Analysis of Generalized Bessel–Maitland Function and Its Properties

Abstract

In this article, we introduce the generalized Bessel–Maitland function (EGBMF) using the extended beta function and some important properties obtained. Thus, we first show interesting relationships of this function with Laguerre polynomials and the Whittaker functions. We also introduce and prove some properties of the derivatives associated with EGBMF. In this sense, we establish a result relative to the extended fractional derivatives of Riemann–Liouville. Furthermore, the Mellin transform of this function is evaluated in terms of the generalized Wright hypergeometric function, and its Euler transform is also obtained. Finally, we derive several graphical representations using the Gauss quadrature and the Laguerre–Gauss quadrature methods, which show that the numerical and theoretical simulations are consistent. The results derived from this research can be potentially useful in applications in several fields, in particular, physics, applied mathematics, and engineering.

1. Introduction

The theory of the Bessel function is closely related to the theory of certain types of differential equations. A detailed description of the applications of the Bessel function in engineering or natural sciences is provided in Watson [1]. On the other hand, the Bessel–Maitland function is a generalization of the Bessel function introduced by Edward Maitland Wright [2] and which is given by

$$J_{\nu }^{\mu }\left(z\right)=\sum _{n=0}^{\infty }\frac{{(-z)}^n}{\Gamma (\mu n+\nu +1)n!}.$$

(1)

where $\mu ,\nu \in \mathbb{C};\Re \left(\mu \right)\ge 0,\Re \left(\nu \right)\ge -1.$

In recent years, the research of many mathematicians has focused on establishing some generalizations (or unifications) and introducing various integral transforms of the Bessel function (see [3,4,5,6,7,8,9,10,11]). In this sense, M. Singh et al. [12] introduced and studied properties of the generalized Bessel–Maitland function defined as follows

$$J_{\nu ,q}^{\mu ,\gamma }\left(z\right)=\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_{qn}{(-z)}^n}{\Gamma (\mu n+\nu +1)n!},$$

(2)

where $\mu ,\nu ,\gamma \in \mathbb{C};\Re \left(\mu \right)\ge 0,\Re \left(\nu \right)\ge -1,\Re \left(\gamma \right)\ge 0$ and $q\in (0,1)\cup \mathbb{N}\mathrm{and}{\left(\gamma \right)}_0=1,{\left(\gamma \right)}_{qn}=\frac{\Gamma (\gamma +qn)}{\Gamma \left(\gamma \right)}$ denotes the generalized Pochhammer symbol (see [13]).

In this paper, using the definition of the extended beta function, we extend the generalized Bessel–Maitland function $J_{\nu ,q}^{\mu ,\gamma }\left(z\right)$ as follows

$$J_{\nu ,q}^{\mu ,\gamma ;c}(z;p)=\sum _{n=0}^{\infty }\frac{B_p(\gamma +qn,c-\gamma ){\left(c\right)}_{qn}{(-z)}^n}{B(\gamma ,c-\gamma )\Gamma (\mu n+\nu +1)n!},$$

(3)

$$(p>0,q\in \mathbb{N},\Re (c)>\Re (\gamma )>0)$$

which will be known as the extended generalized Bessel–Maitland function (EGBMF).

The above equation is obtained using the fact that $\frac{{\left(\gamma \right)}_{qn}}{{\left(c\right)}_{qn}}=\frac{B(\gamma +qn,c-\gamma )}{B(\gamma ,c-\gamma )},$ where

$$B_p(x,y)={\int }_0^1{t}^{x-1}{(1-t)}^{y-1}{e}^{\frac{-p}{t(1-t)}}dt,(\Re \left(p\right)>0,\Re \left(x\right)>0,\Re \left(y\right)>0).$$

(4)

Please note that for $p=0$, we obtain the classical beta function

$$B(x,y)={\int }_0^1{t}^{x-1}{(1-t)}^{y-1}dt=\frac{\Gamma \left(x\right)\Gamma \left(y\right)}{\Gamma (x+y)}.$$

(5)

Recent research established important properties of and results from the generalized Bessel–Maitland function. Thus, in [14], new integrals involving the generalized Bessel–Maitland function are investigated in terms of the hypergeometric and beta functions. Furthermore, some integral transforms and generalized integration formulas for the generalized Bessel–Maitland function are proposed in [15]. In one paper [16], a new integral with a generalized Bessel–Maitland kernel is introduced, and its properties are examined. Another interesting approach to the generalized Bessel–Maitland function is proposed in [17]. In this recent research, the integral representation of the function is established by applying the Gauss multiplication theorem and the representation of the beta function, as well as the Mellin-Barnes representation through the residue theorem.

