Bicomplex k-Mittag-Leffler Functions with Two Parameters: Theory and Applications to Fractional Kinetic Equations

Abstract

In this paper, we aim to extend the bicomplex two-parameter Mittag-Leffler (M-L) function by introducing a new k-parameter. This results in the definition of the bicomplex k-M-L function with two parameters. This generalization offers more flexibility and broader applicability in modeling complex fractional systems. We explore its key properties, develop new theorems, and establish the corresponding k-Riemann–Liouville fractional calculus within the bicomplex setting for the extended function. Furthermore, we solve several fractional differential equations using the bicomplex k-M-L function with two parameters. The results prove the enhanced flexibility and generality of the proposed function, particularly in deriving fractional kinetic equations, offering novel insights beyond existing bicomplex fractional models.

1. Introduction

Bicomplex numbers and fractional calculus have undergone significant advancements in recent years. The bicomplex number system has yielded diverse results within complex analysis, while fractional calculus has revolutionized the field of calculus. $\mathbb{BC}$ is a set of bicomplex numbers that originates from Segre’s work and is defined as (see [1,2,3]):

$$\begin{array}{c}\hfill \mathbb{BC}=\{\zeta ={\rho }_1+j{\rho }_2,{\rho }_1,{\rho }_2\in \mathbb{C}\}.\end{array}$$

(1)

Recently, authors have defined the bicomplex M-L functions as

Definition 1.

The bicomplex M-L function with one parameter is defined as [4]

$$\begin{array}{c}\hfill {\mathbf{E}}_M\left(\zeta \right)=\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_2(Mu+1)}},\end{array}$$

(2)

where ${\Gamma }_2$ is the bicomplex Gamma function, defined as (see [5])

$$\begin{array}{c}\hfill {\Gamma }_2\left(\zeta \right)={\int }_{{\rm Y}}{e}^{-T}\otimes T^{\zeta -1}\otimes dT,\end{array}$$

(3)

where $\zeta ,M,T\in \mathbb{BC}$, $T=t_1{e}_1+t_2{e}_2$, $t_1,t_2\in {\mathbb{R}}^{+}$ and ${\rm Y}=({\gamma }_1,{\gamma }_2)$, with ${\gamma }_1\equiv {\gamma }_1\left(t_1\right)$ and ${\gamma }_2\equiv {\gamma }_2\left(t_2\right)$.

Definition 2.

The bicomplex M-L function with two parameters is defined as [6]

$$\begin{array}{c}\hfill {\mathbf{E}}_{M,N}\left(\zeta \right)=\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_2(Mu+N)}},\end{array}$$

(4)

where $\zeta ,M,N\in \mathbb{BC}$.

Compared with previous work, namely the contribution of Sharma et al. [6], who introduced the bicomplex two-parameter M-L function, we introduce a deformation parameter k into the definition, resulting in a more general and flexible function. This generalization leads to a broader analytical form and enables more accurate modeling of fractional systems with complex dynamics. Furthermore, we situate our results within the context of recent research on generalized M-L functions by employing analytical and computational methods (see [4,7,8]), thereby establishing our research within the existing literature.

The parameter k plays a key role in modifying the behavior of the bicomplex M-L function. It affects the growth rate, decay rate, and convergence properties of the function, which are crucial for modeling memory-dependent or anomalous behavior in physical systems. Unlike symbolic generalizations, the k-extension introduces practical flexibility for a variety of applied applications, such as signal processing and viscoelastic modeling.

Recent developments in special functions with variable parameters, such as generalized M-L functions and k-fractional generalizations, have shown excellent potential for modeling complex phenomena in physics and engineering. Applications to anomalous diffusion, viscoelasticity, and memory-dependent systems often require flexible kernel functions, such as those introduced by Sharma et al. (2023) [6], Saxena et al. (2004) [7], and Dorrego Cerutti (2012) [8]. These studies demonstrate that analytical models incorporating tunable parameters, like k, are highly relevant for characterizing multiscale or fractional dynamics. Therefore, our work contributes to this growing body of literature by presenting a bicomplex generalization with enhanced modeling usefulness.

In this article, we introduce a new extension of the bicomplex M-L function with two parameters. We explore the fundamental aspects of the k-bicomplex M-L function in two parameters and discuss some important properties and theoretical results. We also establish the k-Riemann–Liouville fractional calculus in the bicomplex setting for this function. We also use the bicomplex k-M-L function to derive solutions for certain fractional differential equations. The paper is organized as follows:

In Section 2, we extend the M-L function to the bicomplex domain with two parameters. In Section 3, we obtain the k-bicomplex Riemann–Liouville fractional calculus of the bicomplex k-M-L function.Section 4 provides the bicomplex kinetic equation formulated using the bicomplex k-M-L function in two parameters. Section 5 contains the concluding remarks.

2. k-M-L Function of Two Parameter in Bicomplex Setting

In this section, we introduce the two-parameter bicomplex k-M-L function, along with some basic concepts and crucial features of these functions.

Definition 3.

Assume that $M,N,\zeta \in \mathbb{BC}$. The two parameter bicomplex k-M-L function is defined by

$$\begin{array}{c}\hfill {\mathbf{E}}_{k,M,N}\left(\zeta \right)=\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}(Mu+N)}},k\in {\mathbb{R}}^{+},\end{array}$$

(5)

where ${\Gamma }_{2,k}$ is explained in [9] as

$$\begin{array}{c}\hfill {\Gamma }_{2,k}\left(\zeta \right)={\int }_{{\rm Y}}{e}^{-{\displaystyle \frac{T^k}{k}}}\otimes T^{\zeta -1}\otimes dT,\end{array}$$

(6)

where $\zeta ={\rho }_1+j{\rho }_2={\xi }_1{e}_1+{\xi }_2{e}_2$, $M={\mu }_1{e}_1+{\mu }_2{e}_2=m_1+j{m}_{2}andN={\upsilon }_1{e}_1+{\upsilon }_2{e}_2=n_1+i{n}_2$, with $\left|{\mathrm{Im}}_{\mathrm{i}}\left(m_2\right)\right|<Re\left(m_1\right)\mathit{and}\left|{\mathrm{Im}}_{\mathrm{i}}\left(n_2\right)\right|<Re\left(n_1\right).$

Remark 1.

For specific values of the parameters $k, M, and N$, we can derive different bicomplex functions from the bicomplex k-M-L function (5), as follows

(i)       

Setting $k=1$, we obtain the bicomplex M-L function defined in Equation (4).

(ii)      

Setting $N=1$, we obtain the bicomplex k-M-L with one parameter, as defined in [10].

(iii)     

Setting $k=1,N=1$, we recover the bicomplex M-L function defined in (2).

(iv)      

Setting $k=1,N=1,and M=1$, we obtain the exponential function in the bicomplex domain

$${\displaystyle {\mathbf{E}}_{1,1,1}\left(\zeta \right)=\sum _{u=0}^{\infty }\frac{{\zeta }^u}{{\Gamma }_2(u+1)}=e^{\zeta }.}$$

(v)       

Setting $k=1,N=2,and M=1$, we have ${\displaystyle {\mathbf{E}}_{1,1,2}\left(\zeta \right)=\sum _{u=0}^{\infty }\frac{{\zeta }^u}{{\Gamma }_2(u+2)}=\frac{e^{\zeta }-1}{\zeta }.}$

(vi)      

Setting $k=1,N=1,and M=2$, we obtain ${\displaystyle {\mathbf{E}}_{1,2,1}(-{\zeta }^2)=\sum _{u=0}^{\infty }\frac{{(-1)}^u{\zeta }^{2u}}{{\Gamma }_2(2u+1)}=cos\left(\zeta \right).}$

(vii)     

Setting $k=1,N=2,and M=2$, we have ${\displaystyle {\mathbf{E}}_{1,2,2}\left(\zeta \right)=\sum _{u=0}^{\infty }\frac{{\zeta }^u}{{\Gamma }_2(2u+2)}=\frac{sinh\left(\sqrt{\zeta }\right)}{\sqrt{\zeta }}.}$

(viii)    

Setting $k=1,N=3,and M=1$, we have ${\displaystyle {\mathbf{E}}_{1,1,3}(-{\zeta }^2)=\sum _{u=0}^{\infty }\frac{{(-1)}^u{\zeta }^{2u}}{{\Gamma }_2(u+3)}=\frac{e^{\zeta }-1-\zeta }{{\zeta }^2}.}$

Theorem 1.

Let $\zeta ,M,N\in \mathbb{BC},k\in {\mathbb{R}}^{+}$, where $\zeta ={\rho }_1+j{\rho }_2={\xi }_1{e}_1+{\xi }_2{e}_2$, $M={\mu }_1{e}_1+{\mu }_2{e}_2=m_1+j{m}_2,m_1=u_1+i{u}_2,m_2=u_3+i{u}_4,N={\upsilon }_1{e}_1+{\upsilon }_2{e}_2=n_1+j{n}_2,n_1=w_1+i{w}_2,n_2=w_3+i{w}_4,$ then ${\mathbf{E}}_{k,M,N}\left(\zeta \right)$ is convergent for $\left|{\mathrm{Im}}_{\mathrm{i}}\left(m_2\right)\right|<Re\left(m_1\right)\mathit{and}\left|{\mathrm{Im}}_{\mathrm{i}}\left(n_2\right)\right|<Re\left(n_1\right).$

Proof. 

