral formula holds true:

\begin{aligned}[b]& \biggl(I^{\sigma}_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a;\frac{1}{t} \biggr) \biggr\} \biggr) (x) \\ & \quad=x^{\rho+\sigma-1}\frac{\Gamma(1-\rho-\sigma )}{\Gamma(1-\rho)}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a; \frac {1}{x} \biggr)*{}_{1}F_{1} \left [ \textstyle\begin{array}{c}1-\rho-\sigma; \\ 1-\rho; \end{array}\displaystyle \frac{1}{x} \right ]. \end{aligned}
(3.19)

Fractional differentiation of extended generalized Mathieu series

In this section we present certain fractional differential formulas involving the extended generalized Mathieu series $$S_{\mu,\lambda }^{(\alpha, \beta)}(r,a;z)$$ by using fractional differential operators.

Theorem 3

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho+\xi n)>\max\{0,\Re(\eta-\sigma-\sigma'-\nu '),\Re(\nu-\sigma)\}$$ then the following fractional derivative formula holds true:

\begin{aligned}[b]& \bigl(D^{\sigma,\sigma',\nu,\nu ',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta )} \bigl(r,a;t^{\xi} \bigr) \bigr\} \bigr) (x)\\ &\quad=x^{\rho-\eta+\sigma+\sigma'-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;x^{\xi} \bigr) \\ & \qquad *{}_{3}\Psi_{3} \left [ \textstyle\begin{array}{c} (\rho, \xi), (\rho-\eta+\sigma+\sigma'+\nu', \xi), (\rho-\nu+\sigma, \xi); \\ (\rho-\nu, \xi), (\rho-\eta+\sigma+\sigma', \xi), (\rho-\eta+\sigma+\nu', \xi); \end{array}\displaystyle x^{\xi} \right ]. \end{aligned}
(4.1)

Proof

For convenience, we denote the left-hand side of the result (4.1) by $$\mathscr{D}$$. Then by using (2.8) and then changing the order of differentiation and summation, we get

\begin{aligned}[b]& \mathscr{D}=\sum_{n=1}^{\infty } \frac{2a_{n}^{\beta}(\lambda)_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}}\frac {1}{n!} \bigl(D^{\sigma,\sigma',\nu,\nu',\eta}_{0,x}t^{\rho+\xi n-1} \bigr) (x) .\end{aligned}
(4.2)

Applying the result (1.27), Eq. (4.2) reduces to

\begin{aligned}[b]& \mathscr{D}=\sum_{n=1}^{\infty } \frac{2a_{n}^{\beta}(\lambda)_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}}\frac {1}{n!} \\ & \quad{}\times\frac{\Gamma(\rho+\xi n)\Gamma(\rho-\eta+\sigma+\sigma '+\nu'+\xi n)\Gamma(\rho-\nu+\sigma+\xi n)}{\Gamma(\rho-\nu+\xi n)\Gamma (\rho-\eta+\sigma+\sigma'+\xi n)\Gamma(\rho-\eta+\sigma+\nu'+\xi n)} \\ &\quad{} \times x^{\rho+\xi n-\eta+\sigma+\sigma'-1}; \end{aligned}
(4.3)

after simplification, Eq. (4.3) reduces to

\begin{aligned}[b]& \mathscr{D}=x^{\rho-\eta+\sigma +\sigma'-1}\sum _{n=1}^{\infty}\frac{2a_{n}^{\beta}(\lambda )_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}} \\ & \quad{}\times\frac{\Gamma(\rho+\xi n)\Gamma (\rho-\eta+\sigma+\sigma'+\nu'+\xi n)\Gamma(\rho-\nu+\sigma+\xi n)}{\Gamma(\rho-\nu+\xi n)\Gamma(\rho-\eta+\sigma+\sigma'+\xi n)\Gamma (\rho-\eta+\sigma+\nu'+\xi n)}\frac{x^{\xi n}}{n!},\end{aligned}
(4.4)

and interpreting the above equation, from the point of view of (2.8), (2.13) and (2.15), we have the required result. □

Theorem 4

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho-\xi n)<1+\min\{\Re(\nu'),\Re(\eta-\sigma -\sigma'),\Re(\eta-\sigma'-\nu)\}$$ then the following fractional derivative formula holds true:

\begin{aligned}[b]& \biggl(D^{\sigma,\sigma',\nu,\nu ',\eta}_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta )} \biggl(r,a;\frac{1}{t^{\xi}} \biggr) \biggr\} \biggr) (x)\\ &\quad=x^{\rho-\eta +\sigma+\sigma'-1}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a;\frac {1}{x^{\xi}} \biggr) \\ &\qquad{} *{}_{3}\Psi_{3} \left [ \textstyle\begin{array}{c} (1-\rho+\nu', \xi), (1-\rho+\eta-\sigma-\sigma', \xi ), (1-\rho+\eta-\sigma'-\nu, \xi); \\ (1-\rho, \xi), (1-\rho+\eta-\sigma -\sigma'-\nu, \xi), (1-\rho-\sigma'+\nu', \xi); \end{array}\displaystyle \frac{1}{x^{\xi}} \right ]. \end{aligned}
(4.5)

Proof

The proof of Theorem 4 is similar to that of Theorem 3. □

Special cases

Here we present some special cases by choosing suitable values of the parameters σ, $$\sigma'$$, ν, $$\nu'$$ and η. If we put $$\sigma=\sigma+\nu$$, $$\sigma'=\nu'=0$$, $$\nu=-\eta$$, $$\eta=\sigma$$ in Theorems 3 and 4, we get certain interesting results concerning the Saigo fractional differential operator given in the following corollaries.

Corollary 15

Let $$x>0$$, $$\sigma, \nu, \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho+\xi n)>-\min\{0,\Re(\sigma+\nu+\eta)\}$$ then the following fractional derivative formula holds true:

\begin{aligned}[b]& \bigl(D^{\sigma,\nu,\eta }_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;t^{\xi } \bigr) \bigr\} \bigr) (x) \\ &\quad =x^{\rho+\nu-1}S_{\mu,\lambda}^{(\alpha, \beta )} \bigl(r,a;x^{\xi} \bigr)*{}_{2}\Psi_{2} \left [ \textstyle\begin{array}{c}(\rho, \xi), (\rho+\eta+\sigma+\nu, \xi); \\ (\rho+\eta , \xi), (\rho+\nu, \xi); \end{array}\displaystyle x^{\xi} \right ]. \end{aligned}
(4.6)

Corollary 16

Let $$x>0,\sigma, \nu, \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}; |1/t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho-\xi n)<1+\min\{\Re(-\nu-n^{*}),\Re(\eta +\sigma)\}$$ where $$n^{*}=[\Re(\sigma)]+1$$ then the following fractional derivative formula holds true:

\begin{aligned}[b]& \biggl(D^{\sigma,\nu,\eta }_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a;\frac{1}{t^{\xi}} \biggr) \biggr\} \biggr) (x) \\ & \quad=x^{\rho+\nu-1}S_{\mu ,\lambda}^{(\alpha, \beta)} \biggl(r,a; \frac{1}{x^{\xi}} \biggr)*{}_{2}\Psi _{2} \left [ \textstyle\begin{array}{c}(1-\rho-\nu, \xi), (1-\rho+\sigma+\eta, \xi); \\ (1-\rho, \xi), (1-\rho+\eta-\nu, \xi); \end{array}\displaystyle \frac{1}{x^{\xi}} \right ]. \end{aligned}
(4.7)

Further, if we put $$\nu=0$$ in (4.6) and (4.7) then these Saigo fractional differential formulas reduce to the following fractional differential formulas.

Corollary 17

Let $$x>0$$, $$\sigma, \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho+\xi n)>-\Re(\eta+\sigma)$$ then the following fractional derivative formula holds true:

\begin{aligned}[b]& \bigl(D^{\sigma, \eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;t^{\xi} \bigr) \bigr\} \bigr) (x) \\ & \quad=x^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;x^{\xi } \bigr)*{}_{1}\Psi_{1} \left [ \textstyle\begin{array}{c}(\rho+\eta+\sigma, \xi); \\ (\rho+\eta, \xi); \end{array}\displaystyle x^{\xi} \right ]. \end{aligned}
(4.8)

Corollary 18

Let $$x>0$$, $$\sigma, \rho\in\mathbb{C}$$ and $$r,\alpha ,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho-\xi n)<1+\Re(\eta+\sigma)-n^{*}$$ where $$n^{*}=[\Re(\sigma)]+1$$ then the following fractional derivative formula holds true:

\begin{aligned}[b]& \biggl(D^{\sigma,\eta}_{x,\infty } \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a;\frac {1}{t^{\xi}} \biggr) \biggr\} \biggr) (x) \\ & \quad=x^{\rho-1}S_{\mu,\lambda }^{(\alpha, \beta)} \biggl(r,a; \frac{1}{x^{\xi}} \biggr)*{}_{1}\Psi_{1} \left [ \textstyle\begin{array}{c}(1-\rho+\sigma+\eta, \xi); \\ (1-\rho+\eta, \xi); \end{array}\displaystyle \frac{1}{x^{\xi}} \right ]. \end{aligned}
(4.9)

Further, if we put $$\nu=-\sigma$$ in (4.6) and (4.7), then these Saigo fractional derivatives reduce to the following Riemann–Liouville and the Weyl type fractional derivative formulas.

