On the Expansion of Legendre Polynomials in Bicomplex Space and Coupling with Fractional Operators

Abstract

In this paper, we introduce a novel version of the Legendre polynomials in the bicomplex system. We investigate the essential properties of the Legendre polynomial, focusing on its bicomplex structure, generating functions, orthogonality, and recurrence relations. We present a solution to the Legendre differential equation in bicomplex space. Additionally, we discuss both theoretical and practical contributions, especially in bicomplex Riemann Liouville fractional calculus. We numerically study the construction of bicomplex Legendre polynomials, orthogonality, spectral projection, coefficient decay, and spectral convergence in bicomplex space. The findings contribute to a deeper insight into bicomplex functions, paving the way for further developments in science and mathematical analysis, and providing a foundation for future research on special functions and fractional operators within the bicomplex setting.

1. Introduction

In 1892, Segre first introduced the notion of bicomplex numbers as an extension of the classical complex number system. They can be represented as ordered pairs of complex numbers and naturally define a four-dimensional vector space over the reals. Due to their structure, bicomplex numbers have significant applications in various fields, including digital signal processing, quantum mechanics, Dirac theory, and electromagnetism [1]. Over the past few years, numerous studies have laid the groundwork for bicomplex functions, with particular emphasis on the idempotent representation that enables operations to be carried out term by term [2,3,4,5].

Special functions are integral to bicomplex analysis, and extending classical functions to bicomplex variables has proven particularly fruitful, as exemplified by the binomial theorem, Gauss multiplication theorem, and Legendre duplication formula. Among the most prominent special functions are the Gamma and Beta functions, the Pochhammer symbol, and hypergeometric functions, which have played a central role in mathematical analysis and theoretical physics. Their utility spans a wide spectrum of applications, ranging from quantum mechanics and differential equations to number theory and mathematical statistics [4,6,7]. As bicomplex numbers emerged as a natural extension of the complex system to a four-dimensional setting, it became necessary to recast these functions in the bicomplex domain. This extension goes beyond preserving the core properties of special functions; it also facilitates the discovery of new recurrence relations and broader analytical representations, creating opportunities for pioneering applications in quantum mechanics, wave theory, and advanced computation.

Research on Legendre polynomials has maintained a prominent position in mathematical physics and applied mathematics due to their crucial role in solving differential equations with spherical symmetry [8,9]. They naturally arise in areas such as potential theory, quantum mechanics, and electromagnetism, where they offer elegant and efficient representations of physical phenomena. Their key properties, including orthogonality, recurrence relations, and generating functions make them indispensable tools in both theoretical studies and practical applications. Recent research has further broadened their scope through generalizations and higher-dimensional analysis, reinforcing their connections with fractional calculus and other special functions.

Consequently, extending Legendre polynomials to the bicomplex space is more than a purely formal generalization, but necessary analytical method enabling the orthogonal decomposition of bicomplex-valued fields, bicomplex differential equations and deriving new solutions for hypergeometric structures. Using idempotent decomposition provides an effective mechanism for handling two interacting complex components, makes bicomplex Legendre polynomials valuable for applications ranging the analysis of bicomplex harmonic, biharmonic functions to the formulation of hypercomplex signal transforms and fractional calculus expansions.

Coloma [10] developed fractional bicomplex calculus based on the Riemann Liouville derivative. Integrating classical functions. Although fractional operators were initially met with skepticism, they have gradually gained acceptance, as differential equations involving them have demonstrated strong capabilities in modeling real-world phenomena. This framework has since been applied in fields such as ecology, diffusion dynamics, control engineering, signal processing, and the analysis of biological and viscoelastic materials [11,12].

In this work, we present new results in the bicomplex space by extending and analyzing the Legendre polynomials. To provide a rigorous theoretical foundation, we establish several fundamental theorems and introduce key concepts associated with these functions. The study further investigates important properties, including the solution of the bicomplex Legendre differential equation, generating functions, orthogonality, recurrence relations, and special representations. Moreover, we broaden the framework by incorporating the fractional Riemann Liouville derivative and integral into the four-dimensional bicomplex setting for Legendre polynomials. In addition, we present a numerical study of the Legendre polynomial in bicomplex space and explain its construction and orthogonality verification. We focus on spectral projection and coefficient decay using bicomplex-valued test functions, highlighting their role in efficiently representing bicomplex functions, assessing numerical accuracy and ensuring rapid convergence in bicomplex systems.

This paper aims to present new findings relating to Legendre polynomials in bicomplex space, focusing on fractional calculus and numerical analysis. This highlights the importance of bicomplex numbers in advancing mathematical theory and multidimensional applications.

The organization of this paper is as follows: we display the main concepts and features of bicomplex numbers in Section 2. In Section 3, we present a new version of the Legendre polynomial in bicomplex space. In Section 4, we introduce the fractional Riemann Liouville operator in the setting of bicomplex analysis. In Section 5, we present numerical investigations of bicomplex Legendre polynomials. The last Section 6 provides concluding remarks and outlines potential directions for future research.

2. Preliminaries

This part outlines the fundamental terms and concepts necessary for presenting the main results.