Since the new formulation of EGBFM was introduced in (3), the objective of our research is to obtain interesting results, such as its characterization as an integral, its relationship with the Laguerre polynomials or the Whittaker function, its properties associated with the fractional derivative, and its integral transform. As we have just indicated, one of the objectives of our research is the study of EGBMF in the context of Fractional Calculus, given the relevance that this theory has currently reached in the field of mathematics, both from the theoretical point of view, and for its applications in various fields of science and engineering. From a theoretical point of view, the various definitions of derived fractional proposals are based on two conceptions: global (non-local) and local. In the first conception, the fractional derivative is defined by an integral transformation, so its nature is non-local and has “memory”. This global formulation is associated with the appearance of the Fractional Calculus itself, from the pioneering works of Euler, Laplace, Lacroix, Fourier, Abel, Liouville etc., to the establishment of the well-known Riemann–Liouville and Caputo definitions. More information about this definition of fractional derivative, its physical meaning, and its applications can be found in (see [18,19,20]). In the second conception of the fractional derivative (local), it is formulated using certain quotients of increments (see [21,22,23,24,25,26]). Recently, the theory and applications of ordinary differential equations and equations in partial derivatives, which involve derivatives defined in this local sense, constitute one of the fields that researchers have focused on most.

Finally, we will recall some notions necessary to develop the results that we will establish in the following sections.

The Wright generalized hypergeometric function is defined as (see [13])

$${}_p{\Psi }_q\left(z\right)={}_p{\Psi }_q\left[\begin{array}{cccc}(a_1,{\alpha }_1),& (a_2,{\alpha }_2),& \cdots ,& (a_p,{\alpha }_p);\\ & & & \\ (b_1,{\beta }_1),& (b_2,{\beta }_2),& \cdots ,& (b_q,{\beta }_q);\end{array}z\right],$$

(6)

$$=\sum _{k=0}^{\infty }\frac{{\prod }_{i=1}^p\Gamma (a_i,{\alpha }_{i}k)z^k}{{\prod }_{j=1}^q\Gamma (b_j,{\beta }_{j}k)k!},$$

where the coefficients ${\alpha }_i(i=1,2,\cdots ,p)$ and ${\beta }_j(j=1,2,\cdots ,q)$ are positive real numbers such that

$$1+\sum _{j=1}^q{\beta }_j-\sum _{i=1}^p{\alpha }_i\ge 0.$$

The Mellin transform (see [27]) of the function $f\left(z\right)$ is defined as

$$M[f\left(z\right);s]={\int }_0^{\infty }{z}^{s-1}f\left(z\right)dz=f^{*}\left(s\right),\Re \left(s\right)>0.$$

(7)

The inverse of Mellin transform is defined as

$$M^{-1}[f^{*}\left(s\right);z]=\frac{1}{2\pi i}\int f^{*}\left(s\right)z^{-s}ds.$$

The Euler (Beta) (see [27]) transform of the function $f\left(z\right)$ is defined as

$$B[f\left(z\right):a,b]={\int }_0^1{z}^{a-1}{(1-z)}^{b-1}f\left(z\right)dz.$$

(8)

In this paper, we use the extended Riemann–Liouville fractional derivative introduced by Özarslan and Özergin [10] which is defined as

$$D_x^{\mu ,p}\left\{f\left(x\right)\right\}=\frac{1}{\Gamma (-\mu )}{\int }_0^{x}f\left(t\right){(x-t)}^{-\mu -1}{e}^{\frac{-p{x}^2}{t(x-t)}}dt,\Re \left(\mu \right)<0,\Re \left(p\right)>0.$$

(9)

where the path of integration is a line from 0 to z in the complex t-plane. For the case $p=0$, we obtain the classical Riemann–Liouville fractional derivative operator as follows

$$D_x^{\mu }\left\{f\left(x\right)\right\}=\frac{1}{\Gamma (-\mu )}{\int }_0^{x}f\left(t\right){(x-t)}^{-\mu -1}dt,\Re \left(\mu \right)<0.$$

(10)

Some authors have found interesting results and applications of new extensions of this fractional derivative operator (see [28,29,30]).