Corollary 19

Let $$x>0$$, $$\sigma, \rho\in\mathbb{C}$$ and $$r,\alpha ,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho+\xi n)>\Re(\sigma)>0$$ then the following fractional derivative formula holds true:

\begin{aligned}[b]& \bigl(D^{\sigma}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;t^{\xi} \bigr) \bigr\} \bigr) (x) \\ & \quad=x^{\rho-\sigma-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;x^{\xi } \bigr)*{}_{1}\Psi_{1} \left [ \textstyle\begin{array}{c}(\rho, \xi); \\ (\rho-\sigma, \xi); \end{array}\displaystyle x^{\xi} \right ]. \end{aligned}
(4.10)

Corollary 20

Let $$x>0$$, $$\sigma, \rho\in\mathbb{C}$$ and $$r,\alpha ,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\rho-\xi n)<1+\Re(\sigma)-n^{*}$$ where $$n^{*}=[\Re(\sigma)]+1$$ then the following fractional derivative formula holds true:

\begin{aligned}[b]& \biggl(D^{\sigma}_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a;\frac{1}{t^{\xi }} \biggr) \biggr\} \biggr) (x) \\ & \quad=x^{\rho-\sigma-1}S_{\mu,\lambda }^{(\alpha, \beta)} \biggl(r,a; \frac{1}{x^{\xi}} \biggr)*{}_{1}\Psi_{1} \left [ \textstyle\begin{array}{c}(1-\rho+\sigma, \xi); \\ (1-\rho, \xi); \end{array}\displaystyle \frac{1}{x^{\xi}} \right ]. \end{aligned}
(4.11)

If we put $$\xi=1$$ in Theorems 3, 4 and corollaries (4.6), (4.7), (4.8), (4.9), (4.10) and (4.11), we get interesting results given in the following corollaries.

Corollary 21

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta, \rho\in \mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho+n)>\max\{0,\Re(\eta-\sigma-\sigma'-\nu'),\Re (\nu-\sigma)\}$$ then the following fractional derivative formula holds true:

\begin{aligned}[b]& \bigl(D^{\sigma,\sigma',\nu,\nu ',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta )}(r,a;t) \bigr\} \bigr) (x) \\ &\quad =x^{\rho-\eta+\sigma+\sigma'-1} \frac {\Gamma(\rho)\Gamma(\rho-\eta+\sigma+\sigma'+\nu')\Gamma(\rho-\nu+\sigma )}{\Gamma(\rho-\nu)\Gamma(\rho-\eta+\sigma+\sigma')\Gamma(\rho-\eta +\sigma+\nu')}S_{\mu,\lambda}^{(\alpha, \beta )}(r,a;x) \\ & \qquad{}*{}_{3}F_{3} \left [ \textstyle\begin{array}{c} \rho, \rho-\eta+\sigma+\sigma'+\nu', \rho-\nu+\sigma; \\ \rho-\nu, \rho-\eta+\sigma+\sigma', \rho-\eta+\sigma+\nu'; \end{array}\displaystyle x \right ]. \end{aligned}
(4.12)

Corollary 22

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta, \rho\in \mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho-n)<1+\min\{\Re(\nu'),\Re(\eta-\sigma-\sigma '),\Re(\eta-\sigma'-\nu)\}$$ then the following fractional integral formula holds true:

\begin{aligned}[b]& \biggl(D^{\sigma,\sigma',\nu,\nu ',\eta}_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta )} \biggl(r,a;\frac{1}{t} \biggr) \biggr\} \biggr) (x) \\ & \quad=x^{\rho-\eta+\sigma +\sigma'-1}\frac{\Gamma(1-\rho+\nu')\Gamma(1-\rho+\eta-\sigma-\sigma ')\Gamma(1-\rho+\eta-\sigma'-\nu)}{\Gamma(1-\rho)\Gamma(1-\rho+\eta -\sigma-\sigma'-\nu)\Gamma(1-\rho-\sigma'+\nu')} S_{\mu ,\lambda}^{(\alpha, \beta)} \biggl(r,a;\frac{1}{x} \biggr) \\ & \qquad *{}_{3}F_{3} \left [ \textstyle\begin{array}{c}1-\rho+\nu', 1-\rho+\eta-\sigma-\sigma', 1-\rho+\eta -\sigma'-\nu; \\ 1-\rho, 1-\rho+\eta-\sigma-\sigma'-\nu, 1-\rho-\sigma '+\nu'; \end{array}\displaystyle \frac{1}{x} \right ]. \end{aligned}
(4.13)

Corollary 23

Let $$x>0$$, $$\sigma, \nu, \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho+n)>-\min\{0,\Re(\eta+\sigma+\nu)\}$$ then the following fractional derivative formula holds true:

\begin{aligned}[b]& \bigl(D^{\sigma,\nu,\eta }_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)}(r,a;t) \bigr\} \bigr) (x) \\ & \quad=x^{\rho+\nu-1}\frac{\Gamma(\rho)\Gamma(\rho+\eta+\sigma +\nu)}{\Gamma(\rho+\eta)\Gamma(\rho+\nu)}S_{\mu,\lambda}^{(\alpha, \beta )}(r,a;x)*{}_{2}F_{2} \left [ \textstyle\begin{array}{c}\rho, \rho+\eta+\sigma+\nu; \\ \rho+\eta, \rho+\nu; \end{array}\displaystyle x \right ]. \end{aligned}
(4.14)

Corollary 24

Let $$x>0$$, $$\sigma, \nu, \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho-n)<1+\min\{\Re(-\nu-n^{*}),\Re(\eta+\sigma )\}$$ where $$n^{*}=[\Re(\sigma)]+1$$ then the following fractional derivative formula holds true:

\begin{aligned}[b]& \biggl(D^{\sigma,\nu,\eta }_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a;\frac{1}{t} \biggr) \biggr\} \biggr) (x) \\ & \quad=x^{\rho+\nu-1}\frac{\Gamma (1-\rho-\nu)\Gamma(1-\rho+\sigma+\eta)}{\Gamma(1-\rho)\Gamma(1-\rho+\eta -\nu)} S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a;\frac {1}{x} \biggr) \\ &\qquad{}*{}_{2}F_{2} \left [ \textstyle\begin{array}{c} 1-\rho-\nu, 1-\rho+\sigma+\eta; \\ 1-\rho, 1-\rho+\eta -\nu; \end{array}\displaystyle \frac{1}{x} \right ]. \end{aligned}
(4.15)

Corollary 25

Let $$x>0$$, $$\sigma, \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho+n)>-\Re(\eta+\sigma)$$ then the following fractional derivative formula holds true:

\begin{aligned}[b]& \bigl(D^{\sigma, \eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)}(r,a;t) \bigr\} \bigr) (x) \\ &\quad =x^{\rho-1}\frac{\Gamma(\rho+\eta+\sigma)}{\Gamma(\rho+\eta)}S_{\mu ,\lambda}^{(\alpha, \beta)}(r,a;x)*{}_{1}F_{1} \left [ \textstyle\begin{array}{c}\rho+\eta+\sigma; \\ \rho+\eta; \end{array}\displaystyle x \right ]. \end{aligned}
(4.16)

Corollary 26

Let $$x>0$$, $$\sigma, \rho\in\mathbb{C}$$ and $$r,\alpha ,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho-n)<1+\Re(\eta+\sigma)-n^{*}$$ where $$n^{*}=[\Re (\sigma)]+1$$ then the following fractional derivative formula holds true:

\begin{aligned}[b]& \biggl(D^{\sigma,\eta}_{x,\infty } \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a;\frac {1}{t} \biggr) \biggr\} \biggr) (x) \\ &\quad =x^{\rho-1}\frac{\Gamma(1-\rho+\sigma +\eta)}{\Gamma(1-\rho+\eta)}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a; \frac{1}{x} \biggr)*{}_{1}F_{1} \left [ \textstyle\begin{array}{c}1-\rho+\sigma+\eta; \\ 1-\rho+\eta; \end{array}\displaystyle \frac{1}{x} \right ]. \end{aligned}
(4.17)