2.1. Bicomplex Number

The concept of bicomplex numbers was originally presented by Segre $\mathbb{BC}$ [1,2] as

$$\begin{array}{c}\hfill \mathbb{BC}:=\{\mathsf{\Psi }={\psi }_1+j{\psi }_2,{\psi }_1,{\psi }_2\in \mathbb{C}\},\end{array}$$

(1)

where ${\psi }_1=u_1+i{v}_1$, ${\psi }_2=u_2+i{v}_2$, of which $i,j$ are independent imaginary units defined as

$$ji=ij,i^2=j^2=-1.$$

If $\mathsf{\Psi }={\psi }_1+j{\psi }_2$, $\mathsf{\Phi }={\varphi }_1+j{\varphi }_2$, we get

$$\mathsf{\Psi }+\mathsf{\Phi }=({\psi }_1+{\varphi }_1)+j({\psi }_2+{\varphi }_2),$$

$$\mathsf{\Psi }\otimes \mathsf{\Phi }=({\psi }_1{\varphi }_1-{\psi }_2{\varphi }_2)+j({\psi }_1{\varphi }_2+{\psi }_2{\varphi }_1).$$

2.1.1. Zero Divisors

If $\mathsf{\Psi }={\psi }_1+j{\psi }_2\ne 0$, then $\mathsf{\Psi }$ is named zero divisor. If both ${\psi }_1$ and ${\psi }_2$ are nonzero, then the set of all zero divisors in $\mathbb{BC}$ is determined [3] as follows:

$$\begin{array}{c}\hfill {\mathcal{H}}_2=\left\{\mathsf{\Psi }={\psi }_1+j{\psi }_2=\mathsf{\Psi }(1\pm ij)\mid \mathsf{\Psi }\in \mathbb{C}\setminus \left\{0\right\},{\psi }_1=\pm i{\psi }_2,{\psi }_1^2+{\psi }_2^2=0\right\}.\end{array}$$

(2)

Zero divisor numbers, represented by $e_1={\displaystyle \frac{1+ij}{2}}$ and $e_2={\displaystyle \frac{1-ij}{2}}$ have the properties

$$e_1+e_2=1,e_1-e_2=ij,$$

$$e_1·e_2=e_2·e_1=0,$$

$$e_1^s=e_1,e_2^s=e_2,s\in N.$$

Thus, the bicomplex number can be written in the form

$$\begin{array}{c}\hfill \mathsf{\Psi }={\psi }_1+j{\psi }_2={\mathsf{\Psi }}_1{e}_1+{\mathsf{\Psi }}_2{e}_2,\end{array}$$

(3)

such that ${\mathsf{\Psi }}_1={\psi }_1-i{\psi }_2,{\mathsf{\Psi }}_2={\psi }_1+i{\psi }_2.$

2.1.2. Idempotent Representations

Idempotent representations simplify calculations using complex numbers in standard complex numbers. Due to the identities, certain properties of idempotents may be described as follows [4]. If $\mathsf{\Psi }={\mathsf{\Psi }}_1{e}_1+{\mathsf{\Psi }}_2{e}_2$, $\mathsf{\Phi }={\mathsf{\Phi }}_1{e}_1+{\mathsf{\Phi }}_2{e}_2$, then we have

(i)

$\mathsf{\Psi }+\mathsf{\Phi }=({\mathsf{\Psi }}_1+{\mathsf{\Phi }}_1)e_1+({\mathsf{\Psi }}_2+{\mathsf{\Phi }}_2)e_2,$

(ii)

$\mathsf{\Psi }\otimes \mathsf{\Phi }={\mathsf{\Psi }}_1{\mathsf{\Phi }}_1{e}_1+{\mathsf{\Psi }}_2{\mathsf{\Phi }}_2{e}_2,$

(iii)

${\mathsf{\Psi }}^s={\mathsf{\Psi }}_1^s{e}_1+{\mathsf{\Psi }}_2^s{e}_2,s\in N,$

(iv)

$e^{\mathsf{\Psi }}=e^{{\mathsf{\Psi }}_1}{e}_1+e^{{\mathsf{\Psi }}_2}{e}_2,$

(v)

${\displaystyle \frac{1}{\mathsf{\Psi }}}={\displaystyle \frac{1}{{\mathsf{\Psi }}_1{e}_1+{\mathsf{\Psi }}_2{e}_2}}={\displaystyle \frac{1}{{\mathsf{\Psi }}_1}}{e}_1+{\displaystyle \frac{1}{{\mathsf{\Psi }}_2}}{e}_2.$

2.1.3. Bicomplex Differentiation and Integration

Assume that a function $W:A\to \mathbb{BC}$ is defined by

$W\left(\mathsf{\Psi }\right)=w_1\left({\mathsf{\Psi }}_1\right){\mathbf{e}}_1+w_2\left({\mathsf{\Psi }}_2\right){\mathbf{e}}_2$. The function W is said to be differentiable at ${\mathsf{\Omega }}_0\in A$ if the following limit exists [2,5]:

$$\begin{array}{c}\hfill W^{\prime }\left(\mathsf{\Psi }\right)=\underset{\mathsf{\Psi }\to {\mathsf{\Psi }}_0}{lim}{\displaystyle \frac{W\left(\mathsf{\Psi }\right)-W\left({\mathsf{\Psi }}_0\right)}{\mathsf{\Psi }-{\mathsf{\Psi }}_0}},\mathsf{\Psi }-{\mathsf{\Psi }}_0\notin {\mathcal{H}}_2,\mathrm{yields}\mathrm{a}\mathrm{finite}\mathrm{value}.\end{array}$$

(4)

where $\mathsf{\Psi }-{\mathsf{\Psi }}_0\notin {\mathcal{H}}_2,$ yields a finite value.