Consider the function

$$\begin{array}{c}\hfill {\mathbf{E}}_{k,M,N}\left(\zeta \right)=\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}(Mu+N)}},\end{array}$$

using idempotent representation (see [11]), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbf{E}}_{k,M,N}\left(\zeta \right)& =\sum _{u=0}^{\infty }{\displaystyle \frac{{\xi }_1^u}{{\Gamma }_k\left({\mu }_{1}u+{\upsilon }_1\right)}}{e}_1+\sum _{u=0}^{\infty }{\displaystyle \frac{{\xi }_2^u}{{\Gamma }_k\left({\mu }_{2}u+{\upsilon }_2\right)}}{e}_2\hfill \\ & ={\mathbf{E}}_{k,{\mu }_1,{\upsilon }_1}\left({\xi }_1\right)e_1+{\mathbf{E}}_{k,{\mu }_2,{\upsilon }_2}\left({\xi }_2\right)e_2.\hfill \end{array}\end{array}$$

As $E_{k,{\mu }_1,{\upsilon }_1}\left({\xi }_1\right)$, and $E_{k,{\mu }_2,{\upsilon }_2}\left({\xi }_2\right)$ are complex k-M-L functions convergent for $Re\left({\mu }_1\right)>0,Re\left({\mu }_2\right)>0,Re\left({\upsilon }_1\right)>0,Re\left({\upsilon }_2\right)>0$ (see [8]), which ${\mu }_1,{\mu }_2,{\upsilon }_1,{\upsilon }_2\in \mathbb{C}, \mathrm{then}{E}_{k,M,N}\left(\zeta \right)$ is also a convergent function;

• thus, if $M=m_1+j{m}_2=(u_1+i{u}_2)+j(u_3+i{u}_4)$, since $Re\left({\mu }_1\right)>0\mathrm{and}Re\left({\mu }_2\right)>0$
• $⟹u_1+u_4>0,u_1-u_4>0.$ Thus, $-u_4<u_1<u_4⟺\left|{Im}_i\left(m_2\right)\right|<Re\left(m_1\right).$
• And if $N=n_1+j{n}_2=(w_1+i{w}_2)+j(w_3+i{w}_4)$, since $Re\left({\upsilon }_1\right)>0\mathrm{and}Re\left({\upsilon }_2\right)>0$ $⟹w_1-w_4>0,w_1+w_4>0,-w_4<w_1<w_4⟺\left|{\mathrm{Im}}_{\mathrm{i}}\left(n_2\right)\right|<Re\left(n_1\right).$

Therefore, the bicomplex k-M-L function ${\displaystyle {\mathbf{E}}_{k,M,N}\left(\xi \right)=\sum _{u=0}^{\infty }\frac{{\zeta }^u}{{\Gamma }_{2,k}(Mu+N)}}$ is a convergent function under the conditions $\left|{Im}_{\mathrm{i}}\left(m_2\right)\right|<Re\left(m_1\right)\mathit{and}\left|{Im}_{\mathrm{i}}\left(n_2\right)\right|<Re\left(n_1\right).$

This is the proof. □

Definition 4.

Let $\zeta ,M,N\in \mathbb{BC},\zeta ={\xi }_1{e}_1+{\xi }_2{e}_2,M={\mu }_1{e}_1+{\mu }_2{e}_2,N={\upsilon }_1{e}_1+{\upsilon }_2{e}_2,\left|Im\left({\mu }_2\right)\right|<Re\left({\mu }_1\right),|Im\left({\upsilon }_2\right)|<Re\left({\upsilon }_1\right)$, then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbf{E}}_{k,M,N}\left(\zeta \right)& =\sum _{u=0}^{\infty }{\zeta }^u\otimes k^{{\displaystyle \frac{-\zeta }{k}}}\otimes (Mu+N)\otimes e^{{\displaystyle \frac{\gamma \left(Mu+N\right)}{k}}}\hfill \\ & \otimes \left[\prod _{n=0}^{\infty }\left(1+{\displaystyle \frac{(Mn+N)}{nk}}\right)\otimes exp\left(-{\displaystyle \frac{(Mu+N)}{nk}}\right)\right].\hfill \end{array}\end{array}$$

(7)

The definition of the k-bicomplex Gamma function is given as follows (see [9])

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{\Gamma }_{2,k}\left(\zeta \right)}}=\zeta \otimes k^{-{\displaystyle \frac{\zeta }{k}}}\otimes e^{\gamma {\displaystyle \frac{\zeta }{k}}}\otimes \prod _{n=1}^{\infty }\left[\left(1+{\displaystyle \frac{\zeta }{nk}}\right)\otimes e^{-{\displaystyle \frac{\zeta }{nk}}}\right].\end{array}$$

(8)

Definition 5.

Given $\zeta ,M,N\in \mathbb{BC}$, we obtain a bicomplex k-Gamma function integral representation of the bicomplex k-M-L function

$$\begin{array}{c}\hfill {\mathbf{E}}_{k,M,N}\left(\zeta \right)=\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\int }_{{\rm Y}}{e}^{-\frac{T^k}{k}}\otimes T^{uM+N-1}\otimes dT}},\end{array}$$

(9)

where ${\rm Y}=({\gamma }_1,{\gamma }_2)$, as defined with ${\gamma }_1\equiv {\gamma }_1\left(t_1\right)$ and ${\gamma }_2\equiv {\gamma }_2\left(t_2\right)$.

Theorem 2.

Bicomplex Cauchy–Riemann equations are satisfied by the bicomplex k-M-L function, which is defined in Equation (5).

Proof. 

$\mathrm{Let}\zeta ,M,N\in \mathbb{BC},\zeta ={\rho }_1+j{\rho }_2={\xi }_1{e}_1+{\xi }_2{e}_2,M={\mu }_1{e}_1+{\mu }_2{e}_2,N={\upsilon }_1{e}_1+{\upsilon }_2{e}_2$, by using the idempotent represention of the two-parameter k-M-L function, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbf{E}}_{k,M,N}\left(\zeta \right)=& E_{k,{\mu }_1,{\upsilon }_1}\left({\xi }_1\right)e_1+E_{k,{\mu }_2,{\upsilon }_2,{\upsilon }_2}\left({\xi }_2\right)e_2\hfill \\ \hfill =& E_{k,{\mu }_1,{\upsilon }_1}\left({\rho }_1-i{\rho }_2\right)·\left({\displaystyle \frac{1+ij}{2}}\right)+E_{k,{\mu }_2,{\upsilon }_2}\left({\rho }_1+i{\rho }_2\right)·\left({\displaystyle \frac{1-ij}{2}}\right)\hfill \\ \hfill =& \left({\displaystyle \frac{1}{2}}\left[(E_{k,{\mu }_1,{\upsilon }_1}\left({\rho }_1-i{\rho }_2\right)+E_{k,{\mu }_2,{\upsilon }_2}\left({\rho }_1+i{\rho }_2\right)\right]\right)\hfill \\ & +j\left({\displaystyle \frac{i}{2}}\left[E_{k,{\mu }_1,{\upsilon }_1}\left({\rho }_1-i{\rho }_2\right)-E_{k,{\mu }_2,{\upsilon }_2}\left({\rho }_1+i{\rho }_2\right)\right]\right)\hfill \\ \hfill =& {\Xi }_1({\rho }_1,{\rho }_2)+j{\Xi }_2({\rho }_1,{\rho }_2),\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathrm{where}{\Xi }_1({\rho }_1,{\rho }_2)=& \left({\displaystyle \frac{1}{2}}\left[(E_{k,{\mu }_1,{\upsilon }_1}\left({\rho }_1-i{\rho }_2\right)+E_{k,{\mu }_2,{\upsilon }_2}\left({\rho }_1+i{\rho }_2\right)\right]\right),\hfill \\ \hfill {\Xi }_2({\rho }_1,{\rho }_2)=& \left({\displaystyle \frac{i}{2}}\left[E_{k,{\mu }_1,{\upsilon }_1}\left({\rho }_1-i{\rho }_2\right)-E_{k,{\mu }_2,{\upsilon }_2}\left({\rho }_1+i{\rho }_2\right)\right]\right).\hfill \end{array}\end{array}$$