Corollary 27

Let $$x>0$$, $$\sigma, \rho\in\mathbb{C}$$ and $$r,\alpha ,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\sigma)>0$$ and $$\Re(\rho+n)>\Re(\sigma)>0$$ then the following fractional derivative formula holds true:

\begin{aligned}[b]& \bigl(D^{\sigma}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)}(r,a;t) \bigr\} \bigr) (x) \\ &\quad =x^{\rho-\sigma-1}\frac{\Gamma(\rho)}{\Gamma(\rho-\sigma)}S_{\mu,\lambda }^{(\alpha, \beta)}(r,a;x)*{}_{1}F_{1} \left [ \textstyle\begin{array}{c}\rho; \\ \rho-\sigma; \end{array}\displaystyle x \right ]. \end{aligned}
(4.18)

Corollary 28

Let $$x>0$$, $$\sigma, \rho\in\mathbb{C}$$ and $$r,\alpha ,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\rho-n)<1+\Re(\sigma)-n^{*}$$ where $$n^{*}=[\Re(\sigma)]+1$$ then the following fractional derivative formula holds true:

\begin{aligned}[b]& \biggl(D^{\sigma}_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a;\frac{1}{t} \biggr) \biggr\} \biggr) (x) \\ & \quad=x^{\rho-\sigma-1}\frac{\Gamma(1-\rho+\sigma )}{\Gamma(1-\rho)}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a; \frac {1}{x} \biggr)*{}_{1}F_{1} \left [ \textstyle\begin{array}{c}1-\rho+\sigma; \\ 1-\rho; \end{array}\displaystyle \frac{1}{x} \right ]. \end{aligned}
(4.19)

Integral transform formulas of the extended generalized Mathieu series

In this section, we establish certain theorems involving the results obtained in the previous sections associated with the integral transforms like the beta transform, the Laplace transform and the Whittaker transform.

Definition 9

The beta transform of the function $$f(z)$$ is defined as :

$$B \bigl\{ f(z):l,m \bigr\} = \int_{0}^{1}z^{l-1}(1-z)^{m-1}f(z)\,dz.$$
(5.1)

Theorem 5

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho+\xi n)>\max\{0,\Re(\sigma+\sigma'+\nu-\eta ),\Re(\sigma'-\nu')\}$$ then the following formula holds:

\begin{aligned}[b]& B \bigl\{ \bigl(I^{\sigma,\sigma ',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta )} \bigl(r,a;(tz)^{\xi} \bigr) \bigr\} \bigr) (x): l,m \bigr\} \\ &\quad =x^{\rho+\eta -\sigma-\sigma'-1}\Gamma(m) S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;x^{\xi } \bigr) \\ & \qquad{}*{}_{4}\Psi_{4} \left [ \textstyle\begin{array}{c}(l,\xi),(\rho,\xi), (\rho+\eta-\sigma-\sigma'-\nu, \xi ), (\rho+\nu'-\sigma', \xi); \\ (l+m,\xi),(\rho+\nu', \xi), (\rho+\eta -\sigma-\sigma', \xi), (\rho+\eta-\sigma'-\nu, \xi); \end{array}\displaystyle x^{\xi} \right ]. \end{aligned}
(5.2)

Proof

In order to prove (5.2), we use the definition of the beta transform as given in Eq. (5.1), to get

\begin{aligned}[b]& B \bigl\{ \bigl(I^{\sigma,\sigma ',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta )} \bigl(r,a;(tz)^{\xi} \bigr) \bigr\} \bigr) (x): l,m \bigr\} \\ &\quad = \int _{0}^{1}z^{l-1}(1-z)^{m-1} \bigl\{ \bigl(I^{\sigma,\sigma',\nu,\nu',\eta }_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;(tz)^{\xi } \bigr) \bigr\} \bigr) (x) \bigr\} \,dz. \end{aligned}
(5.3)

Applying the result (1.25), Eq. (5.3) reduces to

\begin{aligned}[b]& = \int_{0}^{1}z^{l+\xi n-1}(1-z)^{m-1} \Biggl(x^{\rho+\eta-\sigma-\sigma'-1}\sum_{n=1}^{\infty } \frac{2a_{n}^{\beta}(\lambda)_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}} \\ &\quad{} \times\frac{\Gamma(\rho+\xi n)\Gamma(\rho+\eta-\sigma-\sigma'-\nu +\xi n)\Gamma(\rho+\nu'-\sigma'+\xi n)}{\Gamma(\rho+\nu'+\xi n)\Gamma (\rho+\eta-\sigma-\sigma'+\xi n)\Gamma(\rho+\eta-\sigma'-\nu+\xi n)}\frac{x^{\xi n}}{n!} \Biggr)\,dz. \end{aligned}
(5.4)

Interchanging the order of integration and summation, we have

\begin{aligned}[b]& =x^{\rho+\eta-\sigma-\sigma '-1}\sum _{n=1}^{\infty} \frac{2a_{n}^{\beta}(\lambda)_{n}}{(a_{n}^{\alpha }+r^{2})^{\mu}} \\ &\quad{} \times\frac{\Gamma(\rho+\xi n)\Gamma(\rho+\eta-\sigma -\sigma'-\nu+\xi n)\Gamma(\rho+\nu'-\sigma'+\xi n)}{\Gamma(\rho+\nu '+\xi n)\Gamma(\rho+\eta-\sigma-\sigma'+\xi n)\Gamma(\rho+\eta-\sigma '-\nu+\xi n)}\frac{x^{\xi n}}{n!} \\ & \quad{}\times \int_{0}^{1}z^{l+\xi n-1}(1-z)^{m-1}\,dz. \end{aligned}
(5.5)

After a little simplification, we have

\begin{aligned}[b]& =x^{\rho+\eta-\sigma-\sigma '-1}\Gamma(m)\sum _{n=1}^{\infty}\frac{2a_{n}^{\beta}(\lambda )_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}} \\ & \quad\times\frac{\Gamma(l+\xi n)\Gamma (\rho+\xi n)\Gamma(\rho+\eta-\sigma-\sigma'-\nu+\xi n)\Gamma(\rho+\nu '-\sigma'+\xi n)}{\Gamma(l+m+\xi n)\Gamma(\rho+\nu'+\xi n)\Gamma(\rho +\eta-\sigma-\sigma'+\xi n)\Gamma(\rho+\eta-\sigma'-\nu+\xi n)}\frac {x^{\xi n}}{n!}.\end{aligned}
(5.6)

By applying the Hadamard product (2.13) in Eq. (5.6), which in view of (2.8) and (2.15), gives the required result (5.2). □

Theorem 6

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho-\xi n)<1+\min\{\Re(-\nu),\Re(\sigma+\sigma '-\eta),\Re(\sigma+\nu'-\eta)\}$$ then the following formula holds:

\begin{aligned}[b]& B \biggl\{ \biggl(I^{\sigma,\sigma ',\nu,\nu',\eta}_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a; \biggl( \frac{z}{t} \biggr)^{\xi} \biggr) \biggr\} \biggr) (x): l,m \biggr\} \\ &\quad =x^{\rho+\eta-\sigma-\sigma'-1}\Gamma(m) S_{\mu ,\lambda}^{(\alpha, \beta)} \biggl(r,a; \frac{1}{x^{\xi}} \biggr) \\ &\qquad{} *{}_{4}\Psi_{4} \left [ \textstyle\begin{array}{c}(l,\xi),(1-\rho-\nu,\xi), (1-\rho-\eta+\sigma+\sigma', \xi), (1-\rho-\eta+\sigma+\nu', \xi); \\ (l+m,\xi),(1-\rho, \xi), (1-\rho-\eta+\sigma+\sigma'+\nu', \xi), (1-\rho+\sigma-\nu, \xi); \end{array}\displaystyle \frac{1}{x^{\xi}} \right ]. \end{aligned}
(5.7)

Proof

The proof of Theorem 6 is similar to as that of Theorem 5, therefore, we omit the details. □

Theorem 7

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho+\xi n)>\max\{0,\Re(\eta-\sigma-\sigma'-\nu '),\Re(\nu-\sigma)\}$$ then the following formula holds:

\begin{aligned}[b]& B \bigl\{ \bigl(D^{\sigma,\sigma ',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta )} \bigl(r,a;(tz)^{\xi} \bigr) \bigr\} \bigr) (x): l,m \bigr\} \\ & \quad=x^{\rho-\eta +\sigma+\sigma'-1}\Gamma(m) S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;x^{\xi } \bigr) \\ &\qquad{} *{}_{4}\Psi_{4} \left [ \textstyle\begin{array}{c}(l,\xi),(\rho,\xi), (\rho-\eta+\sigma+\sigma'+\nu', \xi ), (\rho-\nu+\sigma, \xi); \\ (l+m,\xi),(\rho-\nu, \xi), (\rho-\eta +\sigma+\sigma', \xi), (\rho-\eta+\sigma+\nu', \xi); \end{array}\displaystyle x^{\xi} \right ]. \end{aligned}
(5.8)