Bicomplex integration is defined as the evaluation of a line integral along a curve in four-dimensional bicomplex space $\mathbb{O}$ in $\mathbb{BC}$. In particular, It is formulated as

$$\begin{array}{c}\hfill {\int }_{\mathbb{O}}W\left(\mathsf{\Psi }\right)\otimes d\mathsf{\Psi }={\int }_{n_1}^{n_2}W\left(\mathsf{\Psi }\left(t\right)\right)\otimes {\mathsf{\Psi }}^{\prime }\left(t\right)dt,d\mathsf{\Psi }=(d{\mathsf{\Psi }}_1,d{\mathsf{\Psi }}_2),\end{array}$$

(5)

where $\mathbb{O}$ has the parametric form

$$\mathbb{O}:\mathsf{\Psi }\left(t\right)\equiv ({\mathsf{\Psi }}_1\left(t\right),{\mathsf{\Psi }}_2\left(t\right)),\mathrm{for}{n}_1\le t\le n_2.$$

which can be regarded as a curve formed by two component curves $d_1$ and $d_2$ in $\mathbb{C}$, specifically $\mathbb{O}\equiv (o_1,o_2)$. Thus

$$\begin{array}{c}\hfill {\int }_{\mathbb{O}}W\left(\mathsf{\Psi }\right)\otimes d\mathsf{\Psi }=\left\{{\int }_{o_1}W\left({\mathsf{\Psi }}_1\right)d{\mathsf{\Psi }}_1\right\}{e}_1+\left\{{\int }_{o_2}W\left({\mathsf{\Psi }}_2\right)d{\mathsf{\Psi }}_2\right\}{e}_2.\end{array}$$

(6)

2.2. Bicomplex Special Functions

The polynomials in bicomplex space is defined as [13]

$$\begin{array}{c}\hfill P\left(\psi \right)=\sum _{r=0}^{\infty }{A}_r\otimes {\mathsf{\Phi }}^r,\end{array}$$

(7)

where $\psi ,A\in \mathbb{BC}$, $\psi ={\psi }_1{e}_1+{\psi }_2{e}_2$ and $A=a_1{e}_1+a_2{e}_2$. Then, we can write bicomplex polynomials as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill P\left(\psi \right)& =\sum _{r=0}^{\infty }{(a_1{e}_1+a_2{e}_2)}_r{({\psi }_1{e}_1+{\psi }_2{e}_2)}^r\hfill \\ & =\sum _{r=0}^{\infty }{\left(a_1\right)}_r{\left({\psi }_1\right)}^r{e}_1+\sum _{r=0}^{\infty }{\left(a_2\right)}_r{\left({\psi }_2\right)}^r{e}_2=P\left({\psi }_1\right)e_1+P\left({\psi }_2\right)e_2.\hfill \end{array}\end{array}$$

The bicomplex functions of gamma and beta were defined by Mathur and Goyal as follows [7]

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_2\left(\nu \right)={\int }_{\mathbb{O}}{e}^{-\lambda }\otimes {\lambda }^{\nu -1}\otimes d\lambda ,\end{array}$$

(8)

where $\nu ={\nu }_1{e}_1+{\nu }_2{e}_2$, $\lambda ={\lambda }_1{e}_1+{\lambda }_2{e}_2$, ${\lambda }_1,{\lambda }_2,{\nu }_1,{\nu }_2\in \mathbb{C}$.

Let $\mathbb{O}$ be a domain in $\mathbb{BC}$, that is, $\mathbb{O}\equiv (o_1,o_2)$ for $o_1\equiv o_1\left({\lambda }_1\right)$, $o_2\equiv o_2\left({\lambda }_2\right)$; then

$$\begin{array}{c}\hfill {\mathit{\beta }}_2(\nu ,\mu )={\int }_{\mathbb{O}}{\lambda }^{\mu -1}\otimes {(1-\lambda )}^{\nu -1}\otimes d\lambda ,\end{array}$$

(9)

$$\begin{array}{c}\hfill {\mathit{\beta }}_2(\nu ,\mu )={\displaystyle \frac{{\mathsf{\Gamma }}_2\left(\nu \right)\otimes {\mathsf{\Gamma }}_2\left(\mu \right)}{{\mathsf{\Gamma }}_2(\mu +\nu )}},\end{array}$$

(10)

where $\mu ,\nu ,\lambda \in \mathbb{BC}$, $\lambda ={\lambda }_1{e}_1+{\lambda }_2{e}_2$, ${\lambda }_1,{\lambda }_2\in \mathbb{C}$.

We represent the bicomplex Pochhammer symbol by [7] by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left(\mu \right)}_s& =\mu \otimes (\mu +1)\otimes (\mu +2)\otimes (\mu +3)\otimes \dots \otimes (\mu +s-1)\hfill \\ & ={\left({\mu }_1\right)}_s{e}_1+{\left({\mu }_2\right)}_s{e}_2.\hfill \end{array}\end{array}$$

(11)

Equivalently, the Pochhammer symbol is written using the gamma function and factorial as

$$\begin{array}{c}\hfill {\left(\mu \right)}_s={\displaystyle \frac{{\mathsf{\Gamma }}_2(\mu +s)}{{\mathsf{\Gamma }}_2\left(\mu \right)}}.\end{array}$$

(12)

The bicomplex Pochhammer symbol has several important properties

(i)

${\left(\mu \right)}_0=1.$

(ii)

${\left(\mu \right)}_{s+r}={\left(\mu \right)}_s{\left(\mu +s\right)}_r.$

(iii)

${\left({\displaystyle \frac{1}{2}}\right)}_s={\displaystyle \frac{\left(2s\right)!}{4^s{(s!)}^2}}={\displaystyle \frac{\mathsf{\Gamma }(s+\frac{1}{2})}{\mathsf{\Gamma }\left(\frac{1}{2}\right)}}.$

where $\mu \in \mathbb{BC}\mathrm{such}\mathrm{that}\mu ={\mu }_1{e}_1+{\mu }_2{e}_2\mathrm{and}s,r\in N$.