2. Characterization of EGBMF as Integral

In this section, we establish the integral representation of EGBMF in Theorem 1. In addition, by giving special values to the parameters, new results are obtained as corollaries. Throughout the section, we will make the following assumptions: Let $p>0,q\in \mathbb{N},\Re \left(c\right)>\Re \left(\gamma \right)>0,\Re \left(\mu \right)>0,\Re \left(\nu \right)>-1$.

Theorem 1.

The following integral representation for the extended generalized Bessel–Maitland function holds

$$J_{\nu ,q}^{\mu ,\gamma ;c}(z;p)=\frac{1}{B(\gamma ,c-\gamma )}{\int }_0^1{t}^{\gamma -1}{(1-t)}^{c-\gamma -1}{e}^{\frac{-p}{t(1-t)}}{J}_{\nu ,q}^{\mu ,c}\left(t^{q}z\right)dt.$$

(11)

Proof. 

By substituting Equation (4) in Equation (3), we obtain

$$J_{\nu ,q}^{\mu ,\gamma ;c}(z;p)={\int }_0^1{t}^{\gamma +qn-1}{(1-t)}^{c-\gamma -1}{e}^{\frac{-p}{t(1-t)}}dt\sum _{n=0}^{\infty }\frac{{\left(c\right)}_{qn}{(-z)}^n}{B(\gamma ,c-\gamma )\Gamma (\mu n+\nu +1)n!}.$$

By the above assumptions, summation and integration can be interchanged, then we obtain

$$J_{\nu ,q}^{\mu ,\gamma ;c}(z;p)=\frac{1}{B(\gamma ,c-\gamma )}{\int }_0^1{t}^{\gamma -1}{(1-t)}^{c-\gamma -1}{e}^{\frac{-p}{t(1-t)}}\sum _{n=0}^{\infty }\frac{{\left(c\right)}_{qn}{(-t^{q}z)}^n}{\Gamma (\mu n+\nu +1)n!}dt,$$

Finally using (2), our result follows.  □

Corollary 1.

By setting $t=\frac{u}{1+u}$ in (11), we obtain the following special case of Theorem 1:

$$J_{\nu ,q}^{\mu ,\gamma ;c}(z;p)=\frac{1}{B(\gamma ,c-\gamma )}{\int }_0^{\infty }\frac{u^{\gamma -1}}{{(1+u)}^c}{e}^{\frac{-p{(1+u)}^2}{u}}{J}_{\nu ,q}^{\mu ,c}\left(\frac{u^{q}z}{{(1+u)}^q}\right)du.$$

(12)

Corollary 2.

By setting $t={sin}^2\theta$ in (11), we obtain the following special case of Theorem 1:

$$J_{\nu ,q}^{\mu ,\gamma ;c}(z;p)=\frac{2}{B(\gamma ,c-\gamma )}{\int }_0^{\frac{\pi }{2}}{(sin\theta )}^{2\gamma -1}{(cos\theta )}^{2c-2\gamma -1}{e}^{\frac{-p}{{sin}^2\theta {cos}^2\theta }}{J}_{\nu ,q}^{\mu ,c}\left(z{sin}^{2q}\theta \right)d\theta .$$

(13)

Corollary 3.

The recurrence relation for the EGBMF is given by the following formula:

$$J_{\nu ,q}^{\mu ,\gamma ;c}(z;p)=(\nu +1)J_{\nu +1,q}^{\mu ,\gamma ;c}(z;p)+\mu z\frac{d}{dz}{J}_{\nu +1,q}^{\mu ,\gamma ;c}(z;p).$$

(14)

Proof. 