Proof

In order to prove (5.8), we use definition of beta transform as given in Eq. (5.1), we get

\begin{aligned}[b]& B \bigl\{ \bigl(D^{\sigma,\sigma ',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta )} \bigl(r,a;(tz)^{\xi} \bigr) \bigr\} \bigr) (x): l,m \bigr\} \\ & \quad= \int _{0}^{1}z^{l-1}(1-z)^{m-1} \bigl\{ \bigl(D^{\sigma,\sigma',\nu,\nu',\eta }_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;(tz)^{\xi } \bigr) \bigr\} \bigr) (x) \bigr\} \,dz, \end{aligned}
(5.9)

applying the result (1.27), Eq. (5.9) reduces to

\begin{aligned}[b]& = \int_{0}^{1}z^{l+\xi n-1}(1-z)^{m-1}x^{\rho-\eta+\sigma+\sigma'-1} \sum_{n=1}^{\infty}\frac {2a_{n}^{\beta}(\lambda)_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}} \\ & \quad{}\times\frac {\Gamma(\rho+\xi n)\Gamma(\rho-\eta+\sigma+\sigma'+\nu'+\xi n)\Gamma (\rho-\nu+\sigma+\xi n)}{\Gamma(\rho-\nu+\xi n)\Gamma(\rho-\eta+\sigma +\sigma'+\xi n)\Gamma(\rho-\eta+\sigma+\nu'+\xi n)}\frac{x^{\xi n}}{n!}. \end{aligned}
(5.10)

Interchanging the order of integration and summation, we have

\begin{aligned}[b]& =x^{\rho-\eta+\sigma+\sigma '-1}\sum _{n=1}^{\infty} \frac{2a_{n}^{\beta}(\lambda)_{n}}{(a_{n}^{\alpha }+r^{2})^{\mu}} \\ &\quad{} \times\frac{\Gamma(\rho+\xi n)\Gamma(\rho-\eta+\sigma +\sigma'+\nu'+\xi n)\Gamma(\rho-\nu+\sigma+\xi n)}{\Gamma(\rho-\nu+\xi n)\Gamma(\rho-\eta+\sigma+\sigma'+\xi n)\Gamma(\rho-\eta+\sigma+\nu '+\xi n)}\frac{x^{\xi n}}{n!} \\ & \quad{}\times \int_{0}^{1}z^{l+\xi n-1}(1-z)^{m-1}\,dz. \end{aligned}
(5.11)

After a little simplification, we have

\begin{aligned}[b]& =x^{\rho-\eta+\sigma+\sigma '-1}\Gamma(m)\sum _{n=1}^{\infty}\frac{2a_{n}^{\beta}(\lambda )_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}} \\ &\quad{} \times\frac{\Gamma(l+\xi n) \Gamma (\rho+\xi n)\Gamma(\rho-\eta+\sigma+\sigma'+\nu'+\xi n)\Gamma(\rho-\nu +\sigma+\xi n)}{\Gamma(l+m+\xi n) \Gamma(\rho-\nu+\xi n)\Gamma(\rho -\eta+\sigma+\sigma'+\xi n)\Gamma(\rho-\eta+\sigma+\nu'+\xi n)}\frac {x^{\xi n}}{n!}.\end{aligned}
(5.12)

Applying the Hadamard product (2.13) in Eq. (5.12), in view of (2.8) and (2.15), gives the required result (5.8). □

Theorem 8

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho-\xi n)<1+\min\{\Re(\nu'),\Re(\eta-\sigma -\sigma'),\Re(\eta-\sigma'-\nu)\}$$ then the following formula holds:

\begin{aligned}[b]& B \biggl\{ \biggl(D^{\sigma,\sigma ',\nu,\nu',\eta}_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a; \biggl( \frac{z}{t} \biggr)^{\xi} \biggr) \biggr\} \biggr) (x): l,m \biggr\} \\ & \quad=x^{\rho-\eta+\sigma+\sigma'-1}\Gamma(m) S_{\mu ,\lambda}^{(\alpha, \beta)} \biggl(r,a; \frac{1}{x^{\xi}} \biggr) \\ & \qquad{}*{}_{4}\Psi_{4} \left [ \textstyle\begin{array}{c}(l,\xi),(1-\rho+\nu',\xi), (1-\rho+\eta-\sigma-\sigma ', \xi), (1-\rho+\eta-\sigma'-\nu, \xi); \\ (l+m,\xi),(1-\rho, \xi), (1-\rho+\eta-\sigma-\sigma'-\nu, \xi), (1-\rho-\sigma'+\nu', \xi); \end{array}\displaystyle \frac{1}{x^{\xi}} \right ]. \end{aligned}
(5.13)

Proof

The proof of Theorem 8 is similar to as that of Theorem 7. Therefore, we omit the details. □

Definition 10

The Laplace transform of $$f(z)$$ is defined as [70, 71]:

$$L \bigl\{ f(z) \bigr\} = \int_{0}^{\infty}e^{-sz}f(z)\,dz.$$
(5.14)

Theorem 9

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta, \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho+\xi n)>\max\{0,\Re(\sigma+\sigma'+\nu-\eta ),\Re(\sigma'-\nu')\}$$ then the following formula holds:

\begin{aligned}[b]& L \bigl\{ z^{l-1} \bigl(I^{\sigma ,\sigma',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;(tz)^{\xi} \bigr) \bigr\} \bigr) (x) \bigr\} \\ & \quad=\frac{x^{\rho+\eta -\sigma-\sigma'-1}}{s^{l}}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a; \biggl( \frac{x}{s} \biggr)^{\xi} \biggr) \\ &\qquad{} *{}_{4}\Psi_{3} \left [ \textstyle\begin{array}{c}(l,\xi),(\rho,\xi), (\rho+\eta-\sigma-\sigma'-\nu, \xi ), (\rho+\nu'-\sigma', \xi); \\ (\rho+\nu', \xi), (\rho+\eta-\sigma -\sigma', \xi), (\rho+\eta-\sigma'-\nu, \xi); \end{array}\displaystyle \biggl(\frac{x}{s} \biggr)^{\xi} \right ]. \end{aligned}
(5.15)

Proof

In order to prove (5.15), we use definition of the Laplace transform as given in Eq. (5.14), to get

\begin{aligned}[b]& L \bigl\{ z^{l-1} \bigl(I^{\sigma ,\sigma',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;(tz)^{\xi} \bigr) \bigr\} \bigr) (x) \bigr\} \\ &\quad = \int_{0}^{\infty }e^{-sz}z^{l-1} \bigl\{ \bigl(I^{\sigma,\sigma',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;(tz)^{\xi} \bigr) \bigr\} \bigr) (x) \bigr\} \,dz \end{aligned}
(5.16)

and applying the result (1.25) and interchanging the order of integration and summation, Eq. (5.16) reduces to

\begin{aligned}[b]& =x^{\rho+\eta-\sigma-\sigma '-1}\sum _{n=1}^{\infty} \frac{2a_{n}^{\beta}(\lambda)_{n}}{(a_{n}^{\alpha }+r^{2})^{\mu}} \\ &\quad{} \times\frac{\Gamma(\rho+\xi n)\Gamma(\rho+\eta-\sigma -\sigma'-\nu+\xi n)\Gamma(\rho+\nu'-\sigma'+\xi n)}{\Gamma(\rho+\nu '+\xi n)\Gamma(\rho+\eta-\sigma-\sigma'+\xi n)\Gamma(\rho+\eta-\sigma '-\nu+\xi n)}\frac{x^{\xi n}}{n!} \\ &\quad{} \times \int_{0}^{\infty}z^{l+\xi n-1}e^{-sz}\,dz. \end{aligned}
(5.17)

After a little simplification, we have

\begin{aligned}[b]& =\frac{x^{\rho+\eta-\sigma -\sigma'-1}}{s^{l}}\sum _{n=1}^{\infty} \frac{2a_{n}^{\beta}(\lambda )_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}} \\ & \quad{}\times\frac{\Gamma(l+\xi n)\Gamma (\rho+\xi n)\Gamma(\rho+\eta-\sigma-\sigma'-\nu+\xi n)\Gamma(\rho+\nu '-\sigma'+\xi n)}{\Gamma(\rho+\nu'+\xi n)\Gamma(\rho+\eta-\sigma-\sigma '+\xi n)\Gamma(\rho+\eta-\sigma'-\nu+\xi n)}\frac{(x/s)^{\xi n}}{n!},\end{aligned}
(5.18)

and interpreting the above equation, from the point of view of (2.13), (2.8) and (2.15), we have the required result. □