Thus

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{\partial {\Xi }_1}{\partial {\rho }_1}}=& {\displaystyle \frac{1}{2}}\left(E_{k,{\mu }_1,{\upsilon }_1}^{\prime }\left({\rho }_1-i{\rho }_2\right)+E_{k,{\mu }_2,{\upsilon }_2}^{\prime }\left({\rho }_1+i{\rho }_2\right)\right)\hfill \\ \hfill {\displaystyle \frac{\partial {\Xi }_1}{\partial {\rho }_2}}=& -{\displaystyle \frac{i}{2}}\left(E_{k,{\mu }_1,{\upsilon }_1}^{\prime }\left({\rho }_1-i{\rho }_2\right)-E_{k,{\mu }_2,{\upsilon }_2}^{\prime }\left({\rho }_1+i{\rho }_2\right)\right)\hfill \\ \hfill {\displaystyle \frac{\partial {\Xi }_2}{\partial {\rho }_1}}=& {\displaystyle \frac{i}{2}}\left(E_{k,{\mu }_1,{\upsilon }_1}^{\prime }\left({\rho }_1-i{\rho }_2\right)-E_{k,{\mu }_2,{\upsilon }_2}^{\prime }\left({\rho }_1+i{\rho }_2\right)\right)\hfill \\ \hfill {\displaystyle \frac{\partial {\Xi }_1}{\partial {\rho }_2}}=& {\displaystyle \frac{1}{2}}\left(E_{k,{\mu }_1,{\upsilon }_1}^{\prime }\left({\rho }_1-i{\rho }_2\right)+E_{k,{\mu }_2,{\upsilon }_2}^{\prime }\left({\rho }_1+i{\rho }_2\right)\right),\hfill \end{array}\end{array}$$

thus, we conclude that

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial {\Xi }_1}{\partial {\rho }_1}}={\displaystyle \frac{\partial {\Xi }_2}{\partial {\rho }_2}},{\displaystyle \frac{\partial {\Xi }_1}{\partial {\rho }_2}}=-{\displaystyle \frac{\partial {\Xi }_2}{\partial {\rho }_1}}.\end{array}$$

Hence, the proof is completed. □

Theorem 3.

Let $\zeta ,M,N\in \mathbb{BC},\zeta ={\xi }_1{e}_1+{\xi }_2{e}_2,M={\mu }_1{e}_1+{\mu }_2{e}_2,N={\upsilon }_1{e}_1+{\upsilon }_2{e}_2,\mathrm{with}\left|{\mathrm{Im}}_{\mathrm{i}}\left(m_2\right)\right|<Re\left(m_1\right),\left|{\mathrm{Im}}_{\mathrm{i}}\left(n_2\right)\right|<Re\left(n_1\right)$. Under these assumptions, we establish some properties of the bicomplex k-M-L function in two parameters.

(i)       

${\displaystyle {\mathbf{E}}_{k,M,0}\left(\zeta \right)=\zeta \otimes {\mathbf{E}}_{k,M,M}\left(\zeta \right)}$.

(ii)      

${\displaystyle {\mathbf{E}}_{k,0,N}\left(\zeta \right)=\frac{1}{{\Gamma }_{2,k}\left(N\right)}\otimes \frac{1}{1-\zeta },\parallel \zeta \parallel <1}$.

(iii)     

${\displaystyle {\mathbf{E}}_{k,M,N}\left(\zeta \right)=\frac{1}{{\Gamma }_{2,k}\left(N\right)}+\zeta \otimes {\mathbf{E}}_{k,M,M+N}\left(\zeta \right)}$.

(iv)      

${\displaystyle {\mathbf{E}}_{k,M,N}\left(\zeta \right)=N\otimes {\mathbf{E}}_{k,M,N+1}\left(\zeta \right)+M\otimes \zeta \otimes \frac{d}{d\zeta }{\mathbf{E}}_{k,M,N+1}\left(\zeta \right)}$.

Proof. 

(i) Since

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \underset{\zeta \to 0}{lim}{\displaystyle \frac{1}{{\Gamma }_{2,k}\left(\zeta \right)}}=\underset{{\xi }_1\to 0}{lim}{\displaystyle \frac{1}{{\Gamma }_k\left({\xi }_1\right)}}{e}_1+\underset{{\xi }_2\to 0}{lim}{\displaystyle \frac{1}{{\Gamma }_k\left({\xi }_2\right)}}{e}_2=0.e_1+0.e_2=0.\end{array}\end{array}$$

According to the bicomplex k-M-L definition (5), setting $N=0$, then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbf{E}}_{k,M,0}\left(\zeta \right)& =\sum _{u=1}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}\left(Mu\right)}}=\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^{u+1}}{{\Gamma }_{2,k}(Mu+M)}}=\zeta \otimes {\mathbf{E}}_{k,M,M}\left(\zeta \right).\hfill \end{array}\end{array}$$

(ii) Setting $M=0$ in Equation (5) of the bicomplex k-M-L function, then

$$\begin{array}{c}\hfill {\mathbf{E}}_{k,0,N}\left(\zeta \right)=\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}\left(N\right)}}={\displaystyle \frac{1}{{\Gamma }_{2,k}\left(N\right)}}\otimes \sum _{u=0}^{\infty }{\zeta }^u={\displaystyle \frac{1}{{\Gamma }_{2,k}\left(N\right)}}\otimes {\displaystyle \frac{1}{1-\zeta }},\parallel \zeta \parallel <1.\end{array}$$

(iii) From Equation (5), then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathrm{L}.\mathrm{H}.\mathrm{S}=& {\mathbf{E}}_{k,M,N}\left(\zeta \right)=\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,K}(Mu+N)}}={\displaystyle \frac{1}{{\Gamma }_{2,k}\left(N\right)}}+\sum _{u=1}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}(Mu+N)}}\hfill \\ \hfill =& {\displaystyle \frac{1}{{\Gamma }_{2,k}\left(N\right)}}\otimes \sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^{u+1}}{{\Gamma }_{2,k}(Mu+M+N)}}={\displaystyle \frac{\zeta }{{\Gamma }_{2,k}\left(N\right)}}\otimes \sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}(Mu+M+N)}}\hfill \\ \hfill =& {\displaystyle \frac{\zeta }{{\Gamma }_{2,k}\left(N\right)}}\otimes {\mathbf{E}}_{k,M,M+N}\left(\zeta \right).\hfill \end{array}\end{array}$$

(iv)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathrm{R}.\mathrm{H}.\mathrm{S}=& N\otimes {\mathbf{E}}_{k,M,N+1}\left(\zeta \right)+M\otimes \zeta \otimes {\displaystyle \frac{d}{d\zeta }}{\mathbf{E}}_{k,M,N+1}\left(\zeta \right)\hfill \\ \hfill =& N\otimes \sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}(Mu+N+1)}}+M\otimes \zeta \otimes {\displaystyle \frac{d}{d\zeta }}\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}(Mu+N+1)}}\hfill \\ \hfill =& N\otimes \sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}(Mu+N+1)}}+M\otimes \sum _{u=0}^{\infty }{\displaystyle \frac{u{\zeta }^u}{{\Gamma }_{2,k}(Mu+N+1)}}\hfill \\ \hfill =& \sum _{u=0}^{\infty }{\displaystyle \frac{(Mu+N)\otimes {\zeta }^u}{{\Gamma }_{2,k}(Mu+N+1)}}=\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}(Mu+N)}}\hfill \\ \hfill =& {\mathbf{E}}_{k,M,N}\left(\zeta \right).\hfill \end{array}\end{array}$$

The proof has been finished. □

Theorem 4.

Let $\zeta ,M,N\in \mathbb{BC}$, $\zeta ={\xi }_1{e}_1+{\xi }_2{e}_2$, $M={\mu }_1{e}_1+{\mu }_2{e}_2$, $N={\upsilon }_1{e}_1+{\upsilon }_2{e}_2$, with $\left|{\mathrm{Im}}_{\mathrm{i}}\left(m_2\right)\right|<Re\left(m_1\right),\left|{\mathrm{Im}}_{\mathrm{i}}\left(n_2\right)\right|<Re\left(n_1\right)$, then

(i)       

${\mathbf{E}}_{k,M,N}\left(\zeta \right)+{\mathbf{E}}_{k,M,N}(-\zeta )=2{\mathbf{E}}_{k,2M,N}\left({\zeta }^2\right)$.

(ii)      

${\mathbf{E}}_{k,M,N}\left(\zeta \right)-{\mathbf{E}}_{k,M,N}(-\zeta )=2\zeta \otimes {\mathbf{E}}_{k,2M,M+N}\left({\zeta }^2\right)$.

Proof. 