Starting with right hand side of (14) and using (3), we obtain

$$(\nu +1)J_{\nu +1,q}^{\mu ,\gamma ;c}(z;p)+\mu z\frac{d}{dz}{J}_{\nu +1,q}^{\mu ,\gamma ;c}(z;p)=(\nu +1)\sum _{n=0}^{\infty }\frac{B_p(\gamma +qn,c-\gamma ){\left(c\right)}_{qn}{(-z)}^n}{B(\gamma ,c-\gamma )\Gamma (\mu n+\nu +2)n!}$$

$$+\mu z\frac{d}{dz}\sum _{n=0}^{\infty }\frac{B_p(\gamma +qn,c-\gamma ){\left(c\right)}_{qn}{(-z)}^n}{B(\gamma ,c-\gamma )\Gamma (\mu n+\nu +2)n!}$$

$$=\sum _{n=0}^{\infty }\frac{B_p(\gamma +qn,c-\gamma ){\left(c\right)}_{qn}{(-z)}^n(\mu n+\nu +1)}{B(\gamma ,c-\gamma )\Gamma (\mu n+\nu +2)n!}$$

$$=J_{\nu ,q}^{\mu ,\gamma ;c}(z;p).$$

 □

3. Properties of the EGBMF and Derivative

In this section, we demonstrate some interesting derivative properties associated with the generalized Bessel–Maitland function. In the following result, we consider the extended Riemann–Liouville fractional derivative defined in (9).

Theorem 2.

Let $p>0,q\in \mathbb{N},\Re \left(\lambda \right)>0,\Re \left(\mu \right)>0,\Re \left(\gamma \right)>0,\Re \left(\nu \right)>-1$, then

$$D_z^{\lambda -c,p}\left(z^{\lambda -1}{J}_{\nu ,q}^{\mu ,c}\left(z^q\right)\right)=\frac{z^{c-1}}{\Gamma (c-\lambda )}B(\lambda ,c-\lambda )J_{\nu ,q}^{\mu ,\lambda ;c}(z^q;p).$$

(15)

Proof. 

Replace $\mu$ by $\lambda -c$ in the definition of the extended fractional derivative operators, we obtain

$$D_z^{\lambda -c,p}\left(z^{\lambda -1}{J}_{\nu ,q}^{\mu ,c}\left(z^q\right)\right)=\frac{1}{\Gamma (c-\lambda )}{\int }_0^z{t}^{\lambda -1}{(z-t)}^{c-\lambda -1}{J}_{\nu ,q}^{\mu ,c}\left(t^q\right)e^{\frac{-p{z}^2}{t(z-t)}}dt$$

$$=\frac{z^{c-\lambda -1}}{\Gamma (c-\lambda )}{\int }_0^z{t}^{\lambda -1}{\left(1-\frac{t}{z}\right)}^{c-\lambda -1}{J}_{\nu ,q}^{\mu ,c}\left(t^q\right)e^{\frac{-p{z}^2}{t(z-t)}}dt$$

introducing $u=\frac{t}{z}$ in above equation, we have

$$=\frac{z^{c-1}}{\Gamma (c-\lambda )}{\int }_0^1{u}^{\lambda -1}{(1-u)}^{c-\lambda -1}{J}_{\nu ,q}^{\mu ,c}\left(u^q{z}^q\right)e^{\frac{-p}{u(1-u)}}du.$$

Finally, using (11), we obtain the desired result.  □

Theorem 3.

The derivative formula of the extended generalized Bessel–Maitland function is as follows:

$$\frac{d^n}{d{z}^n}{J}_{\nu ,q}^{\mu ,\gamma ;c}(z;p)={\left(c\right)}_q{(c+q)}_q\cdots {(c+(n-1)q)}_q{J}_{\nu +n\mu }^{\mu ,\nu +nq;c+nq}(z;p).$$

(16)

Proof. 

Taking derivative as regards ${}^{\prime }{z}^{\prime }$ in (3), we obtain

$$\frac{d}{dz}{J}_{\nu ,q}^{\mu ,\gamma ;c}(z;p)={\left(c\right)}_q{J}_{\nu +\mu }^{\mu ,\nu +q;c+q}(z;p).$$

(17)

So, by applying the derivative as regards ${}^{\prime }{z}^{\prime }$ in (17), we obtain

$$\frac{d^2}{d{z}^2}{J}_{\nu ,q}^{\mu ,\gamma ;c}(z;p)={\left(c\right)}_q{(c+q)}_q{J}_{\nu +2\mu }^{\mu ,\nu +2q;c+2q}(z;p).$$

Finally, if we apply the derivation as regards ${}^{\prime }{z}^{\prime }$n times, our result follows easily.  □

Theorem 4.