According to [4], the bicomplex hypergeometric functions are defined by Rekha and Ajit in 2022 as

$$\begin{array}{c}\hfill {}_{2}M_1\left(\mu ,\delta ;\nu ;\mathsf{\Xi }\right)=\sum _{h=0}^{\infty }{\displaystyle \frac{{\left(\mu \right)}_h{\left(\delta \right)}_h}{{\left(\nu \right)}_h}}\otimes {\displaystyle \frac{{\mathsf{\Xi }}^h}{h!}},\end{array}$$

(13)

where $\mathsf{\Xi },\delta ,\mu \mathrm{and}\nu \in \mathbb{BC}\mathrm{such}\mathrm{that}h\in N$.

3. Bicomplex Legendre Polynomials

In this section, we explore the expansion of legendre polynomials in the space of bicomplex numbers and present some fundamental concepts Pertaining to these functions. We investigate several properties of these functions, including their generating functions, orthogonality and representations involving recurrence relations and special relations.

Definition 1.

$Let\mathsf{\Xi }\in \mathbb{BC}where\mathsf{\Xi }=a_1+j{a}_2={\mathsf{\Xi }}_1{e}_1+{\mathsf{\Xi }}_2{e}_2,r,h\in N,$ then the bicomplex first legendre polynomials are given by

$$\begin{array}{c}\hfill {\mathcal{P}}_h\left(\mathsf{\Xi }\right)=\sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}\otimes {\left(\mathsf{\Xi }\right)}^{(h-2r).}\end{array}$$

(14)

At this stage, we clarify the methodological framework adopted throughout the paper. All subsequent results and proofs are derived by systematically employing the idempotent decomposition of bicomplex numbers. This technique reduces bicomplex-valued operators and identities to their componentwise counterparts, which naturally leads to proofs that formally resemble those of the classical Legendre polynomials. This resemblance is intrinsic to the bicomplex setting and reflects a deliberate and standard methodological choice rather than a mere repetition of known arguments. For completeness and transparency, we present full proofs of all results to explicitly demonstrate how the classical theory is consistently extended to the bicomplex domain.

Theorem 1.

$Let\mathsf{\Xi }\in \mathbb{BC}where\mathsf{\Xi }=a_1+j{a}_2={\mathsf{\Xi }}_1{e}_1+{\mathsf{\Xi }}_2{e}_2,h\in N,$ then the independent form of the bicomplex first Legendre polynomials is given by

$$\begin{array}{c}\hfill {\mathcal{P}}_h\left(\mathsf{\Xi }\right)={\mathcal{P}}_h\left({\mathsf{\Xi }}_1\right)e_1+{\mathcal{P}}_h\left({\mathsf{\Xi }}_2\right)e_2.\end{array}$$

(15)

Proof. 

By using Equation (14) and idempotent representations, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{P}}_h\left(\mathsf{\Xi }\right)& =\sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}\otimes {\left(\mathsf{\Xi }\right)}^{(h-2r)}\hfill \\ & =\sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}{({\mathsf{\Xi }}_1{e}_1+{\mathsf{\Xi }}_2{e}_2)}^{(h-2r)}\hfill \\ & =\left(\sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}{\left({\mathsf{\Xi }}_1\right)}^{(h-2r)}\right)e_1\hfill \\ & +\left(\sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}{\left({\mathsf{\Xi }}_2\right)}^{(h-2r)}\right)e_2\hfill \\ & ={\mathcal{P}}_h\left({\mathsf{\Xi }}_1\right)e_1+{\mathcal{P}}_h\left({\mathsf{\Xi }}_2\right)e_2.\hfill \end{array}\end{array}$$

Thus, the proof is verified. □

Theorem 2.

$Let\mathsf{\Xi }\in \mathbb{BC}where\mathsf{\Xi }=a_1+j{a}_2={\mathsf{\Xi }}_1{e}_1+{\mathsf{\Xi }}_2{e}_2,r,h\in N$ then, another form of the bicomplex first Legendre polynomials using generating function can be written as

$$\begin{array}{c}\hfill {\mathcal{P}}_h\left(\mathsf{\Xi }\right)={}_{2}M_1\left[\begin{array}{c}-h,h+1\hfill \\ 1\hfill \end{array};{\displaystyle \frac{1-\mathsf{\Xi }}{2}}\right]\end{array}$$

(16)

where ${}_{2}M_1$ is defined in Equation (13).

Proof. 

From the generating function Equation (21), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{h=0}^{\infty }{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathsf{\Omega }}^h& ={(1-2\mathsf{\Xi }\mathsf{\Omega }+{\mathsf{\Omega }}^2)}^{{\displaystyle \frac{-1}{2}}}\hfill \\ & ={(1-2{\mathsf{\Xi }}_1{\mathsf{\Omega }}_1+{\mathsf{\Omega }}_1^2)}^{{\displaystyle \frac{-1}{2}}}{e}_1+{(1-2{\mathsf{\Xi }}_2{\mathsf{\Omega }}_2+\mathsf{\Omega }-2^2)}^{{\displaystyle \frac{-1}{2}}}{e}_2\hfill \\ & ={\left({(1-{\mathsf{\Omega }}_1)}^2-2{\mathsf{\Xi }}_1({\mathsf{\Omega }}_1-1)\right)}^{{\displaystyle \frac{-1}{2}}}{e}_1+{\left({(1-{\mathsf{\Omega }}_2)}^2-2{\mathsf{\Xi }}_2({\mathsf{\Omega }}_2-1)\right)}^{{\displaystyle \frac{-1}{2}}}{e}_2\hfill \\ & ={(1-{\mathsf{\Omega }}_1)}^{-1}{\left[1-{\displaystyle \frac{2{\mathsf{\Omega }}_1({\mathsf{\Xi }}_1-1)}{{(1-{\mathsf{\Omega }}_1)}^2}}\right]}^{{\displaystyle \frac{-1}{2}}}{e}_1+{(1-{\mathsf{\Omega }}_2)}^{-1}{\left[1-{\displaystyle \frac{2{\mathsf{\Omega }}_2({\mathsf{\Xi }}_2-1)}{{(1-{\mathsf{\Omega }}_2)}^2}}\right]}^{{\displaystyle \frac{-1}{2}}}{e}_2.\hfill \end{array}\end{array}$$