(i) Using the bicomplex k-M-L definition (5), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbf{E}}_{k,M,N}\left(\zeta \right)& +{\mathbf{E}}_{k,M,N}(-\zeta )\hfill \\ & =\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}(Mu+N)}}+\sum _{u=0}^{\infty }{\displaystyle \frac{{(-1)}^u{\zeta }^u}{{\Gamma }_{2,k}(Mu+N)}}=\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u\left(1+{(-1)}^u\right)}{{\Gamma }_{2,k}(Mu+N)}}\hfill \\ & ={\displaystyle \frac{2}{{\Gamma }_{2,k}\left(N\right)}}+{\displaystyle \frac{2{\zeta }^2}{{\Gamma }_{2,k}(2M+N)}}+{\displaystyle \frac{2{\left({\zeta }^2\right)}^2}{{\Gamma }_{2,k}(2\left(2M\right)+N)}}+{\displaystyle \frac{2{\left({\zeta }^2\right)}^3}{{\Gamma }_{2,k}(3\left(2M\right)+N)}}+\dots \hfill \\ & =2\left({\displaystyle \frac{1}{{\Gamma }_{2,k}\left(N\right)}}+{\displaystyle \frac{{\zeta }^2}{{\Gamma }_{2,k}(2M+N)}}+{\displaystyle \frac{{\left({\zeta }^2\right)}^2}{{\Gamma }_{2,k}(2\left(2M\right)+N)}}+{\displaystyle \frac{{\left({\zeta }^2\right)}^3}{{\Gamma }_{2,k}(3\left(2M\right)+N)}}+\dots \right)\hfill \\ & =2\sum _{u=0}^{\infty }{\displaystyle \frac{{\left({\zeta }^2\right)}^u}{{\Gamma }_{2,k}(2Mu+N)}}=2{\mathbf{E}}_{k,2M,N}\left({\zeta }^2\right).\hfill \end{array}\end{array}$$

(ii) Using the bicomplex k-M-L definition (5), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbf{E}}_{k,M,N}\left(\zeta \right)-{\mathbf{E}}_{k,M,N}(-\zeta )=& \sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}(Mu+N)}}-\sum _{u=0}^{\infty }{\displaystyle \frac{{(-1)}^u{\zeta }^u}{{\Gamma }_{2,k}(Mu+N)}}=\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u\left(1-{(-1)}^u\right)}{{\Gamma }_{2,k}(Mu+N)}}\hfill \\ \hfill =& {\displaystyle \frac{2\zeta }{{\Gamma }_{2,k}(M+N)}}+{\displaystyle \frac{2{\zeta }^3}{{\Gamma }_{2,k}(3M+N)}}+{\displaystyle \frac{2{\zeta }^5}{{\Gamma }_{2,k}(5M+N)}}+\dots \hfill \\ \hfill =& 2\zeta \otimes \left({\displaystyle \frac{1}{{\Gamma }_{2,k}(M+N)}}+{\displaystyle \frac{{\zeta }^2}{{\Gamma }_{2,k}(3M+N)}}+{\displaystyle \frac{{\zeta }^4}{{\Gamma }_{2,k}(5M+N)}}+\dots \right)\hfill \\ \hfill =& 2\zeta \otimes \sum _{u=0}^{\infty }{\displaystyle \frac{{\left({\zeta }^2\right)}^u}{{\Gamma }_{2,k}(2Mu+M+N)}}=2\zeta \otimes {\mathbf{E}}_{k,2M,M+N}\left({\zeta }^2\right).\hfill \end{array}\end{array}$$

The proof is finished. □

Theorem 5

(Differential Relation). Let $\zeta ,M,N\in \mathbb{BC},\zeta ={\xi }_1{e}_1+{\xi }_2{e}_2,N={\upsilon }_1{e}_1+{\upsilon }_2{e}_2$,with $\left|{\mathrm{Im}}_{\mathrm{i}}\left(m_2\right)\right|<Re\left(m_1\right),\left|{\mathrm{Im}}_{\mathrm{i}}\left(n_2\right)\right|<Re\left(n_1\right)$, $M>0$, then

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{d}{d\zeta }}\right)}^{\left(n\right)}\left({\zeta }^{N-1}\otimes {\mathbf{E}}_{k,M,N}\left({\zeta }^M\right)\right)={\zeta }^{N-n-1}\otimes {\mathbf{E}}_{k,M,N-n}\left({\zeta }^M\right),n\ge 1.\end{array}$$

(10)

Proof. 

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathrm{L}.\mathrm{H}.\mathrm{S}=& {\left({\displaystyle \frac{d}{d\zeta }}\right)}^{\left(n\right)}\left({\zeta }^{N-1}\otimes {\mathbf{E}}_{k,M,N}\left({\zeta }^M\right)\right)={\left({\displaystyle \frac{d}{d\zeta }}\right)}^{\left(n\right)}\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^{Mu+N-1}}{{\Gamma }_{2,k}\left(Mu+N\right)}}\hfill \\ \hfill =& \sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^{Mu+N-n-1}}{{\Gamma }_{2,k}\left(Mu+N-n\right)}}={\zeta }^{N-n-1}\otimes \sum _{n=0}^{\infty }{\displaystyle \frac{{\left({\zeta }^M\right)}^u}{{\Gamma }_{2,k}\left(Mu+N-n\right)}}\hfill \\ \hfill =& {\zeta }^{N-n-1}\otimes {\mathbf{E}}_{k,M,N-n}\left({\zeta }^M\right).\hfill \end{array}\end{array}$$

The proof is completed. □

Theorem 6.

Let $\zeta ,M,N\in \mathbb{BC},\zeta ={\xi }_1{e}_1+{\xi }_2{e}_2,M={\mu }_1{e}_1+{\mu }_2{e}_2,N={\upsilon }_1{e}_1+{\upsilon }_2{e}_2,\mathit{with}\left|{\mathrm{Im}}_{\mathrm{i}}\left(m_2\right)\right|<Re\left(m_1\right),\left|{\mathrm{Im}}_{\mathrm{i}}\left(n_2\right)\right|<Re\left(n_1\right)$, $n\in \mathbb{N}$, then

$$\begin{array}{c}\hfill {\zeta }^n\otimes {\mathbf{E}}_{k,M,Mn+N}\left(\zeta \right)={\mathbf{E}}_{k,M,N}\left(\zeta \right)-\sum _{u=0}^{n-1}{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}\left(Mu+N\right)}}.\end{array}$$

(11)

Proof. 

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathrm{R}.\mathrm{H}.\mathrm{S}=& {\mathbf{E}}_{k,M,N}\left(\zeta \right)-\sum _{u=0}^{n-1}{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}\left(Mu+N\right)}}=\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}(Mu+N)}}-\sum _{u=0}^{n-1}{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}\left(Mu+N\right)}}\hfill \\ \hfill =& \sum _{u=n}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}\left(Mu+N\right)}}\left(\mathrm{put}u=s+n\right)\hfill \\ \hfill =& \sum _{s=0}^{\infty }{\displaystyle \frac{{\zeta }^{s+n}}{{\Gamma }_{2,k}\left(Ms+n+N\right)}}={\zeta }^n\otimes \sum _{s=0}^{\infty }{\displaystyle \frac{{\zeta }^s}{{\Gamma }_{2,k}\left(Ms+Mn+N\right)}}\hfill \\ \hfill =& {\zeta }^n\otimes {\mathbf{E}}_{k,M,Mn+N}\left(\zeta \right)\hfill \end{array}\end{array}$$

The proof is finished. □

Remark 2.

From Theorem 6, we obtain the following special cases for n = 2, 3, 4, respectively, where $\zeta ,M,N\in \mathbb{BC},\zeta ={\xi }_1{e}_1+{\xi }_2{e}_2,M={\mu }_1{e}_1+{\mu }_2{e}_2,N={\upsilon }_1{e}_1+{\upsilon }_2{e}_2$, with $\left|{\mathrm{Im}}_{\mathrm{i}}\left(m_2\right)\right|<Re\left(m_1\right),\left|{\mathrm{Im}}_{\mathrm{i}}\left(n_2\right)\right|<Re\left(n_1\right)$, $n\in \mathbb{N}$

(i)        

${\displaystyle {\zeta }^2{\mathbf{E}}_{k,M,2M+N}\left(\zeta \right)={\mathbf{E}}_{k,M,N}\left(\zeta \right)-\frac{1}{{\Gamma }_{2,k}\left(N\right)}-\frac{\zeta }{{\Gamma }_{2,k}(M+N)}.}$

(ii)       

${\displaystyle {\zeta }^3{\mathbf{E}}_{k,M,3M+N}\left(\zeta \right)={\mathbf{E}}_{k,M,N}\left(\zeta \right)-\frac{1}{{\Gamma }_{2,k}\left(N\right)}-\frac{\zeta }{{\Gamma }_{2,k}(M+N)}-\frac{{\zeta }^2}{{\Gamma }_{2,k}(2M+N)}.}$

(iii)      

${\displaystyle {\zeta }^4{\mathbf{E}}_{k,M,4M+N}\left(\zeta \right)={\mathbf{E}}_{k,M,N}\left(\zeta \right)-\frac{1}{{\Gamma }_{2,k}\left(N\right)}-\frac{\zeta }{{\Gamma }_{2,k}(M+N)}}$

$${\displaystyle -\frac{{\zeta }^2}{{\Gamma }_{2,k}(2M+N)}-\frac{{\zeta }^3}{{\Gamma }_{2,k}(3M+N)}.}$$

3. Fractional Calculus’s Effect on the Bicomplex $\mathit{k}$-M-L Function

The authors of [12] defined k-Bicomplex Riemann–Liouville in 2024. The following is an introduction to fractional calculus:

The k-Riemann–Liouville Fractional integration in a bicomplex setting is provided by

$$\begin{array}{c}\hfill {}_{0}I_{k,M}^{\delta }F\left(M\right)={\displaystyle \frac{1}{k{\Gamma }_{2,k}\left(\delta \right)}}\otimes {\int }_0^M{(M-\Psi )}^{{\displaystyle \frac{M}{k}}-1}\otimes F(\Psi )\otimes d\Psi ,k>0,\end{array}$$

(12)

where $M,\delta ,\Psi \in \mathbb{BC}$, $M={\mu }_1{e}_1+{\mu }_2{e}_2=m_1+j{m}_2$, with $Re\left(m_1\right)>|Im\left(m_2\right)|$, $\delta ={ϵ}_1+j{ϵ}_2={\delta }_1{e}_1+{\delta }_2{e}_2$, with $Re\left({\delta }_1\right)>0,Re\left({\delta }_2\right)>0$. $\Psi ={\psi }_1{e}_1+{\psi }_2{e}_2$, ${\psi }_1,{\psi }_2\in {\mathbb{R}}^{+}$.