Theorem 10

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta , \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho-\xi n)<1+\min\{\Re(-\nu),\Re(\sigma+\sigma '-\eta),\Re(\sigma+\nu'-\eta)\}$$ then the following formula holds:

\begin{aligned}[b]& L \biggl\{ z^{l-1} \biggl(I^{\sigma ,\sigma',\nu,\nu',\eta}_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda }^{(\alpha, \beta)} \biggl(r,a; \biggl( \frac{z}{t} \biggr)^{\xi} \biggr) \biggr\} \biggr) (x) \biggr\} \\ & \quad=\frac{x^{\rho+\eta-\sigma-\sigma '-1}}{s^{l}}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a; \biggl( \frac {1}{sx} \biggr)^{\xi} \biggr) \\ & \quad\quad{}*{}_{4}\Psi_{3} \left [ \textstyle\begin{array}{c}(l,\xi), (1-\rho-\nu,\xi), (1-\rho-\eta+\sigma+\sigma ', \xi), (1-\rho-\eta+\sigma+\nu', \xi); \\ (1-\rho, \xi), (1-\rho-\eta +\sigma+\sigma'+\nu', \xi), (1-\rho+\sigma-\nu, \xi); \end{array}\displaystyle \biggl(\frac{1}{sx} \biggr)^{\xi} \right ]. \end{aligned}
(5.19)

Proof

The proof of Theorem 10 is similar to that of Theorem 9. Therefore, we omit the details. □

Theorem 11

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta , \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho+\xi n)>\max\{0,\Re(\eta-\sigma-\sigma'-\nu '),\Re(\nu-\sigma)\}$$ then the following formula holds:

\begin{aligned}[b]& L \bigl\{ z^{l-1} \bigl(D^{\sigma ,\sigma',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;(tz)^{\xi} \bigr) \bigr\} \bigr) (x) \bigr\} \\ &\quad =\frac{x^{\rho-\eta +\sigma+\sigma'-1}}{s^{l}} S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a; \biggl( \frac{x}{s} \biggr)^{\xi} \biggr) \\ & \qquad{}*{}_{4}\Psi_{3} \left [ \textstyle\begin{array}{c} (l, \xi), (\rho, \xi), (\rho-\eta+\sigma+\sigma'+\nu ', \xi), (\rho-\nu+\sigma, \xi); \\ (\rho-\nu, \xi), (\rho-\eta+\sigma +\sigma', \xi), (\rho-\eta+\sigma+\nu', \xi); \end{array}\displaystyle \biggl(\frac{x}{s} \biggr)^{\xi} \right ]. \end{aligned}
(5.20)

Proof

In order to prove (5.20), we use definition of the Laplace transform as given in Eq. (5.14), we get

\begin{aligned}[b]& L \bigl\{ z^{l-1} \bigl(D^{\sigma ,\sigma',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;(tz)^{\xi} \bigr) \bigr\} \bigr) (x) \bigr\} \\ &\quad = \int_{0}^{\infty }e^{-sz} \bigl\{ z^{l-1} \bigl(D^{\sigma,\sigma',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;(tz)^{\xi} \bigr) \bigr\} \bigr) (x) \bigr\} \,dz \end{aligned}
(5.21)

applying the result (1.27), Eq. (5.21) reduces to

\begin{aligned}[b]& = \int_{0}^{\infty}z^{l+\xi n-1}e^{-sz}x^{\rho-\eta+\sigma+\sigma'-1} \sum_{n=1}^{\infty}\frac {2a_{n}^{\beta}(\lambda)_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}} \\ & \quad{}\times\frac {\Gamma(\rho+\xi n)\Gamma(\rho-\eta+\sigma+\sigma'+\nu'+\xi n)\Gamma (\rho-\nu+\sigma+\xi n)}{\Gamma(\rho-\nu+\xi n)\Gamma(\rho-\eta+\sigma +\sigma'+\xi n)\Gamma(\rho-\eta+\sigma+\nu'+\xi n)}\frac{x^{\xi n}}{n!}\,dz. \end{aligned}
(5.22)

Interchanging the order of integration and summation, we have

\begin{aligned}[b]& =x^{\rho-\eta+\sigma+\sigma '-1}\sum _{n=1}^{\infty} \frac{2a_{n}^{\beta}(\lambda)_{n}}{(a_{n}^{\alpha }+r^{2})^{\mu}} \\ & \quad{}\times\frac{\Gamma(\rho+\xi n)\Gamma(\rho-\eta+\sigma +\sigma'+\nu'+\xi n)\Gamma(\rho-\nu+\sigma+\xi n)}{\Gamma(\rho-\nu+\xi n)\Gamma(\rho-\eta+\sigma+\sigma'+\xi n)\Gamma(\rho-\eta+\sigma+\nu '+\xi n)}\frac{x^{\xi n}}{n!} \\ & \quad{}\times \int_{0}^{\infty}z^{l+\xi n-1}e^{-sz}\,dz. \end{aligned}
(5.23)

After a little simplification we have

\begin{aligned}[b]& =\frac{x^{\rho-\eta+\sigma +\sigma'-1}}{s^{l}}\sum _{n=1}^{\infty} \frac{2a_{n}^{\beta}(\lambda )_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}} \\ &\quad{} \times\frac{\Gamma(l+\xi n)\Gamma (\rho+\xi n)\Gamma(\rho-\eta+\sigma+\sigma'+\nu'+\xi n)\Gamma(\rho-\nu +\sigma+\xi n)}{\Gamma(\rho-\nu+\xi n)\Gamma(\rho-\eta+\sigma+\sigma '+\xi n)\Gamma(\rho-\eta+\sigma+\nu'+\xi n)}\frac{(x/s)^{\xi n}}{n!},\end{aligned}
(5.24)

and interpreting the above equation, in the view of of (2.13), (2.8) and (2.15), we have the required result. □

Theorem 12

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta , \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho-\xi n)<1+\min\{\Re(\nu'),\Re(\eta-\sigma -\sigma'),\Re(\eta-\sigma'-\nu)\}$$ then the following formula holds:

\begin{aligned}[b]& L \biggl\{ z^{l-1} \biggl(D^{\sigma ,\sigma',\nu,\nu',\eta}_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda }^{(\alpha, \beta)} \biggl(r,a; \biggl( \frac{z}{t} \biggr)^{\xi} \biggr) \biggr\} \biggr) (x) \biggr\} \\ &\quad =\frac{x^{\rho-\eta+\sigma+\sigma '-1}}{s^{l}}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a; \biggl( \frac {1}{sx} \biggr)^{\xi} \biggr) \\ & \qquad{}*{}_{3}\Psi_{3} \left [ \textstyle\begin{array}{c} (1-\rho+\nu', \xi), (1-\rho+\eta-\sigma-\sigma', \xi ), (1-\rho+\eta-\sigma'-\nu, \xi); \\ (1-\rho, \xi), (1-\rho+\eta-\sigma -\sigma'-\nu, \xi), (1-\rho-\sigma'+\nu', \xi); \end{array}\displaystyle \biggl(\frac{1}{sx} \biggr)^{\xi} \right ]. \end{aligned}
(5.25)

Proof

The proof of the Theorem 12 would run parallel to Theorem 11. Therefore, we omit the details. □

Definition 11

The Whittaker transform is defined as 

\begin{aligned}[b]& \int_{0}^{\infty} t^{l-1}e^{-t/2}W_{\tau, \zeta}(t)\,dt= \frac{\Gamma(1/2+\zeta+l)\Gamma (1/2-\zeta+l)}{\Gamma(1/2-\tau+l)}.\end{aligned}
(5.26)

Theorem 13

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta , \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho+\xi n)>\max\{0,\Re(\sigma+\sigma'+\nu-\eta ),\Re(\sigma'-\nu')\}$$ then the following formula holds:

\begin{aligned}[b]& \int_{0}^{\infty }z^{l-1}e^{-\delta z/2}W_{\tau,\zeta}( \delta z) \bigl\{ \bigl(I^{\sigma ,\sigma',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;(wzt)^{\xi} \bigr) \bigr\} \bigr) (x) \bigr\} \,dz \\ & \quad=\frac{x^{\rho +\eta-\sigma-\sigma'-1}}{\delta^{l}}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a; \biggl( \frac{wx}{\delta} \biggr)^{\xi} \biggr)\\ &\qquad{}*{}_{5} \Psi_{4} \left [ \textstyle\begin{array}{c}(1/2+\zeta+l, \xi), (1/2-\zeta+l,\xi),\\ (1/2-\tau+l, \xi), \end{array}\displaystyle \right . \\ &\qquad{} \left . \textstyle\begin{array}{c} (\rho, \xi), (\rho+\eta-\sigma-\sigma'-\nu, \xi), (\rho+\nu'-\sigma', \xi); \\ (\rho+\nu', \xi), (\rho+\eta-\sigma-\sigma ', \xi), (\rho+\eta-\sigma'-\nu, \xi); \end{array}\displaystyle \biggl(\frac{wx}{\delta} \biggr)^{\xi} \right ]. \end{aligned}
(5.27)