Using the generalized binomial series,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{h=0}^{\infty }{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathsf{\Omega }}^h& ={(1-{\mathsf{\Omega }}_1)}^{-1}\sum _{r=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{2}\right)}_r}{r!}}{\left({\displaystyle \frac{2{\mathsf{\Omega }}_1({\mathsf{\Xi }}_1-1)}{{(1-{\mathsf{\Omega }}_1)}^2}}\right)}^r{e}_1+{(1-{\mathsf{\Omega }}_2)}^{-1}\sum _{r=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{2}\right)}_r}{r!}}{\left({\displaystyle \frac{2{\mathsf{\Omega }}_2({\mathsf{\Xi }}_2-1)}{{(1-{\mathsf{\Omega }}_2)}^2}}\right)}^r{e}_2\hfill \\ & =\left(\sum _{r=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{2}\right)}_r}{r!}}{\displaystyle \frac{2^r{\mathsf{\Omega }}_1^r{({\mathsf{\Xi }}_1-1)}^r}{{(1-{\mathsf{\Omega }}_1)}^{2r+1}}}\right)e_1+\left(\sum _{r=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{2}\right)}_r}{r!}}{\displaystyle \frac{2^r{\mathsf{\Omega }}_2^r{({\mathsf{\Xi }}_2-1)}^r}{{(1-{\mathsf{\Omega }}_2)}^{2r+1}}}\right)e_2\hfill \\ & =\left(\sum _{r,h=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{2}\right)}_r{2}^r{(2r+1)}_h{({\mathsf{\Xi }}_1-1)}^r{\mathsf{\Omega }}_1^{r+h}}{r!h!}}\right)e_1+\left(\sum _{r,h=0}^{\infty }{\displaystyle \frac{{\left(\frac{1}{2}\right)}_r{2}^r{(2r+1)}_h{({\mathsf{\Xi }}_2-1)}^r{\mathsf{\Omega }}_2^{r+h}}{r!h!}}\right)e_2.\hfill \end{array}\end{array}$$

Using properties of bicomplex Pochhammer symbol by putting ${{\displaystyle \frac{1}{2}}}_r={\displaystyle \frac{\left(2r\right)!}{4^{r}r!}}$ and ${(2r+1)}_h={\displaystyle \frac{(2r+h)!}{\left(2r\right)!}}$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{h=0}^{\infty }{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathsf{\Omega }}^h& =\left(\sum _{r,h=0}^{\infty }{\displaystyle \frac{(h+2r)!{({\mathsf{\Xi }}_1-1)}^r{\mathsf{\Omega }}_1^{r+h}}{2^{r}h!{(r!)}^2}}\right)e_1+\left(\sum _{r,h=0}^{\infty }{\displaystyle \frac{(h+2r)!{({\mathsf{\Xi }}_2-1)}^r{\mathsf{\Omega }}_2^{r+h}}{2^{r}h!{(r!)}^2}}\right)e_2.\hfill \end{array}\end{array}$$

putting $h=h+r$, $(r-h)!={\displaystyle \frac{{(-1)}^{r}h!}{{(-h)}_r}}$ and $(r+h)!={(h+1)}_{r}h!$, then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{h=0}^{\infty }{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathsf{\Omega }}^h& =\left(\sum _{r=0}^h\sum _{h=0}^{\infty }{\displaystyle \frac{(r+h)!{({\mathsf{\Xi }}_1-1)}^r{\mathsf{\Omega }}_1^h}{2^r(h-r)!{(r!)}^2}}\right)e_1\hfill \\ & +\left(\sum _{r=0}^h\sum _{h=0}^{\infty }{\displaystyle \frac{(r+h)!{({\mathsf{\Xi }}_2-1)}^r{\mathsf{\Omega }}_2^h}{2^r(h-r)!{(r!)}^2}}\right)e_2\hfill \\ & =\left(\sum _{r=0}^h\sum _{h=0}^{\infty }{\displaystyle \frac{{(-1)}^r{(-h)}_r{(h+1)}_r{(1-{\mathsf{\Xi }}_1)}^r{\mathsf{\Omega }}_1^h}{2^r{(r!)}^2}}\right)e_1\hfill \\ & +\left(\sum _{r=0}^h\sum _{h=0}^{\infty }{\displaystyle \frac{{(-1)}^r{(-h)}_r{(h+1)}_r{(1-{\mathsf{\Xi }}_2)}^r{\mathsf{\Omega }}_2^h}{2^r{(r!)}^2}}\right)e_2.\hfill \end{array}\end{array}$$

Then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{P}}_h\left(\mathsf{\Xi }\right)=& {}_{2}m_1\left[\begin{array}{c}-h,h+1\hfill \\ 1\hfill \end{array};{\displaystyle \frac{1-{\mathsf{\Xi }}_1}{2}}\right]e_1+{}_{2}m_1\left[\begin{array}{c}-h,h+1\hfill \\ 1\hfill \end{array};{\displaystyle \frac{1-{\mathsf{\Xi }}_2}{2}}\right]e_2\hfill \\ \hfill =& {}_{2}M_1\left[\begin{array}{c}-h,h+1\hfill \\ 1\hfill \end{array};{\displaystyle \frac{1-\mathsf{\Xi }}{2}}\right].\hfill \end{array}\end{array}$$

The proof has been completed. □

Theorem 3.