The expression for a bicomplex function’s k-Riemann–Liouville fractional derivative F of order $\delta$ is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left({}_{0}D_{k,M}^{\delta }F\right)\left(M\right)& ={}_{0}D_{k,M}^{\alpha }\left({}_{0}D_{k,M}^{-(\alpha -\delta )}F\left(M\right)\right)\hfill \\ & ={\displaystyle \frac{1}{k{\Gamma }_{2,k}(\alpha -\delta )}}{\displaystyle \frac{d^{\alpha }}{d{M}^{\alpha }}}{\int }_0^M{(M-\Psi )}^{{\displaystyle \frac{\alpha -\delta }{k}}-1}\otimes F(\Psi )\otimes d\Psi ,\hfill \end{array}\end{array}$$

(13)

where $M,\delta ,\Psi \in \mathbb{BC}$, $M={\mu }_1{e}_1+{\mu }_2{e}_2=m_1+j{m}_2$, with $Re\left(m_1\right)>|Im\left(m_2\right)|$, $\delta ={ϵ}_1+j{ϵ}_2={\delta }_1{e}_1+{\delta }_2{e}_2$, with $Re\left({\delta }_1\right)>0,Re\left({\delta }_2\right)>0$ and $\alpha =\left[Re\left({ϵ}_1\right)\right]+1$. $\Psi ={\psi }_1{e}_1+{\psi }_2{e}_2$, ${\psi }_1,{\psi }_2\in {\mathbb{R}}^{+}$.

Definition 6.

Assume that $\zeta \in \mathbb{BC}$. The k-bicomplex Fox–Wright function is given by (see [10])

$$\begin{array}{c}\hfill {}_p\Psi _p^k={}_p\Psi _q^k\left[\begin{array}{c}{\left(X_n,V_n\right)}_{1,p}\hfill \\ {\left(Y_m,W_m\right)}_{1,q}\hfill \end{array};\zeta \right]=\sum _{u=0}^{\infty }{\displaystyle \frac{{\prod }_{n=1}^p{\Gamma }_{2,k}(X_{n}u+V_n)}{{\prod }_{m=1}^q{\Gamma }_{2,k}(Y_{m}u+W_m)}}\otimes {\displaystyle \frac{{\zeta }^u}{u!}},\end{array}$$

(14)

where the numerators and denominators of the function are indicated by p and q, respectively. $X_n,Y_m,V_n,W_m\in \mathbb{BC},m=1,2,\dots ,q;n=1,2,\dots ,p.$ S.t ${\displaystyle 1+\sum _{m=1}^q{Y}_m-\sum _{n=1}^p{X}_n\ge 0}$.

Theorem 7.

Let $\zeta ,\delta ,{\rm Y},M,N\in \mathbb{BC}$, $\zeta ={\rho }_1+j{\rho }_2={\xi }_1{e}_1+{\xi }_2{e}_2$, with $Re\left({\rho }_1\right)>|Im\left({\rho }_2\right)|$, $\delta ={\delta }_1{e}_1+{\delta }_2{e}_2$ and $\Psi ={\psi }_1{e}_1+{\psi }_2{e}_2$, ${\psi }_1,{\psi }_2\in {\mathbb{R}}^{+}$. Then, the k-Riemann–Liouville Fractional integration to the k-M-L function in a bicomplex setting given as

$$\begin{array}{c}\hfill {}_{0}I_{k,\zeta }^{\delta }\left[{\mathbf{E}}_{k,M,N}\left(\zeta \right)\right]={\left(\zeta \right)}^{{\displaystyle \frac{\delta }{k}}}\otimes {}_2\Psi _2^k\left[\begin{array}{c}(k,k),(k,k)\hfill \\ (M,N),(k,\delta +k)\hfill \end{array};{\displaystyle \frac{\zeta }{k}}\right].\end{array}$$

(15)

Proof. 

Using the definition of the k-bicomplex Riemann–Liouville fractional integration in Equation (12) and applying it to the bicomplex k-M-L function, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_{0}I_{k,\zeta }^{\delta }\left[{\mathbf{E}}_{k,M,N}\left(\zeta \right)\right]=& I_{k,\zeta }^{\delta }\left(\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}(Mu+N)}}\right)\hfill \\ \hfill =& {\displaystyle \frac{1}{k{\Gamma }_{2,k}\left(\delta \right)}}\otimes {\int }_0^{\zeta }{(\zeta -\Psi )}^{{\displaystyle \frac{\delta }{k}}-1}\otimes \sum _{u=0}^{\infty }{\displaystyle \frac{{\Psi }^u}{{\Gamma }_{2,k}(Mu+N)}}\otimes d\Psi \hfill \\ \hfill =& {\displaystyle \frac{1}{k{\Gamma }_{2,k}\left(\delta \right)}}\otimes \sum _{u=0}^{\infty }{\displaystyle \frac{1}{{\Gamma }_{2,k}(Mu+N)}}\otimes {\int }_0^{\zeta }{(\zeta -\Psi )}^{{\displaystyle \frac{\delta }{k}}-1}\otimes {\Psi }^u\otimes d\Psi .\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \mathrm{Let}I={\int }_0^{\zeta }{(\zeta -\Psi )}^{\frac{\delta }{k}-1}\otimes {\Psi }^u\otimes d\Psi =}& \left\{{\int }_0^{{\xi }_1}{({\xi }_1-{\psi }_1)}^{{\displaystyle \frac{{\delta }_1}{k}}-1}{\psi }_1^{u}d{\psi }_1\right\}{e}_1\hfill \\ & +\left\{{\int }_0^{{\xi }_2}{({\xi }_2-{\psi }_2)}^{{\displaystyle \frac{{\delta }_2}{k}}-1}{\psi }_2^{u}d{\psi }_2\right\}{e}_2\hfill \\ \hfill =& I_1{e}_1+I_2{e}_2,\hfill \end{array}\end{array}$$

where ${\displaystyle I_1={\int }_0^{{\xi }_1}{({\xi }_1-{\psi }_1)}^{\frac{{\delta }_1}{k}-1}{\psi }_1^{u}d{\psi }_1,I_2={\int }_0^{{\xi }_2}{({\xi }_2-{\psi }_2)}^{\frac{{\delta }_2}{k}-1}{\psi }_2^{u}d{\psi }_2.}$

Let $t_1={\displaystyle \frac{{\psi }_1}{{\xi }_1}},{\psi }_1=t_1{\xi }_1.$ When ${\psi }_1=0\Rightarrow t_1=0;when{\psi }_1={\xi }_1\Rightarrow t_1=1$.

Then,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\displaystyle I_1={\int }_0^1{\left(1-t_1\right)}^{\frac{{\delta }_1}{k}-1}{\xi }_1^{\frac{{\delta }_1}{k}+u}{t}_1^{u}d{t}_1,I_2={\int }_0^1{\left(1-t_2\right)}^{\frac{{\delta }_2}{k}-1}{\xi }_2^{\frac{{\delta }_2}{k}+u}{t}_2^{u}d{t}_2.}\end{array}\end{array}$$

Therefore,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_{0}I_{k,\zeta }^{\delta }\left[{\mathbf{E}}_{k,M,N}\left(\zeta \right)\right]=& {\displaystyle \frac{1}{k{\Gamma }_{2,k}\left(\delta \right)}}\otimes \sum _{u=0}^{\infty }{\displaystyle \frac{1}{{\Gamma }_{2,k}(Mu+N)}}\otimes {\int }_0^1{\left(1-T\right)}^{{\displaystyle \frac{\delta }{k}}-1}\otimes {\zeta }^{{\displaystyle \frac{\delta }{k}}+u}\otimes T^u\otimes dT\hfill \\ \hfill =& {\displaystyle \frac{1}{k{\Gamma }_{2,k}\left(\delta \right)}}\otimes \sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^{\frac{\delta }{k}+u}}{{\Gamma }_{2,k}(Mu+N)}}\otimes {\int }_0^1{\left(1-T\right)}^{{\displaystyle \frac{\delta }{k}}-1}\otimes T^u\otimes dT\hfill \\ \hfill =& {\displaystyle \frac{1}{k{\Gamma }_{2,k}\left(\delta \right)}}\otimes \sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^{\frac{\delta }{k}+u}}{{\Gamma }_{2,k}(Mu+N)}}\otimes k{\beta }_{2,k}(\delta ,k(u+1))\hfill \\ \hfill =& \sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^{u+\frac{\delta }{k}}}{{\Gamma }_{2,k}\left(Mu+N\right)}}\otimes {\displaystyle \frac{{\Gamma }_{2,k}\left(k(u+1)\right)}{{\Gamma }_{2,k}(\delta +k(u+1))}}.\hfill \end{array}\end{array}$$