Under mentioned derivative formula of the extended generalized Bessel–Maitland function holds true:

$$\frac{d^n}{d{z}^n}\left[z^{\nu }{J}_{\nu ,q}^{\mu ,\gamma ;c}(\lambda z^{\mu };p)\right]=z^{\nu -n}{J}_{\nu -n,q}^{\mu ,\gamma ;c}(\lambda z^{\mu };p).$$

(18)

Proof.  

To prove (18) it is enough to replace z by $\lambda z^{\mu }$ in (3) and apply the derivative as regards ${}^{\prime }{z}^{\prime }$ n times of the product with $\lambda z^{\mu }$ with $J_{\nu ,q}^{\mu ,\gamma ;c}(\lambda z^{\mu };p)$. □

4. Relation of the EGBMF with Laguerre Polynomial and Whittaker Function

In this section, we establish the representation of the generalized extended Bessel–Maitland function in terms of Laguerre polynomials and the Whittaker function.

Theorem 5.

Let $p>0,q\in \mathbb{N},\Re \left(\mu \right)>0,\Re \left(\gamma \right)>0,\Re \left(\nu \right)>-1$. The following representation of EGBMF in terms of Laguerre polynomials is defined as:

$$e^{2p}{J}_{\nu ,q}^{\mu ,\gamma ;c}(z;p)=\frac{{(-1)}^k}{B(\gamma ,c-\gamma )}\sum _{m,n,k=0}^{\infty }\frac{L_n\left(p\right)L_m\left(p\right){\left(\gamma \right)}_{qk}{z}^k}{\Gamma (\mu k+\nu +1)k!}B(\gamma +m+qk+1,c+n-\gamma +1).$$

(19)

Proof.  

Using the generating function of Laguerre polynomials (see [13])

$$e^{\frac{-pt}{(1-t)}}=\sum _{n=0}^{\infty }{L}_n\left(p\right)t^n(1-t),$$

(20)

which can be expanded as follows

$$e^{\frac{-p}{t(1-t)}}=e^{-2p}\sum _{m,n=0}^{\infty }{L}_n\left(p\right)L_m\left(p\right)t^{m+1}{(1-t)}^{n+1},0<t<1.$$

(21)

Setting (21) in (11), we obtain

$$J_{\nu ,q}^{\mu ,\gamma ;c}(z;p)=\frac{1}{B(\gamma ,c-\gamma )}{\int }_0^1{t}^{\gamma -1}{(1-t)}^{c-\gamma -1}{e}^{-2p}\sum _{m,n=0}^{\infty }{L}_n\left(p\right)L_m\left(p\right)t^{m+1}{(1-t)}^{n+1}{J}_{\nu ,q}^{\mu ,c}\left(t^{q}z\right)dt.$$

Now, if we apply (2), we obtain

$$=\frac{{(-1)}^k{e}^{-2p}}{B(\gamma ,c-\gamma )}{\int }_0^1{t}^{\gamma -1}{(1-t)}^{c-\gamma -1}\sum _{m,n,k=0}^{\infty }\frac{L_n\left(p\right)L_m\left(p\right)t^{m+qk+1}{(1-t)}^{n+1}{\left(\gamma \right)}_{qk}{z}^k}{\Gamma (\mu k+\nu +1)k!}.$$

Interchanging integration and summation in the equation above, we have

$$=\frac{{(-1)}^k{e}^{-2p}}{B(\gamma ,c-\gamma )}\sum _{m,n,k=0}^{\infty }\frac{L_n\left(p\right)L_m\left(p\right){\left(\gamma \right)}_{qk}{z}^k}{\Gamma (\mu k+\nu +1)k!}{\int }_0^1{t}^{\gamma +m+qk}{(1-t)}^{c+n-\gamma }dt.$$

Using the definition of Beta function (5), we obtain

$$=\frac{{(-1)}^k{e}^{-2p}}{B(\gamma ,c-\gamma )}\sum _{m,n,k=0}^{\infty }\frac{L_n\left(p\right)L_m\left(p\right){\left(\gamma \right)}_{qk}{z}^k}{\Gamma (\mu k+\nu +1)k!}B(\gamma +m+qk+1,c+n-\gamma +1).$$

Multiplying on both sides by $e^{2p}$, we obtain the required result.  □

Theorem 6.