Let $\mathsf{\Xi }\in \mathbb{B}\mathbb{C}$ where $\mathsf{\Xi }={\mathsf{\Xi }}_1{e}_1+{\mathsf{\Xi }}_2{e}_2,\mathsf{\Psi }={\mathsf{\Psi }}_1{e}_1+{\mathsf{\Psi }}_2{e}_2,h\in N$, then bicomplex Legendre polynomial ${\mathcal{P}}_h\left(\mathsf{\Xi }\right)$ satisfies the different equation

$$\begin{array}{c}\hfill \left(1-{\mathsf{\Xi }}^2\right)\otimes {\displaystyle \frac{d^2\mathsf{\Psi }}{d{\mathsf{\Xi }}^2}}-2\mathsf{\Xi }\otimes {\displaystyle \frac{d\mathsf{\Psi }}{d\mathsf{\Xi }}}+h(h+1)\mathsf{\Psi }=0\end{array}$$

(17)

The equation is a linear differential equation of second order with two linearly independent solutions. The first is a solution regular on the interval $[-1,1]$, and this solution is known as the Legendre polynomial ${\mathcal{P}}_h\left(\mathsf{\Xi }\right)$; the second solution is typically irregular at $x=\pm 1$.

Proof. 

The expression of the bicomplex Legendre polynomials is obtained via the bicomplex hypergeometric function, as shown in Equation (16).

Equation (16) is a solution of a another equation

$$\begin{array}{c}\hfill \rho \otimes (1-\rho )\otimes {\displaystyle \frac{d^2\mathsf{\Psi }}{d{\rho }^2}}+[c-(a+b+1)\rho ]\otimes {\displaystyle \frac{d\mathsf{\Psi }}{d\rho }}-ab\mathsf{\Psi }=0.\end{array}$$

(18)

By putting $a=-h,b=h+1,c=1$, and $\rho ={\displaystyle \frac{1-\mathsf{\Xi }}{2}}$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left({\displaystyle \frac{1-\mathsf{\Xi }}{4}}\right)& \otimes \left(1-{\displaystyle \frac{1-\mathsf{\Xi }}{2}}\right)\otimes {\displaystyle \frac{d^2\mathsf{\Psi }}{d{\rho }^2}}+\left[1-\left(2\right){\displaystyle \frac{1-\mathsf{\Xi }}{2}}\right]\otimes {\displaystyle \frac{d\mathsf{\Psi }}{d\rho }}-(-h)(h+1)\mathsf{\Psi }=0,\hfill \\ & \left({\displaystyle \frac{1-{\mathsf{\Xi }}^2}{2}}\right)\otimes {\displaystyle \frac{d^2\mathsf{\Psi }}{d{\rho }^2}}+\left(\mathsf{\Xi }\right)\otimes {\displaystyle \frac{d\mathsf{\Psi }}{d\rho }}+\left(h\right)(h+1)\mathsf{\Psi }=0.\hfill \end{array}\end{array}$$

Using the chain rules, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& {\displaystyle \frac{d\mathsf{\Psi }}{d\rho }}={\displaystyle \frac{d\mathsf{\Psi }}{d\mathsf{\Xi }}}\otimes {\displaystyle \frac{d\mathsf{\Xi }}{d\rho }}=-2{\displaystyle \frac{d\mathsf{\Psi }}{d\mathsf{\Xi }}},\hfill \\ & {\displaystyle \frac{d^2\mathsf{\Psi }}{d{\rho }^2}}={\displaystyle \frac{d}{d\rho }}\left({\displaystyle \frac{d\mathsf{\Psi }}{d\rho }}\right)={\displaystyle \frac{d}{d\rho }}\left(-2{\displaystyle \frac{d\mathsf{\Psi }}{d\mathsf{\Xi }}}\right)=4{\displaystyle \frac{d{\mathsf{\Psi }}^2}{d{\mathsf{\Xi }}^2}}.\hfill \end{array}\end{array}$$

Consequently, we have established the desired result

$$\begin{array}{c}\hfill \left(1-{\mathsf{\Xi }}^2\right)\otimes {\displaystyle \frac{d{\mathsf{\Psi }}^2}{d{\mathsf{\Xi }}^2}}+\left(\mathsf{\Xi }\right)-2\mathsf{\Xi }\otimes {\displaystyle \frac{d\mathsf{\Psi }}{d\mathsf{\Xi }}}+\left(h\right)(h+1)\mathsf{\Psi }=0.\end{array}$$

□

Corollary 1.