Then, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle {}_{0}I_{k,\zeta }^{\delta }\left[{\mathbf{E}}_{k,M,N}\left(\zeta \right)\right]=}& {\left(\zeta \right)}^{{\displaystyle \frac{\delta }{k}}}\otimes \sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}\left(Mu+N\right)}}\otimes {\displaystyle \frac{{\Gamma }_{2,k}(ku+k)}{{\Gamma }_{2,k}(\delta +ku+k)}}\hfill \\ \hfill =& {\left(\zeta \right)}^{{\displaystyle \frac{\delta }{k}}}\otimes \sum _{u=0}^{\infty }{\displaystyle \frac{{\Gamma }_{2,k}(ku+k)\otimes {\Gamma }_{2,k}(ku+k)}{{\Gamma }_{2,k}\left(Mu+N\right)\otimes {\Gamma }_{2,k}(\delta +ku+k)}}\otimes {\displaystyle \frac{{\left(\frac{\zeta }{k}\right)}^u}{u!}}\hfill \\ \hfill =& {\left(\zeta \right)}^{{\displaystyle \frac{\delta }{k}}}\otimes {}_2\Psi _2^k\left[\begin{array}{c}(k,k),(k,k)\hfill \\ (M,N),(k,\delta +k)\hfill \end{array};{\displaystyle \frac{\zeta }{k}}\right].\hfill \end{array}\end{array}$$

Hence, the theorem is proved. □

Corollary 1.

Suppose that $\delta ,\zeta ,N,M\in \mathbb{BC}$, $\zeta ={\rho }_1+j{\rho }_2={\xi }_1{e}_1+{\xi }_2{e}_2$, with $Re\left({\rho }_1\right)>|Im\left({\rho }_2\right)|$, $\delta ={\delta }_1{e}_1+{\delta }_2{e}_2$ and $\psi ={\psi }_1{e}_1+{\psi }_2{e}_2$, ${\psi }_1,{\psi }_2\in {\mathbb{R}}^{+}$. When $k=1$ is selected, the Riemann–Liouville fractional integration of the M-L function in bicomplex setting is given:

$$\begin{array}{c}\hfill {}_{0}I_{\zeta }^{\delta }\left[{\mathbf{E}}_{M,N}\left(\zeta \right)\right]={}_{0}I_{\zeta }^{\delta }\left[\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_2(Mu+N)}}\right]={\left(\zeta \right)}^{\delta }\otimes {}_2\Psi _2\left[\begin{array}{c}(1,1),(1,1)\hfill \\ (M,N),(1,\delta +1)\hfill \end{array};\zeta \right].\end{array}$$

(16)

Theorem 8.

Let $\delta ,\Psi ,\zeta ,N,M\in \mathbb{BC}$, the function ${\mathbf{E}}_{k,M,N}\left(\zeta \right)$ be piecewise continuous on $I^{\prime }=(0,\infty ),$ and integrable on any finite subinterval of $I=[0,\infty )$ and let $\alpha =\left[Re\left({ϵ}_1\right)\right]+1$. Then, the k-bicomplex Riemann–Liouville fractional derivative of ${\mathbf{E}}_{k,M,N}\left(\zeta \right)$ of order δ is given as

$$\begin{array}{c}\hfill {}_{0}D_{k,\zeta }^{\delta }\left[{\mathbf{E}}_{k,M,N}\left(\zeta \right)\right]=k^{-\alpha }{\zeta }^{{\displaystyle \frac{\alpha -\delta }{k}}-\alpha }\otimes {}_2\Psi _2^k\left[\begin{array}{c}(k,k),(k,k)\hfill \\ (M,N),(k,\alpha -\delta -k\alpha +k)\hfill \end{array};{\displaystyle \frac{\zeta }{k}}\right].\end{array}$$

(17)

Proof. 

From the definition of the k-bicomplex Riemann–Liouville fractional derivative in Equation (13), applying it to the bicomplex k-M-L function, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_{0}D_{k,\zeta }^{\delta }\left[{\mathbf{E}}_{k,M,N}\left(\zeta \right)\right]=& {}_{0}D_{k,\zeta }^{\delta }\left[{\mathbf{E}}_{k,M,N}\left(\zeta \right)\right]\left[\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^u}{{\Gamma }_{2,k}(Mu+N)}}\right]\hfill \\ \hfill =& {\displaystyle \frac{1}{k{\Gamma }_{2,k}(\alpha -\delta )}}\otimes {\displaystyle \frac{d^{\alpha }}{d{\zeta }^{\alpha }}}{\int }_0^{\zeta }{(\zeta -\Psi )}^{{\displaystyle \frac{\alpha -s}{k}}-1}\otimes {\mathbf{E}}_{k,M,N}(\Psi )\otimes d\Psi \hfill \\ \hfill =& {\displaystyle \frac{1}{k{\Gamma }_{2,k}(\alpha -\delta )}}\otimes {\displaystyle \frac{d^{\alpha }}{d{\zeta }^{\alpha }}}{\int }_0^{\zeta }{(\zeta -\Psi )}^{{\displaystyle \frac{\alpha -s}{k}}-1}\otimes \sum _{u=0}^{\infty }{\displaystyle \frac{{\Psi }^u}{{\Gamma }_{2,k}(Mu+N)}}\otimes d\Psi \hfill \\ \hfill =& {\displaystyle \frac{1}{k{\Gamma }_{2,k}(\alpha -\delta )}}\otimes \sum _{u=0}^{\infty }{\displaystyle \frac{1}{{\Gamma }_{2,k}(Mu+N)}}\otimes {\displaystyle \frac{d^{\alpha }}{d{\zeta }^{\alpha }}}{\int }_0^{\zeta }{(\zeta -\Psi )}^{{\displaystyle \frac{\alpha -s}{k}}-1}\otimes {\Psi }^u\otimes d\Psi \hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \mathrm{Let}I={\int }_0^{\zeta }{(\zeta -\Psi )}^{\frac{\alpha -\delta }{k}-1}\otimes {\Psi }^u\otimes d\Psi =}& \left\{{\int }_0^{{\xi }_1}{({\xi }_1-{\psi }_1)}^{{\displaystyle \frac{\alpha -{\delta }_1}{k}}-1}{\psi }_1^{u}d{\psi }_1\right\}{e}_1\hfill \\ & +\left\{{\int }_0^{{\xi }_2}{({\xi }_2-{\psi }_2)}^{{\displaystyle \frac{\alpha -{\delta }_2}{k}}-1}{\psi }_2^{u}d{\psi }_2\right\}{e}_2\hfill \\ \hfill =& I_1{e}_1+I_2{e}_2,\hfill \end{array}\end{array}$$

where ${\displaystyle I_1={\int }_0^{{\xi }_1}{({\xi }_1-{\psi }_1)}^{\frac{\alpha -{\delta }_1}{k}-1}{\psi }_1^{u}d{\psi }_1,I_2={\int }_0^{{\xi }_2}{({\xi }_2-{\psi }_2)}^{\frac{\alpha -{\delta }_2}{k}-1}{\psi }_2^{u}d{\psi }_2}$

Let $t_1={\displaystyle \frac{{\psi }_1}{{\xi }_1}},{\psi }_1=t_1{\xi }_1.$ When ${\psi }_1=0\Rightarrow t_1=0;when{\psi }_1={\xi }_1\Rightarrow t_1=1$. Then

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\displaystyle I_1={\int }_0^1{\left(1-t_1\right)}^{\frac{\alpha -{\delta }_1}{k}-1}{\xi }_1^{\frac{\alpha -{\delta }_1}{k}+u}{t}_1^{u}d{t}_1,I_2={\int }_0^1{\left(1-t_2\right)}^{\frac{\alpha -{\delta }_2}{k}-1}{\xi }_2^{\frac{\alpha -{\delta }_2}{k}+u}{t}_2^{u}d{t}_2.}\end{array}\end{array}$$