Proof

To prove (5.27), by using the definition of the Whittaker transform and by using the result obtained in Eq. (3.4), we have

\begin{aligned}[b]& \int_{0}^{\infty }z^{l-1}e^{-\delta z/2}W_{\tau,\zeta}( \delta z) \bigl\{ \bigl(I^{\sigma ,\sigma',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;(wzt)^{\xi} \bigr) \bigr\} \bigr) (x) \bigr\} \,dz \\ &\quad =x^{\rho+\eta -\sigma-\sigma'-1}\sum_{n=1}^{\infty} \frac{2a_{n}^{\beta}(\lambda )_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}} \\ &\qquad{} \times\frac{\Gamma(\rho+\xi n)\Gamma (\rho+\eta-\sigma-\sigma'-\nu+\xi n)\Gamma(\rho+\nu'-\sigma'+\xi n)}{\Gamma(\rho+\nu'+\xi n)\Gamma(\rho+\eta-\sigma-\sigma'+\xi n)\Gamma (\rho+\eta-\sigma'-\nu+\xi n)}\frac{(wx)^{\xi n}}{n!} \\ & \qquad{}\times \int _{0}^{\infty}z^{l+\xi n-1}e^{-\delta z/2}W_{\tau,\zeta}( \delta z)\,dz. \end{aligned}
(5.28)

By substituting $$\delta z = y$$ and after a little simplification, we have

\begin{aligned}[b]& \int_{0}^{\infty }z^{l-1}e^{-\delta z/2}W_{\tau,\zeta}( \delta z) \bigl\{ \bigl(I^{\sigma ,\sigma',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;(wzt)^{\xi} \bigr) \bigr\} \bigr) (x) \bigr\} \,dz \\ &\quad =\frac{x^{\rho +\eta-\sigma-\sigma'-1}}{\delta^{l}}\sum_{n=1}^{\infty} \frac{2a_{n}^{\beta }(\lambda)_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}} \\ &\qquad{} \times\frac{\Gamma(\rho+\xi n)\Gamma(\rho+\eta-\sigma-\sigma'-\nu+\xi n)\Gamma(\rho+\nu'-\sigma '+\xi n)}{\Gamma(\rho+\nu'+\xi n)\Gamma(\rho+\eta-\sigma-\sigma'+\xi n)\Gamma(\rho+\eta-\sigma'-\nu+\xi n)} \biggl(\frac{wx}{\delta} \biggr)^{\xi n} \frac{1}{n!} \\ &\qquad{} \times \int_{0}^{\infty}y^{l+\xi n-1}e^{-y/2}W_{\tau,\zeta}(y)\,dy. \end{aligned}
(5.29)

By using the integral formula involving the Whittaker function, we have

\begin{aligned}[b]& \int_{0}^{\infty }z^{l-1}e^{-\delta z/2}W_{\tau,\zeta}( \delta z) \bigl\{ \bigl(I^{\sigma ,\sigma',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;(wzt)^{\xi} \bigr) \bigr\} \bigr) (x) \bigr\} \,dz \\ &\quad =\frac{x^{\rho +\eta-\sigma-\sigma'-1}}{\delta^{l}}\sum_{n=1}^{\infty} \frac{2a_{n}^{\beta }(\lambda)_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}}\frac{\Gamma(1/2+\zeta+l+\xi n)\Gamma(1/2-\zeta+l+\xi n)}{\Gamma(1/2-\tau+l+\xi n)} \\ & \qquad\times\frac {\Gamma(\rho+\xi n)\Gamma(\rho+\eta-\sigma-\sigma'-\nu+\xi n)\Gamma(\rho +\nu'-\sigma'+\xi n)}{\Gamma(\rho+\nu'+\xi n)\Gamma(\rho+\eta-\sigma -\sigma'+\xi n)\Gamma(\rho+\eta-\sigma'-\nu+\xi n)} \biggl(\frac {wx}{\delta} \biggr)^{\xi n} \frac{1}{n!}, \end{aligned}
(5.30)

and interpreting the above equation, from the point of view of (2.13), (2.8) and (2.15), we have the required result. □

Theorem 14

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta , \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho-\xi n)<1+\min\{\Re(-\nu),\Re(\sigma+\sigma '-\eta),\Re(\sigma+\nu'-\eta)\}$$ then the following formula holds:

\begin{aligned}[b]& \int_{0}^{\infty }z^{l-1}e^{-\delta z/2}W_{\tau,\zeta}( \delta z) \biggl\{ \biggl(I^{\sigma ,\sigma',\nu,\nu',\eta}_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda }^{(\alpha, \beta)} \biggl(r,a; \biggl( \frac{wz}{t} \biggr)^{\xi} \biggr) \biggr\} \biggr) (x) \biggr\} \,dz \\ & \quad=\frac{x^{\rho+\eta-\sigma-\sigma '-1}}{\delta^{l}}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a; \biggl( \frac {w}{\delta x} \biggr)^{\xi} \biggr)\\ &\qquad{}*{}_{5} \Psi_{4} \left [ \textstyle\begin{array}{c}(1/2+\zeta+l, \xi), (1/2-\zeta+l,\xi),\\ (1/2-\tau +l,\xi), \end{array}\displaystyle \right . \\ &\qquad \left . \textstyle\begin{array}{c} (1-\rho-\nu,\xi), (1-\rho-\eta+\sigma+\sigma', \xi), (1-\rho-\eta+\sigma+\nu', \xi); \\ (1-\rho, \xi), (1-\rho-\eta+\sigma +\sigma'+\nu', \xi), (1-\rho+\sigma-\nu, \xi); \end{array}\displaystyle \biggl(\frac{w}{\delta x} \biggr)^{\xi} \right ]. \end{aligned}
(5.31)

Proof

The proof of Theorem 14 would run parallel to Theorem 13. Therefore, we omit the details. □

Theorem 15

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta , \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho+\xi n)>\max\{0,\Re(\eta-\sigma-\sigma'-\nu '),\Re(\nu-\sigma)\}$$ then the following formula holds:

\begin{aligned}[b]& \int_{0}^{\infty }z^{l-1}e^{-\delta z/2}W_{\tau,\zeta}( \delta z) \bigl\{ \bigl(D^{\sigma ,\sigma',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;(wzt)^{\xi} \bigr) \bigr\} \bigr) (x) \bigr\} \,dz \\ & \quad=\frac{x^{\rho -\eta+\sigma+\sigma'-1}}{\delta^{l}}S_{\mu,\lambda}^{(\alpha, \beta)} \biggl(r,a; \biggl( \frac{wx}{\delta} \biggr)^{\xi} \biggr)\\&\qquad{}*{}_{5} \Psi_{4} \left [ \textstyle\begin{array}{c}(1/2+\zeta+l, \xi), (1/2-\zeta+l,\xi),\\ (1/2-\tau+l, \xi), \end{array}\displaystyle \right . \\ &\qquad \left . \textstyle\begin{array}{c} (\rho,\xi), (\rho-\eta+\sigma+\sigma'+\nu', \xi), (\rho-\nu+\sigma, \xi); \\ (\rho-\nu, \xi), (\rho-\eta+\sigma+\sigma', \xi), (\rho-\eta+\sigma+\nu', \xi); \end{array}\displaystyle \biggl(\frac{wx}{\delta} \biggr)^{\xi} \right ]. \end{aligned}
(5.32)

Proof

To prove (5.32), by using the definition of the Whittaker transform as given in Eq. (5.26) and by using the result obtained in Eq. (4.4), we have

\begin{aligned}[b]& \int_{0}^{\infty }z^{l-1}e^{-\delta z/2}W_{\tau,\zeta}( \delta z) \bigl\{ \bigl(D^{\sigma ,\sigma',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;(wzt)^{\xi} \bigr) \bigr\} \bigr) (x) \bigr\} \,dz \\ &\quad =x^{\rho-\eta+\sigma+\sigma'-1}\sum_{n=1}^{\infty} \frac {2a_{n}^{\beta}(\lambda)_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}} \\ &\qquad \times\frac {\Gamma(\rho+\xi n)\Gamma(\rho-\eta+\sigma+\sigma'+\nu'+\xi n)\Gamma (\rho-\nu+\sigma+\xi n)}{\Gamma(\rho-\nu+\xi n)\Gamma(\rho-\eta+\sigma +\sigma'+\xi n)\Gamma(\rho-\eta+\sigma+\nu'+\xi n)}\frac{(wx)^{\xi n}}{n!} \\ & \qquad\times \int_{0}^{\infty}z^{l+\xi n-1}e^{-\delta z/2}W_{\tau ,\zeta}( \delta z)\,dz. \end{aligned}
(5.33)