We can find some special cases from Equation (14) when we put $h=0,1,2,\dots$

(i) 

${\mathcal{P}}_0\left(\mathsf{\Xi }\right)=1.$

(ii) 

${\mathcal{P}}_1\left(\mathsf{\Xi }\right)=\mathsf{\Xi }=\left({\mathsf{\Xi }}_1\right)e_1+\left({\mathsf{\Xi }}_2\right)e_2.$

(iii) 

${\mathcal{P}}_2\left(\mathsf{\Xi }\right)={\displaystyle \frac{1}{2}}(3{\mathsf{\Xi }}^2-1)=({\displaystyle \frac{1}{2}}(3{\mathsf{\Xi }}_1^2-1)e_1+\left({\displaystyle \frac{1}{2}}(3{\mathsf{\Xi }}_2^2-1)\right)e_2.$

(iv) 

${\mathcal{P}}_3\left(\mathsf{\Xi }\right)={\displaystyle \frac{1}{2}}(5{\mathsf{\Xi }}^3-3\mathsf{\Xi })=\left({\displaystyle \frac{1}{2}}(5{\mathsf{\Xi }}_1^3-3{\mathsf{\Xi }}_1)\right)e_1+({\displaystyle \frac{1}{2}}(5{\mathsf{\Xi }}_2^3-3{\mathsf{\Xi }}_2)e_2.$

Example 1.

Evaluating the bicomplex value by substituting $\mathsf{\Xi }=2+j3$ into Equation (14) for h = 3. Solution: When we put $h=3$ in Equation (14), we get

$$\begin{array}{c}\hfill {\mathcal{P}}_3\left(\mathsf{\Xi }\right)=\sum _{r=0}^{\lceil {\displaystyle \frac{3}{2}}\rceil }{\displaystyle \frac{{(-1)}^r(6-2r)!}{2^3(3-r)!r!(3-2r)!}}\otimes {\left(\mathsf{\Xi }\right)}^{(3-2r)}.\end{array}$$

(19)

Then we consider the cases $r=0$ and $r=1$ in Equation (19)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{P}}_3\left(\mathsf{\Xi }\right)& ={\displaystyle \frac{{(-1)}^0\left(6\right)!}{2^3\left(3\right)!0!\left(3\right)!}}{\left(\mathsf{\Xi }\right)}^{\left(3\right)}+{\displaystyle \frac{{(-1)}^1\left(4\right)!}{2^3\left(2\right)!1!\left(1\right)!}}{\left(\mathsf{\Xi }\right)}^{\left(1\right)}\hfill \\ & ={\displaystyle \frac{5}{2}}{\left(\mathsf{\Xi }\right)}^{\left(3\right)}-{\displaystyle \frac{3}{2}}\left(\mathsf{\Xi }\right).\hfill \end{array}\end{array}$$

Next, we put $\mathsf{\Xi }=2+j3$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{P}}_3(2+j3)& ={\displaystyle \frac{5}{2}}{(2+j3)}^{\left(3\right)}-{\displaystyle \frac{3}{2}}(2+j3)\hfill \\ & ={\displaystyle \frac{5}{2}}\left(8+j36-54-j27\right)-3-j{\displaystyle \frac{9}{2}}\hfill \\ & ={\displaystyle \frac{5}{2}}\left(-46+j9\right)-3-j{\displaystyle \frac{9}{2}}\hfill \\ & =-118+j18.\hfill \end{array}\end{array}$$

Theorem 4.

$Let\mathsf{\Xi }\in \mathbb{BC}where\mathsf{\Xi }=a_1+j{a}_2={\mathsf{\Xi }}_1{e}_1+{\mathsf{\Xi }}_2{e}_2,h\in N$, then we get

$$\begin{array}{c}\hfill {\mathcal{P}}_h(-\mathsf{\Xi })={(-1)}^h{\mathcal{P}}_h\left(\mathsf{\Xi }\right).\end{array}$$

(20)

Proof. 

Putting $(-\mathsf{\Xi })$ in Equation (14), we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{P}}_h(-\mathsf{\Xi })& =\sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}\otimes {(-\mathsf{\Xi })}^{(h-2r)}\hfill \\ & =\left(\sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}{(-{\mathsf{\Xi }}_1)}^{(h-2r)}\right)e_1\hfill \\ & +\left(\sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}{(-{\mathsf{\Xi }}_2)}^{(h-2r)}\right)e_2\hfill \\ & =\left(\sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^{h-r}(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}{\left({\mathsf{\Xi }}_1\right)}^{(h-2r)}\right)e_1\hfill \\ & +\left(\sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^{h-r}(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}{\left({\mathsf{\Xi }}_2\right)}^{(h-2r)}\right)e_2.\hfill \end{array}\end{array}$$

We know that ${(-1)}^{-r}={(-1)}^r$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{P}}_h(-\mathsf{\Xi })=& \left({(-1)}^h\sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}{\left({\mathsf{\Xi }}_1\right)}^{(h-2r)}\right)e_1\hfill \\ \hfill +& \left({(-1)}^h\sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}{\left({\mathsf{\Xi }}_2\right)}^{(h-2r)}\right)e_2\hfill \\ \hfill =& {(-1)}^h{\mathcal{P}}_h\left({\mathsf{\Xi }}_1\right)e_1+{(-1)}^h{\mathcal{P}}_h\left({\mathsf{\Xi }}_2\right)e_2\hfill \\ \hfill =& {(-1)}^h{\mathcal{P}}_h\left(\mathsf{\Xi }\right).\hfill \end{array}\end{array}$$

The proof has been completed. □

Theorem 5.

$Let\mathsf{\Xi }\in \mathbb{BC}where\mathsf{\Xi }=a_1+j{a}_2={\mathsf{\Xi }}_1{e}_1+{\mathsf{\Xi }}_2{e}_2,\mathsf{\Omega }=o_1+j{o}_2={\mathsf{\Omega }}_1{e}_l+{\mathsf{\Omega }}_2{e}_2,h\in N,$ then the generating function for bicomplex first Legendre polynomials are given by

$$\begin{array}{c}\hfill {(1-2\mathsf{\Xi }\otimes \mathsf{\Omega }+{\mathsf{\Omega }}^2)}^{{\displaystyle \frac{-1}{2}}}=\sum _{h=0}^{\infty }{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathsf{\Omega }}^h.\end{array}$$

(21)

Proof. 