Therefore,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_{0}D_{k,\zeta }^{\delta }\left[{\mathbf{E}}_{k,M,N}\left(\zeta \right)\right]& \\ \hfill =& {\displaystyle \frac{1}{k{\Gamma }_{2,k}(\alpha -\delta )}}\otimes \sum _{u=0}^{\infty }{\displaystyle \frac{1}{{\Gamma }_{2,k}(Mu+N)}}\otimes {\displaystyle \frac{d^{\alpha }}{d{\zeta }^{\alpha }}}\left({\zeta }^{{\displaystyle \frac{\alpha -\delta }{k}}+u}\right){\int }_0^1{\left(1-T\right)}^{{\displaystyle \frac{\alpha -\delta }{k}}-1}\otimes T^u\otimes dT\hfill \\ \hfill =& {\displaystyle \frac{1}{{\Gamma }_{2,k}(\alpha -\delta )}}\otimes \sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^{\frac{\alpha -\delta }{k}+u-\alpha }}{{\Gamma }_{2,k}(Mu+N)}}\otimes {\displaystyle \frac{{\Gamma }_2\left(\frac{\alpha -\delta }{k}+u+1\right)}{{\Gamma }_2(\frac{\alpha -\delta }{k}+u-\alpha +1)}}\otimes {\beta }_{2,k}(\alpha -\delta ,ku+k)\hfill \\ \hfill =& \sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^{\frac{\alpha -\delta }{k}+u-\alpha }}{{\Gamma }_{2,k}\left(Mu+N\right)}}\otimes {\displaystyle \frac{{\Gamma }_2(\frac{\alpha -\delta }{k}+u+1)\otimes {\Gamma }_{2,k}(ku+k)}{{\Gamma }_2\left(\frac{\alpha -\delta }{k}+u-\alpha +1\right)\otimes {\Gamma }_{2,k}\left(\alpha -\delta +ku+k\right)}}\hfill \\ \hfill =& k^{-\alpha }\sum _{u=0}^{\infty }{\displaystyle \frac{{\zeta }^{\frac{\alpha -\delta }{k}+u-\alpha }}{{\Gamma }_{2,k}\left(Mu+N\right)}}\otimes {\displaystyle \frac{{\Gamma }_{2,k}(ku+k)}{{\Gamma }_{2,k}\left(\alpha -\delta +ku-k\alpha +k\right)}}\hfill \\ \hfill =& k^{-\alpha }{\zeta }^{{\displaystyle \frac{\alpha -\delta }{k}}-\alpha }\otimes \sum _{u=0}^{\infty }{\displaystyle \frac{{\Gamma }_{2,k}(ku+k)}{{\Gamma }_{2,k}\left(Mu+N\right)}}\otimes {\displaystyle \frac{{\Gamma }_{2,k}(ku+k)}{{\Gamma }_{2,k}\left(\alpha -\delta +ku-k\alpha +k\right)}}\otimes {\displaystyle \frac{{\left(\frac{\zeta }{k}\right)}^u}{u!}}\hfill \\ \hfill =& k^{-\alpha }{\zeta }^{{\displaystyle \frac{\alpha -\delta }{k}}-\alpha }\otimes {}_2\Psi _2^k\left[\begin{array}{c}(k,k),(k,k)\hfill \\ (M,N),(k,\alpha -\delta -k\alpha +k)\hfill \end{array};{\displaystyle \frac{\zeta }{k}}\right].\hfill \end{array}\end{array}$$

The proof is finally complete. □

Corollary 2.

Let ${\mathbf{E}}_{k,M,N}\left(\zeta \right)$ be piecewise continuous on $I^{\prime }=(0,\infty ),$$\delta ,\zeta ,\psi \in \mathbb{BC}$,

$\delta =\left[Re\left({ϵ}_1\right)\right]+1$ and integrable on any finite subinterval of $I=[0,\infty )$. Assuming $k=1$, the bicomplex Riemann–Liouville fractional derivative of order δ of ${\mathbf{E}}_{M,N}\left(\zeta \right)$ is provided as

$$\begin{array}{c}\hfill {}_{0}D_{\zeta }^{\delta }\left[{\mathbf{E}}_{M,N}\left(\zeta \right)\right]={\zeta }^{-\delta }\otimes {}_2\Psi _2\left[\begin{array}{c}(1,1),(1,1)\hfill \\ (M,N),(1,-\delta +1)\hfill \end{array};\zeta \right].\end{array}$$

(18)

4. Application

The kinetic equation was first provided in its standard form by the authors of [13], who later enlarged it to a fractional version and derived its solutions. The following is the expression for the fractional extension of the kinetic equation

$$\begin{array}{c}\hfill \mathbb{N}\left(t\right)-{\mathbb{N}}_0=-c^{\xi }{D}_t^{\xi }\mathbb{N}\left(t\right),\end{array}$$

(19)

where $D_t^{-\xi }$, and $\xi >0$ represents the Riemann–Liouville fractional integral operator.

In [7], the authors also considered generalized versions of the fractional kinetic equations that included M-L functions.

Kumar [14] was the first to define the bicomplex Laplace transform as

$$\begin{array}{c}\hfill L[G\left(s\right);\phi ]=\widehat{F}\left(\phi \right)={\int }_0^{\infty }F\left(s\right)e^{-\phi s}ds,\end{array}$$

(20)

where $s\ge 0$, $\delta ={ϵ}_1+j{ϵ}_2\in \mathbb{BC}$, and $G\left(s\right)$ is a bicomplex-valued function of exponential order $K\in \mathbb{R}$.

We obtain the bicomplex generalization of Equation (19). Replacing the Riemann–Liouville fractional operator $D_t^{-\xi }$ with the bicomplex-order Riemann–Liouville fractional operator $D_t^{-\zeta }$ results in the following

$$\begin{array}{c}\hfill \mathbb{N}\left(t\right)-{\mathbb{N}}_0=-c^{\zeta }{D}_t^{\zeta }\mathbb{N}\left(t\right),\end{array}$$

(21)

where $\zeta ={\rho }_1+j{\rho }_2\in \mathbb{BC}$, and the condition $Re\left({\rho }_1\right)>|Im\left({\rho }_2\right)|$ is satisfied.

The solution to the bicomplex extension of the fractional kinetic Equation (21) was recently presented by Bera et al. [15].

This section explores how to solve a few fractional differential equations using bicomplex k-M-L functions (5).

Theorem 9.

Suppose that $\zeta ,M,N\in \mathbb{BC},C>0$, with $\left|{\mathrm{Im}}_{\mathrm{i}}\left(m_2\right)\right|<Re\left(m_1\right)\mathit{and}\left|{\mathrm{Im}}_{\mathrm{i}}\left(n_2\right)\right|<Re\left(n_1\right).$ Then, the bicomplex generalization solution of the fractional kinetic equation

$$\begin{array}{c}\hfill \mathbb{N}\left(t\right)-{\mathbb{N}}_0{\mathbf{E}}_{k,M,N}\left(t\right)=-C^{\zeta }\otimes {}_{0}D_t^{-\zeta }\mathbb{N}\left(t\right),\end{array}$$

(22)

is

$$\begin{array}{c}\hfill \mathbb{N}\left(t\right)={\mathbb{N}}_0\sum _{u=0}^{\infty }{\displaystyle \frac{{\Gamma }_2(u+1)}{{\Gamma }_{2,k}(Mu+N)}}\otimes t^u{\mathbf{E}}_{\zeta ,u+1}\left(-{\left(Ct\right)}^{\zeta }\right).\end{array}$$

(23)

Proof. 

Laplace transformation is applied to the boss side of Equation (22) as follows

$$\begin{array}{c}\hfill \begin{array}{cc}& \mathbb{N}\left(\varphi \right)+C^{\zeta }\otimes {\varphi }^{-\zeta }\mathbb{N}\left(\varphi \right)={\mathbb{N}}_0\left(\sum _{u=0}^{\infty }{\displaystyle \frac{1}{{\Gamma }_{2,k}(Mu+N)}}\otimes {\int }_0^{\infty }{e}^{-\varphi t}{t}^{u}dt\right)\hfill \\ & \mathbb{N}\left(\varphi \right)\left[1+C^{\zeta }\otimes {\varphi }^{-\zeta }\right]={\mathbb{N}}_0\left(\sum _{u=0}^{\infty }{\displaystyle \frac{1}{{\Gamma }_{2,k}(Mu+N)}}\otimes {\displaystyle \frac{{\Gamma }_2(u+1)}{{\varphi }^{(u+1)}}}\right)\hfill \\ & \mathbb{N}\left(\varphi \right)={\mathbb{N}}_0\left(\sum _{u=0}^{\infty }{\displaystyle \frac{{\Gamma }_2(u+1)}{{\Gamma }_{2,k}(Mu+N)}}\otimes {\varphi }^{-(u+1)}\right)\otimes \sum _{r=0}^{\infty }{(-1)}^r{\left({\displaystyle \frac{C}{\varphi }}\right)}^{\zeta r}\hfill \\ & \mathrm{Applying}\mathrm{the}\mathrm{Laplace}\mathrm{inverse},\mathrm{obtaining}\hfill \\ & \mathbb{N}\left(t\right)={\mathbb{N}}_0\sum _{u=0}^{\infty }{\displaystyle \frac{{\Gamma }_2(u+1)}{{\Gamma }_{2,k}(Mu+N)}}\otimes \left(\sum _{r=0}^{\infty }{(-1)}^r{\left(C\right)}^{\zeta r}\otimes {\displaystyle \frac{t^{\zeta r+u}}{{\Gamma }_2(\zeta r+u+1)}}\right)\hfill \\ & ={\mathbb{N}}_0\sum _{u=0}^{\infty }{\displaystyle \frac{{\Gamma }_2(u+1)}{{\Gamma }_{2,k}(Mu+N)}}\otimes t^u\left(\sum _{r=0}^{\infty }{\displaystyle \frac{{\left(-{\left(Ct\right)}^{\zeta }\right)}^r}{{\Gamma }_2(\zeta r+u+1)}}\right)\hfill \\ & ={\mathbb{N}}_0\sum _{u=0}^{\infty }{\displaystyle \frac{{\Gamma }_2(u+1)}{{\Gamma }_{2,k}(Mu+N)}}\otimes t^u{\mathbf{E}}_{\zeta ,u+1}\left(-{\left(Ct\right)}^{\zeta }\right),\hfill \end{array}\end{array}$$

where ${\displaystyle {\mathbf{E}}_{\zeta ,u+1}\left(-{\left(Ct\right)}^{\zeta }\right)}$ is the M-L function in a bicomplex setting with two parameters [6].