By putting $$\delta z = y$$ and after a little simplification, we have

\begin{aligned}[b]& \int_{0}^{\infty }z^{l-1}e^{-\delta z/2}W_{\tau,\zeta}( \delta z) \bigl\{ \bigl(D^{\sigma ,\sigma',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;(wzt)^{\xi} \bigr) \bigr\} \bigr) (x) \bigr\} \,dz \\ &\quad =\frac{x^{\rho-\eta+\sigma+\sigma'-1}}{\delta^{l}}\sum_{n=1}^{\infty } \frac{2a_{n}^{\beta}(\lambda)_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}} \\ & \qquad{}\times \frac{\Gamma(\rho+\xi n)\Gamma(\rho-\eta+\sigma+\sigma'+\nu'+\xi n)\Gamma(\rho-\nu+\sigma+\xi n)}{\Gamma(\rho-\nu+\xi n)\Gamma(\rho-\eta +\sigma+\sigma'+\xi n)\Gamma(\rho-\eta+\sigma+\nu'+\xi n)} \biggl(\frac {wx}{\delta} \biggr)^{\xi n} \frac{1}{n!} \\ & \qquad{}\times \int_{0}^{\infty }y^{l+\xi n-1}e^{-y/2}W_{\tau,\zeta}(y)\,dy, \end{aligned}
(5.34)

By using the definition of the Whittaker transform, we have

\begin{aligned}[b]& \int_{0}^{\infty }z^{l-1}e^{-\delta z/2}W_{\tau,\zeta}( \delta z) \bigl\{ \bigl(D^{\sigma ,\sigma',\nu,\nu',\eta}_{0,x} \bigl\{ t^{\rho-1}S_{\mu,\lambda}^{(\alpha, \beta)} \bigl(r,a;(wzt)^{\xi} \bigr) \bigr\} \bigr) (x) \bigr\} \,dz \\ & \quad=\frac{x^{\rho-\eta+\sigma+\sigma'-1}}{\delta^{l}}\sum_{n=1}^{\infty } \frac{2a_{n}^{\beta}(\lambda)_{n}}{(a_{n}^{\alpha}+r^{2})^{\mu}}\frac{\Gamma (1/2+\zeta+l+\xi n)\Gamma(1/2-\zeta+l+\xi n)}{\Gamma(1/2-\tau+l+\xi n)} \\ & \qquad{}\times\frac{\Gamma(\rho+\xi n)\Gamma(\rho-\eta+\sigma+\sigma '+\nu'+\xi n)\Gamma(\rho-\nu+\sigma+\xi n)}{\Gamma(\rho-\nu+\xi n)\Gamma (\rho-\eta+\sigma+\sigma'+\xi n)\Gamma(\rho-\eta+\sigma+\nu'+\xi n)} \biggl(\frac{wx}{\delta} \biggr)^{\xi n} \frac{1}{n!},\end{aligned}
(5.35)

and interpreting the above equation, from the point of view of (2.13), (2.8) and (2.15), we get the required result. □

Theorem 16

Let $$x>0$$, $$\sigma, \sigma', \nu, \nu', \eta , \rho\in\mathbb{C}$$ and $$r,\alpha,\beta,\mu\in\mathbb{R}^{+}$$; $$|1/t|\leq1$$ be such that $$\Re(\eta)>0$$ and $$\Re(\rho-\xi n)<1+\min\{\Re(\nu'),\Re(\eta-\sigma -\sigma'),\Re(\eta-\sigma'-\nu)\}$$. Then the following formula holds:

\begin{aligned}[b]& \int_{0}^{\infty }z^{l-1}e^{-\delta z/2}W_{\tau,\zeta}( \delta z) \biggl\{ \biggl(D^{\sigma ,\sigma',\nu,\nu',\eta}_{x,\infty} \biggl\{ t^{\rho-1}S_{\mu,\lambda }^{(\alpha, \beta)} \biggl(r,a; \biggl( \frac{wz}{t} \biggr)^{\xi} \biggr) \biggr\} \biggr) (x) \biggr\} \,dz \\ & \quad=\frac{x^{\rho-\eta+\sigma+\sigma'-1}}{\delta^{l}}S_{\mu ,\lambda}^{(\alpha, \beta)} \biggl(r,a; \biggl( \frac{w}{\delta x} \biggr)^{\xi} \biggr)\\ &\qquad{}*{}_{5} \Psi_{4} \left [ \textstyle\begin{array}{c}(1/2+\zeta+l, \xi), (1/2-\zeta+l,\xi),\\ (1/2-\tau +l+\xi), \end{array}\displaystyle \right . \\ & \quad\quad \left . \textstyle\begin{array}{c} (1-\rho+\nu',\xi), (1-\rho+\eta-\sigma-\sigma', \xi), (1-\rho+\eta-\sigma'-\nu, \xi); \\ (1-\rho, \xi), (1-\rho+\eta-\sigma -\sigma'-\nu, \xi), (1-\rho-\sigma'+\nu', \xi); \end{array}\displaystyle \biggl(\frac{w}{\delta x} \biggr)^{\xi} \right ]. \end{aligned}
(5.36)

Proof

The proof of Theorem 16 would run parallel to Theorem 15. Therefore, we omit the details. □

Conclusion

The applications of fractional integral and differential formulas in communication theory, probability theory and groundwater pumping modeling were showed by many authors. Therefore, the fractional integral and differential formulas (of Marichev–Saigo–Maeda type) involving the extended generalized Mathieu series established in this paper will be very useful in the application point of view. Also, we expect to find some applications in obtaining the solutions of differential equations.

References

1. 1.

Hilfer, R.: Applications of Fractional Calculus in Physics, pp. 87–130. World Scientific, Singapore (2000)

2. 2.

Magin, R.L.: Fractional calculus in bioengineering, part 1. Crit. Rev. Biomed. Eng. 32(1), 1–104 (2004)

3. 3.

Srivastava, H.M., Kumar, D., Singh, J.: An efficient analytical technique for fractional model of vibration equation. Appl. Math. Model. 45, 192–204 (2017)

4. 4.

Benson, D.A., Meerschaert, M.M., Revielle, J.: Fractional calculus in hydrologic modeling: a numerical perspective. Adv. Water Resour. 51, 479–497 (2013)

5. 5.

Abdelkawy, M.A., Zaky, M.A., Bhrawy, A.H., Baleanu, D.: Numerical simulation of time variable fractional order mobile–immobile advection–dispersion model. Rom. Rep. Phys. 67(3), 773–791 (2015)

6. 6.

Zhao, J., Zheng, L., Chen, X., Zhang, X., Liu, F.: Unsteady Marangoni convection heat transfer of fractional Maxwell fluid with Cattaneo heat flux. Appl. Math. Model. 44, 497–507 (2017)

7. 7.

Moghaddam, B.P., Machado, J.A.T.: A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations. Comput. Math. Appl. 73(6), 1262–1269 (2017)

8. 8.

Sin, C.S., Zheng, L., Sin, J.S., Liu, F., Liu, L.: Unsteady flow of viscoelastic fluid with the fractional K-BKZ model between two parallel plates. Appl. Math. Model. 47, 114–127 (2017)

9. 9.

Razminia, A., Baleanu, D., Majd, V.J.: Conditional optimization problems: fractional order case. J. Optim. Theory Appl. 156(1), 45–55 (2013)

10. 10.

Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

11. 11.

Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

12. 12.

Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods. World Scientific, Berlin (2012)

13. 13.

Huang, L.L., Baleanu, D., Wu, G.C., Zeng, S.D.: A new application of the fractional logistic map. Rom. J. Phys. 61(7–8), 1172–1179 (2016)

14. 14.

Kumar, D., Singh, J., Baleanu, D.: Modified Kawahara equation within a fractional derivative with non-singular kernel. Therm. Sci. (2017). https://doi.org/10.2298/TSCI160826008K

15. 15.

Kumar, D., Singh, J., Baleanu, D.: A new fractional model for convective straight fins with temperature-dependent thermal conductivity. Therm. Sci. (2017). https://doi.org/10.2298/TSCI170129096K

16. 16.

Kumar, D., Singh, J., Baleanu, D., Sushila: Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel. Physica A 492, 155–167 (2018)

17. 17.

Kumar, D., Singh, J., Baleanu, D.: New numerical algorithm for fractional Fitzhugh–Nagumo equation arising in transmission of nerve impulses. Nonlinear Dyn. 91, 307–317 (2018)

18. 18.

Kumar, D., Agarwal, R.P., Singh, J.: A modified numerical scheme and convergence analysis for fractional model of Lienard’s equation. J. Comput. Appl. Math. (2017). https://doi.org/10.1016/j.cam.2017.03.011

19. 19.

Hajipour, M., Jajarmi, A., Baleanu, D.: An efficient nonstandard finite difference scheme for a class of fractional chaotic systems. J. Comput. Nonlinear Dyn. 13(2), Article ID 021013 (2017)

20. 20.