Let $p>0,q\in \mathbb{N},\Re \left(\mu \right)>0,\Re \left(\gamma \right)>0,\Re \left(\nu \right)>-1$. The EGBMF is represented in terms of the Whittaker function as follows:

$$e^{\frac{3p}{2}}{J}_{\nu ,q}^{\mu ,\gamma ;c}(z;p)=\frac{{(-1)}^k\Gamma (c-\gamma +1)}{B(\gamma ,c-\gamma )}\sum _{m,k=0}^{\infty }\frac{L_m\left(p\right){\left(c\right)}_{qk}{z}^k}{\Gamma (\mu k+\nu +1)k!}{p}^{\frac{m+\gamma +qk-1}{2}}{W}_{\frac{\gamma -2c-qk-m-1}{2},\frac{m+\gamma +qk}{2}}\left(p\right).$$

(22)

Proof.  

Using (20), we obtain

$$e^{\frac{-p}{t(1-t)}}=e^{-p}{e}^{\frac{-p}{t}}\sum _{m=0}^{\infty }{L}_m\left(p\right)t^m(1-t).$$

(23)

Introducing (23) in (11), we obtain

$$J_{\nu ,q}^{\mu ,\gamma ;c}(z;p)=\frac{1}{B(\gamma ,c-\gamma )}{\int }_0^1{t}^{\gamma -1}{(1-t)}^{c-\gamma -1}{e}^{-p}{e}^{\frac{-p}{t}}(1-t)\sum _{m=0}^{\infty }{L}_m\left(p\right)t^m{J}_{\nu ,q}^{\mu ,c}\left(t^{q}z\right)dt.$$

Now, we apply (2) in the equation above

$$J_{\nu ,q}^{\mu ,\gamma ;c}(z;p)=\frac{e^{-p}}{B(\gamma ,c-\gamma )}{\int }_0^1{t}^{\gamma -1}{(1-t)}^{c-\gamma }{e}^{\frac{-p}{t}}\sum _{m=0}^{\infty }{L}_m\left(p\right)t^m\sum _{k=0}^{\infty }\frac{{\left(c\right)}_{qk}{(-t^{q}z)}^k}{\Gamma (\mu k+\nu +1)k!}dt.$$

If we interchange integration and summation, we obtain

$$J_{\nu ,q}^{\mu ,\gamma ;c}(z;p)=\frac{{(-1)}^k{e}^{-p}}{B(\gamma ,c-\gamma )}\sum _{m,k=0}^{\infty }\frac{L_m\left(p\right){\left(c\right)}_{qk}{z}^k}{\Gamma (\mu k+\nu +1)k!}{\int }_0^1{t}^{m+qk+\gamma -1}{(1-t)}^{c-\gamma }{e}^{\frac{-p}{t}}dt.$$

Finally, using the following equation, we obtain

$${\int }_0^1{t}^{\mu -1}{(1-t)}^{\nu -1}{e}^{\frac{-p}{t}}dt=\Gamma \left(\nu \right)p^{\frac{\mu -1}{2}}{e}^{\frac{-p}{2}}{W}_{\frac{1-\mu -2\nu }{2},\frac{\mu }{2}}\left(p\right),[\Re \left(\nu \right)>0,\Re \left(p\right)>0].$$

We obtain our result.  □

5. Integral Transforms of EGBMF

In this section, we propose certain integral transforms for the EGBMF. To prove our results, the following assumptions are considered: $\mu ,\nu ,\gamma ,c,s\in \mathbb{C};\Re \left(\mu \right)\ge 0,\Re \left(\nu \right)\ge -1,\Re \left(c\right)>\Re \left(\gamma \right)>0,\Re \left(s\right)>0$ and $q\in \mathbb{N}$ which guarantee convergence.

Theorem 7.

The following Mellin transform of extended generalized Bessel–Maitland function holds true:

$$M\left(J_{\nu ,q}^{\mu ,\gamma ;c}(z;p);s\right)=\frac{\Gamma \left(s\right)\Gamma (s+c-\gamma )}{\Gamma \left(\gamma \right)\Gamma (c-\gamma )}{}_2{\Psi }_2\left[\begin{array}{cc}(c,q),& (\gamma +s,q);\\ & \\ (\nu +1,\mu ),& (c+2s,q);\end{array}-z\right].$$

(24)

Proof. 