Thus, the proof was completed. □

Theorem 10.

Assume that $M,N,\zeta \in \mathbb{BC},C>0$, with $\left|{\mathrm{Im}}_{\mathrm{i}}\left(m_2\right)\right|<Re\left(m_1\right)\mathit{and}\left|{\mathrm{Im}}_{\mathrm{i}}\left(n_2\right)\right|<Re\left(n_1\right).$ Then, the bicomplex generalization solution of the fractional kinetic equation

$$\begin{array}{c}\hfill \mathbb{N}\left(t\right)-{\mathbb{N}}_0{\mathbf{E}}_{k,M,N}(C^{\zeta }\otimes t^{\zeta })=-C^{\zeta }\otimes {}_{0}D_t^{-\zeta }\mathbb{N}\left(t\right),\end{array}$$

(24)

is

$$\begin{array}{c}\hfill \mathbb{N}\left(t\right)={\mathbb{N}}_0\sum _{u=0}^{\infty }{\displaystyle \frac{{\Gamma }_2(\zeta u+1)}{{\Gamma }_{2,k}(Mu+N)}}\otimes {\left(Ct\right)}^{\zeta u}\otimes {\mathbf{E}}_{\zeta ,\zeta u+1}\left(-{\left(Ct\right)}^{\zeta }\right).\end{array}$$

(25)

Proof. 

Since Theorems 9 and 10 are similar, the specifics of the proof are left out. □

Theorem 11.

Assume that $\zeta ,M,N\in \mathbb{BC},C>0,S>0$, with $\left|{\mathrm{Im}}_{\mathrm{i}}\left(m_2\right)\right|<Re\left(m_1\right)\mathit{and}$$|{\mathrm{Im}}_{\mathrm{i}}\left(n_2\right)|<Re\left(n_1\right).$ Then, the bicomplex generalization solution for the fractional kinetic equation

$$\begin{array}{c}\hfill \mathbb{N}\left(t\right)-{\mathbb{N}}_0{\mathbf{E}}_{k,M,N}(C^{\zeta }\otimes t^{\zeta })=-S^{\zeta }\otimes {}_{0}D_t^{-\zeta }\mathbb{N}\left(t\right),\end{array}$$

(26)

is

$$\begin{array}{c}\hfill \mathbb{N}\left(t\right)={\mathbb{N}}_0\sum _{u=0}^{\infty }{\displaystyle \frac{{\Gamma }_2(\zeta u+1)}{{\Gamma }_{2,k}(Mu+N)}}\otimes {\left(Ct\right)}^{\zeta u}\otimes {\mathbf{E}}_{\zeta ,\zeta u+1}\left(-{\left(St\right)}^{\zeta }\right).\end{array}$$

(27)

Proof. 

Since Theorems 9 and 11 are similar, the specifics of the proof are left out. □

Special cases The bicomplex M-L function, as described in (4), is obtained by setting $k=1$ in the bicomplex k-M-L function. The following corollaries result from the reduction in Theorems 9–11.

Corollary 3.

Let $\zeta ,M,N\in \mathbb{BC},C>0$, with $\left|{\mathrm{Im}}_{\mathrm{i}}\left(m_2\right)\right|<Re\left(m_1\right)\mathit{and}\left|{\mathrm{Im}}_{\mathrm{i}}\left(n_2\right)\right|<Re\left(n_1\right).$ Then, the bicomplex generalization solution to the fractional kinetic equation

$$\begin{array}{c}\hfill \mathbb{N}\left(t\right)-{\mathbb{N}}_0{\mathbf{E}}_{M,N}\left(t\right)=-C^{\zeta }\otimes {}_{0}D_t^{-\zeta }\mathbb{N}\left(t\right),\end{array}$$

(28)

is

$$\begin{array}{c}\hfill \mathbb{N}\left(t\right)={\mathbb{N}}_0\sum _{u=0}^{\infty }{\displaystyle \frac{{\Gamma }_2(u+1)}{{\Gamma }_2(Mu+N)}}\otimes t^u{\mathbf{E}}_{\zeta ,u+1}\left(-{\left(Ct\right)}^{\zeta }\right).\end{array}$$

(29)

Corollary 4.

Let $\zeta ,M,N\in \mathbb{BC},C>0$, with $\left|{\mathrm{Im}}_{\mathrm{i}}\left(m_2\right)\right|<Re\left(m_1\right)\mathit{and}\left|{\mathrm{Im}}_{\mathrm{i}}\left(n_2\right)\right|<Re\left(n_1\right).$ Then, the bicomplex generalization solution to the fractional kinetic equation

$$\begin{array}{c}\hfill \mathbb{N}\left(t\right)-{\mathbb{N}}_0{\mathbf{E}}_{M,N}(C^{\zeta }\otimes t^{\zeta })=-C^{\zeta }\otimes {}_{0}D_t^{-\zeta }\mathbb{N}\left(t\right),\end{array}$$

(30)

is

$$\begin{array}{c}\hfill \mathbb{N}\left(t\right)={\mathbb{N}}_0\sum _{u=0}^{\infty }{\displaystyle \frac{{\Gamma }_2(\zeta u+1)}{{\Gamma }_2(Mu+N)}}\otimes {\left(Ct\right)}^{\zeta u}\otimes {\mathbf{E}}_{\zeta ,\zeta u+1}\left(-{\left(Ct\right)}^{\zeta }\right).\end{array}$$

(31)

Corollary 5.

Let $\zeta ,M,N\in \mathbb{BC},C>0$, with $\left|{\mathrm{Im}}_{\mathrm{i}}\left(m_2\right)\right|<Re\left(m_1\right)\mathit{and}\left|{\mathrm{Im}}_{\mathrm{i}}\left(n_2\right)\right|<Re\left(n_1\right).$ Then, the bicomplex generalization solution to the fractional kinetic equation

$$\begin{array}{c}\hfill \mathbb{N}\left(t\right)-{\mathbb{N}}_0{\mathbf{E}}_{M,N}(C^{\zeta }\otimes t^{\zeta })=-S^{\zeta }\otimes {}_{0}D_t^{-\zeta }\mathbb{N}\left(t\right),\end{array}$$

(32)

is

$$\begin{array}{c}\hfill \mathbb{N}\left(t\right)={\mathbb{N}}_0\sum _{u=0}^{\infty }{\displaystyle \frac{{\Gamma }_2(\zeta u+1)}{{\Gamma }_2(Mu+N)}}\otimes {\left(Ct\right)}^{\zeta u}\otimes {\mathbf{E}}_{\zeta ,\zeta u+1}\left(-{\left(St\right)}^{\zeta }\right).\end{array}$$

(33)

The bicomplex exponential function can be obtained by setting $M=1,N=1,k=1$ in the bicomplex k-M-L function (see [10]).

5. Conclusions

This paper introduced a new extension of the bicomplex Mittag-Leffler function by incorporating a deformation parameter k. The study explored its analytical properties, conditions for convergence, and operational behavior within the framework of bicomplex fractional calculus. We proved key properties and theorems for this extended function and applied them in the context of k-Riemann–Liouville fractional calculus in the bicomplex setting. Practical relations were further formed to kinetic equations, showing the applicability of the function to real-world applications.

In addition to advancing the concept of bicomplex special functions, these findings offer practical tools for fractional calculus modeling of complex systems. The potential applications of this function in various areas of physics, engineering, and applied mathematics are promising. Investigating these generalizations and extensions is a fruitful area of future research.Furthermore, the methods devised herein can be extended to other types of special functions, forming new avenues to investigate.

In terms of applications, while we have shown the validity of the proposed function through fractional kinetic equations, this general expression can also be applied to other physical phenomena such as anomalous relaxation processes, viscoelasticity of complex materials, transport processes in disordered media, and modeling signal attenuation in MRI. These areas contain memory effects and nonlocal phenomena, which are typically modeled very accurately using generalized fractional models, for which our k-M-L function can be a very powerful tool of analysis.