Baleanu, D., Jajarmi, A., Hajipour, M.: A new formulation of the fractional optimal control problems involving Mittag-Leffler nonsingular kernel. J. Optim. Theory Appl. 175(3), 718–737 (2017)

21. 21.

Baleanu, D., Jajarmi, A., Asad, J.H., Blaszczyk, T.: The motion of a bead sliding on a wire in fractional sense. Acta Phys. Pol. A 131(6), 1561–1564 (2017)

22. 22.

Jajarmi, A., Hajipour, M., Baleanu, D.: New aspects of the adaptive synchronization and hyperchaos suppression of a financial model. Chaos Solitons Fractals 99, 285–296 (2017)

23. 23.

Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

24. 24.

Baleanu, D., Guvenc, Z.B., Machado, J.A.T. (eds.): New Trends in Nanotechnology and Fractional Calculus Applications. Springer, Dordrecht (2010)

25. 25.

Kiryakova, V.: Generalized Fractional Calculus and Applications. Pitman Research Notes in Mathematics Series, vol. 301. Longman, Harlow (1994)

26. 26.

Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon (1993)

27. 27.

Yang, X.J., Srivastava, H.M., Machado, J.A.T.: A new fractional derivative without singular kernel: application to the modelling of the steady heat flow. Therm. Sci. 20, 753–756 (2016)

28. 28.

Caputo, M.: Elasticita e Dissipazione. Zani-Chelli, Bologna (1969)

29. 29.

Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 73–85 (2015)

30. 30.

Atangana, A.: On the new fractional derivative and application to nonlinear Fisher’s reaction–diffusion equation. Appl. Math. Comput. 273, 948–956 (2016)

31. 31.

Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20, 763–769 (2016)

32. 32.

McBride, A.C.: Fractional powers of a class of ordinary differential operators. Proc. Lond. Math. Soc. (3) 45, 519–546 (1982)

33. 33.

Kalla, S.L.: Integral operators involving Fox’s H-function I. Acta Mex. Cienc. Tecnol. 3, 117–122 (1969)

34. 34.

Kalla, S.L.: Integral operators involving Fox’s H-function II. Acta Mex. Cienc. Tecnol. 7, 72–79 (1969)

35. 35.

Kalla, S.L., Saxena, R.K.: Integral operators involving hypergeometric functions. Math. Z. 108, 231–234 (1969)

36. 36.

Kalla, S.L., Saxena, R.K.: Integral operators involving hypergeometric functions II. Rev. Univ. Nac. Tucumán Ser. A 24, 31–36 (1974)

37. 37.

Saigo, M.: A remark on integral operators involving the Gauss hypergeometric functions. Math. Rep. Kyushu Univ. 11, 135–143 (1978)

38. 38.

Saigo, M.: A certain boundary value problem for the Euler–Darboux equation I. Math. Jpn. 24(4), 377–385 (1979)

39. 39.

Saigo, M.: A certain boundary value problem for the Euler–Darboux equation II. Math. Jpn. 25(2), 211–220 (1980)

40. 40.

Saigo, M., Maeda, N.: More generalization of fractional calculus. In: Transform Methods and Special Functions, pp. 386–400 (1996)

41. 41.

Kiryakova, V.: A brief story about the operators of the generalized fractional calculus. Fract. Calc. Appl. Anal. 11(2), 203–220 (2008)

42. 42.

Baleanu, D., Kumar, D., Purohit, S.D.: Generalized fractional integrals of product of two H-functions and a general class of polynomials. Int. J. Comput. Math. (2015). https://doi.org/10.1080/00207160.2015.1045886

43. 43.

Marichev, O.I.: Volterra equation of Mellin convolution type with a Horn function in the kernel. Izv. Akad. Nauk. BSSR, Ser. Fiz.-Mat. Nauk 1, 128–129 (1974) (in Russian)

44. 44.

Srivastava, H.M., Karlson, P.W.: Multiple Gaussian Hypergeometric Series. Halsted Press, Chichester; Wiley, New York (1985)

45. 45.

Mathai, A.M., Saxena, R.K., Haubold, H.J.: The H-Functions: Theory and Applications. Springer, New York (2010)

46. 46.

Kober, H.: On fractional integrals and derivatives. Q. J. Math. 11, 193–212 (1940)

47. 47.

Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration of Arbitrary Order. Academic Press, New York (1974)

48. 48.

Kiryakova, V.S.: Generalized Fractional Calculus and Applications. Pitman Research Notes in Mathematics Series. Longman, Harlow; Wiley, New York (1993)

49. 49.

Srivastava, H.M., Saxena, R.K.: Operators of fractional integration and their applications. Appl. Math. Comput. 118, 1–52 (2001)

50. 50.

Saxena, R.K., Saigo, M.: Generalized fractional calculus of the H-function associated with the Appell function. J. Fract. Calc. 19, 89–104 (2001)

51. 51.

Mathieu, E.L.: Traité de Physique Mathématique, VI–VII: Théorie de L’élasticité des Corps Solides. Gauthier-Villars, Paris (1890)

52. 52.

Emersleben, O.: Uber die Reihe. Math. Ann. 125, 165–171 (1952)

53. 53.

Pogány, T.K., Srivastava, H.M., Tomovski, Ž.: Some families of Mathieu a-series and alternating Mathieu a-series. Appl. Math. Comput. 173, 69–108 (2006)

54. 54.

Cerone, P., Lenard, C.T.: On integral forms of generalized Mathieu series. JIPAM. J. Inequal. Pure Appl. Math. 4, Article ID 100 (2003)

55. 55.

Diananda, P.H.: Some inequalities related to an inequality of Mathieu. Math. Ann. 250, 95–98 (1980)

56. 56.

Tomovski, Ž., Trencevski, K.: On an open problem of Bai-Ni Guo and Feng Qi. J. Inequal. Pure Appl. Math. 4(2), Article ID 29 (2003)

57. 57.

Tomovski, Ž., Pogány, T.K.: Integral expressions for Mathieu-type power series and for the Butzer–Flocke–Hauss Ω-function. Fract. Calc. Appl. Anal. 14(4), 623–634 (2011)

58. 58.

Milovanović, G.V., Pogány, T.K.: New integral forms of generalized Mathieu series and related applications. Appl. Anal. Discrete Math. 7, 180–192 (2013)

59. 59.

Srivastava, H.M., Tomovski, Ž.: Some problems and solutions involving Mathieu’ series and its generalizations. J. Inequal. Pure Appl. Math. 5(2), Article ID 45 (2004)

60. 60.

Tomovski, Ž.: New double inequality for Mathieu series. Publ. Elektroteh. Fak. Univ. Beogr., Ser. Mat. 15, 79–83 (2004)

61. 61.

Elezovic, N., Srivastava, H.M., Tomovski, Ž.: Integral representations and integral transforms of some families of Mathieu type series. Integral Transforms Spec. Funct. 19(7), 481–495 (2008)

62. 62.

Srivastava, H.M., Tomovski, Ž., Leskovski, D.: Some families of Mathieu-type series and Hurwitz–Lerch Zeta functions and associated probability distributions. Appl. Comput. Math. 14(3), 349–380 (2015)

63. 63.

Tomovski, Ž.: Integral representations of generalized Mathieu series via Mittag-Leffler type functions. Fract. Calc. Appl. Anal. 10(2), 127–138 (2007)

64. 64.

Tomovski, Ž.: New integral and series representations of the generalized Mathieu series. Appl. Anal. Discrete Math. 2(2), 205–212 (2008)

65. 65.

Tomovski, Ž., Mehrez, M.: Some families of generalized Mathieu-type power series, associated probability distributions and related functional inequalities involving complete monotonicity and log-convexity. Math. Inequal. Appl. 20(4), 973–986 (2017)

66. 66.

Kiryakova, V.: On two Saigo’s fractional integral operators in the class of univalent functions. Fract. Calc. Appl. Anal. 9, 159–176 (2006)

67. 67.

Pohlen, T.: The Hadamard product and universal power series. Ph.D. thesis, Universität Trier, Trier, Germany (2009)

68. 68.

Srivastava, H.M., Agarwal, R., Jain, S.: Integral transform and fractional derivative formulas involving the extended generalized hypergeometric functions and probability distributions. Math. Methods Appl. Sci. 40, 255–273 (2017)

69. 69.

Srivastava, H.M., Agarwal, R., Jain, S.: A family of the incomplete hypergeometric functions and associated integral transform and fractional derivative formulas. Filomat 31, 125–140 (2017)

70. 70.

Sneddon, I.N.: The Use of Integral Transforms. Tata McGraw-Hill, New Delhi (1979)

71. 71.

Schiff, J.L.: The Laplace Transform, Theory and Applications. Springer, New York (1999)

Author information

All authors contributed equally to the present investigation. All authors read and approved the final manuscript.

Correspondence to Serkan Araci.

Ethics declarations

Competing interests

The authors declare to have no competing interests. 