Using Mellin transform (7) on extended generalized Bessel–Maitland function

$$M\left(J_{\nu ,q}^{\mu ,\gamma ;c}(z;p);s\right)={\int }_0^{\infty }{p}^{s-1}{J}_{\nu ,q}^{\mu ,\gamma ;c}(z;p)dp.$$

Now using (11)

$$=\frac{1}{B(\gamma ,c-\gamma )}{\int }_0^{\infty }{p}^{s-1}{\int }_0^1{t}^{\gamma -1}{(1-t)}^{c-\gamma -1}{e}^{\frac{-p}{t(1-t)}}{J}_{\nu ,q}^{\mu ,c}\left(t^{q}z\right)dtdp.$$

Reciprocate the order of integrals in the overhead equation, which is acceptable so far as the conditions in the statement of the Theorem, we obtain

$$=\frac{1}{B(\gamma ,c-\gamma )}{\int }_0^1{t}^{\gamma -1}{(1-t)}^{c-\gamma -1}{J}_{\nu ,q}^{\mu ,c}\left(t^{q}z\right){\int }_0^{\infty }{p}^{s-1}{e}^{\frac{-p}{t(1-t)}}dpdt.$$

Now setting $u=\frac{p}{t(1-t)}$ and using the fact that $\Gamma \left(s\right)={\int }_0^{\infty }{u}^{s-1}{e}^{-u}du$, we obtain

$$=\frac{\Gamma \left(s\right)}{B(\gamma ,c-\gamma )}{\int }_0^1{t}^{\gamma +s-1}{(1-t)}^{s+c-\gamma -1}{J}_{\nu ,q}^{\mu ,c}\left(t^{q}z\right)dt.$$

Using (2) in the equation above

$$=\frac{\Gamma \left(s\right)}{B(\gamma ,c-\gamma )}{\int }_0^1{t}^{\gamma +s-1}{(1-t)}^{s+c-\gamma -1}\sum _{n=0}^{\infty }\frac{{\left(c\right)}_{qn}{\left(t^{q}z\right)}^n}{\Gamma (\mu n+\nu +1)n!}dt.$$

If we interchange summation and integration, we have

$$=\frac{\Gamma \left(s\right)}{B(\gamma ,c-\gamma )}\sum _{n=0}^{\infty }\frac{{\left(c\right)}_{qn}{\left(z\right)}^n}{\Gamma (\mu n+\nu +1)n!}{\int }_0^1{t}^{\gamma +s+qn-1}{(1-t)}^{s+c-\gamma -1}dt.$$

Using the classical beta function (5), we obtain

$$=\frac{\Gamma \left(s\right)}{B(\gamma ,c-\gamma )}\sum _{n=0}^{\infty }\frac{{\left(c\right)}_{qn}{\left(z\right)}^n}{\Gamma (\mu n+\nu +1)n!}\frac{\Gamma (\gamma +s+qn)\Gamma (s+c-\gamma )}{\Gamma (2s+c+qn)}.$$

Considering ${\left(\gamma \right)}_{qn}=\frac{\Gamma (\gamma +qn)}{\Gamma \left(\gamma \right)},B(\gamma ,c-\gamma )=\frac{\Gamma \left(\gamma \right)\Gamma (c-\gamma )}{\Gamma \left(c\right)}$, we obtain

$$=\frac{\Gamma \left(s\right)\Gamma (s+c-\gamma )}{\Gamma \left(\gamma \right)\Gamma (c-\gamma )}\sum _{n=0}^{\infty }\frac{\Gamma (c+qn)\Gamma (\gamma +s+qn){\left(z\right)}^n}{\Gamma (\mu n+\nu +1)\Gamma (c+2s+qn)n!}.$$

Finally, using (6), our result follows.  □

Now, by putting $s=1$ in Theorem 7, we can obatin the following corollary.

Corollary 4.

The following result holds true:

$${\int }_0^{\infty }{J}_{\nu ,q}^{\mu ,\gamma ;c}(z;p)dp=\frac{\Gamma (c-\gamma +1)}{\Gamma (c-\gamma )\Gamma \left(\gamma \right)}{}_2{\Psi }_2\left[\begin{array}{cc}(c,q),& (\gamma +1,q);\\ & \\ (\nu +1,\mu ),& (c+2,q);\end{array}z\right].$$

(25)

Corollary 5.

The following inverse Mellin transform of EGBMF holds true:

$$J_{\nu ,q}^{\mu ,\gamma ;c}(z;p)=\frac{1}{2\pi i\Gamma \left(\gamma \right)\Gamma (c-\gamma )}{\int }_{\lambda -\iota \infty }^{\lambda +\iota \infty }\Gamma \left(s\right)\Gamma (s+c-\gamma ){}_2{\Psi }_2\left[\begin{array}{cc}(c,q),& (\gamma +s,q);\\ & \\ (\nu +1,\mu ),& (c+2s,q);\end{array}z\right]p^{-s}ds.$$

(26)

Theorem 8.

The Euler transform of extended generalized Bessel–Maitland function is given by

$$B\left\{J_{\nu ,q}^{\mu ,\gamma ;c}(z^{\mu };p):\nu +1,1\right\}=J_{\nu +1,q}^{\mu ,\gamma ;c}(1;p).$$

(27)

Proof. 

By definition of Euler transform (8) and (3), we obtain

$$B\left\{J_{\nu ,q}^{\mu ,\gamma ;c}(z^{\mu };p):\nu +1,1\right\}={\int }_0^1{z}^{\nu +1-1}{(1-z)}^{1-1}\sum _{n=0}^{\infty }\frac{B_p(\gamma +qn,c-\gamma ){\left(c\right)}_{qn}{(-z^{\mu })}^n}{B(\gamma ,c-\gamma )\Gamma (\mu n+\nu +1)n!}dz.$$

Reciprocating the summation and integral which is guaranteed under convergence condition, we obtain

$$=\sum _{n=0}^{\infty }\frac{{(-1)}^n{B}_p(\gamma +qn,c-\gamma ){\left(c\right)}_{qn}}{B(\gamma ,c-\gamma )\Gamma (\mu n+\nu +1)n!}{\int }_0^1{z}^{\mu n+\nu +1-1}{(1-z)}^{1-1}dz.$$

Applying definition of Beta function (5), we obtain our result.  □

6. Numerical Representations

In this section, we present some numerical simulations of the theoretical results obtained. Specifically, we focus on Theorem 1, Corollary 1, and Theorem 7 in terms of the variable z, to show the compatibility of the numerical solutions and the analytical expressions.

Evaluation of $J_{\nu ,q}^{\mu ,\gamma ;c}\left(z;p\right)$: The following figure presents a graphical simulation of (3), (11) and (12) vs. The variable z. The parameters are taken as $c=8$, $\gamma =6$, $\mu =2$, $\nu =3$, $p=1$ and $q=2$. The integrals in (11) and (12) are evaluated numerically using the Gaussian quadrature and Laguerre–Gauss quadrature methods (see [31]), respectively, and compare this with the series representation of $J_{\nu ,q}^{\mu ,\gamma ;c}\left(z;p\right)$ (Equation (3)). From Figure 1, we can note that the obtained results are identical.

Evaluation of $M\left(J_{\nu ,q}^{\mu ,\gamma ;c}\left(z;p\right);s\right)$: The below figure illustrates the evolution of $M\left(J_{\nu ,q}^{\mu ,\gamma ;c}\left(z;p\right);s\right)$ established from (7) and (24). The Mellin transform of $\left(J_{\nu ,q}^{\mu ,\gamma ;c}\left(z;p\right);s\right)$ is evaluated using th Laguerre–Gauss quadrature method. The values of the other parameters are as follows: $c=10$, $\gamma =6$, $\mu =2$, $\nu =3$, $s=2$ and $q=1$.

We have demonstrated that the numerical solutions derived by the Laguerre–Gauss quadrature method and the exact analytical formulation are compatible as shown in Figure 2.

7. Conclusions

In this article, we introduced a generalized Bessel–Maitland function using the extended beta function and obtained some interesting results. The extended generalized Bessel–Maitland function is expressed in terms of the Mittag–Leffler function, generalized Wright hypergeometric function, and Fox H-function. It is also worth noting that we have established results that connect the generalized extended Bessel–Maitland introduced with the Fractional Calculus or the integral transforms. Furthermore, to compare our theoretical and numerical results, some numerical simulations were performed. The obtained results show that there is an excellent agreement between the numerical solution obtained using the Gaussian quadrature and Laguerre–Gauss quadrature methods and our theoretical results. Finally, we think that the results of our research may have applicability in fields, such as applied sciences, mathematical physics, and engineering.