By applying idempotent units $e_1$, $e_2$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {(1-\mathsf{\Omega }\otimes (2\mathsf{\Xi }-\mathsf{\Omega }))}^{{\displaystyle \frac{-1}{2}}}=& {(1-{\mathsf{\Omega }}_1(2{\mathsf{\Xi }}_1-{\mathsf{\Omega }}_1))}^{{\displaystyle \frac{-1}{2}}}{e}_1+{(1-{\mathsf{\Omega }}_2(2{\mathsf{\Xi }}_2-{\mathsf{\Omega }}_2))}^{{\displaystyle \frac{-1}{2}}}{e}_1.\hfill \end{array}\end{array}$$

From the binomial series expansion, we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {(1-\mathsf{\Omega }\otimes (2\mathsf{\Xi }-\mathsf{\Omega }))}^{{\displaystyle \frac{-1}{2}}}=& (1+{\displaystyle \frac{1}{2}}{\mathsf{\Omega }}_1(2{\mathsf{\Xi }}_1-{\mathsf{\Omega }}_1)+{\displaystyle \frac{3}{8}}{\mathsf{\Omega }}_1^2{(2{\mathsf{\Xi }}_1-{\mathsf{\Omega }}_1)}^2+{\displaystyle \frac{5}{16}}{\mathsf{\Omega }}_1^3{(2{\mathsf{\Xi }}_1-{\mathsf{\Omega }}_1)}^3+......)e_1\hfill \\ \hfill +& (1+{\displaystyle \frac{1}{2}}{\mathsf{\Omega }}_2(2{\mathsf{\Xi }}_1-{\mathsf{\Omega }}_2)+{\displaystyle \frac{3}{8}}{\mathsf{\Omega }}_2^2{(2{\mathsf{\Xi }}_2-{\mathsf{\Omega }}_2)}^2+{\displaystyle \frac{5}{16}}{\mathsf{\Omega }}_2^3{(2{\mathsf{\Xi }}_2-{\mathsf{\Omega }}_2)}^3+......)e_2\hfill \\ \hfill =& (1+{\mathsf{\Xi }}_1{\mathsf{\Omega }}_1+{\displaystyle \frac{1}{2}}(3{\mathsf{\Xi }}_1^2-1){\mathsf{\Omega }}_1^2+{\displaystyle \frac{1}{2}}(5{\mathsf{\Xi }}_1^3-3{\mathsf{\Xi }}_1){\mathsf{\Omega }}_1^3+......)e_1\hfill \\ \hfill +& (1+{\mathsf{\Xi }}_2{\mathsf{\Omega }}_2+{\displaystyle \frac{1}{2}}(3{\mathsf{\Xi }}_2^2-1){\mathsf{\Omega }}_2^2+{\displaystyle \frac{1}{2}}(5{\mathsf{\Xi }}_2^3-3{\mathsf{\Xi }}_2){\mathsf{\Omega }}_2^3+......)e_2.\hfill \end{array}\end{array}$$

Using (i)–(iv) from Corollary (1), we obtain that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {(1-\mathsf{\Omega }\otimes (2\mathsf{\Xi }-\mathsf{\Omega }))}^{{\displaystyle \frac{-1}{2}}}=& ({\mathcal{P}}_0\left({\mathsf{\Xi }}_1\right)+{\mathcal{P}}_1\left({\mathsf{\Xi }}_1\right){\mathsf{\Omega }}_1+{\mathcal{P}}_2\left({\mathsf{\Xi }}_1\right){\mathsf{\Omega }}_1^2+{\mathcal{P}}_3\left({\mathsf{\Xi }}_1\right){\mathsf{\Omega }}_1^3+......)e_1\hfill \\ \hfill +& ({\mathcal{P}}_0\left({\mathsf{\Xi }}_2\right)+{\mathcal{P}}_1\left({\mathsf{\Xi }}_2\right){\mathsf{\Omega }}_2+{\mathcal{P}}_2\left({\mathsf{\Xi }}_2\right){\mathsf{\Omega }}_2^2+{\mathcal{P}}_3\left({\mathsf{\Xi }}_2\right){\mathsf{\Omega }}_2^3+......)e_2\hfill \\ \hfill =& \left(\sum _{h=0}^{\infty }{\mathcal{P}}_h\left({\mathsf{\Xi }}_1\right){\mathsf{\Omega }}_1^h\right)e_1+\left(\sum _{h=0}^{\infty }{\mathcal{P}}_h\left({\mathsf{\Xi }}_2\right){\mathsf{\Omega }}_2^h\right)e_2\hfill \\ \hfill =& (\sum _{h=0}^{\infty }{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathsf{\Omega }}^h).\hfill \end{array}\end{array}$$

□

Theorem 6.

$Let\mathsf{\Xi }\in \mathbb{BC}where\mathsf{\Xi }=a_1+j{a}_2={\mathsf{\Xi }}_1{e}_1+{\mathsf{\Xi }}_2{e}_2,h\in N,$

$$\begin{array}{c}\hfill \left(i\right)(h+1){\mathcal{P}}_{h+1}\left(\mathsf{\Xi }\right)=(2h+1)\mathsf{\Xi }\otimes {\mathcal{P}}_h\left(\mathsf{\Xi }\right)-h{\mathcal{P}}_{h-1}\left(\mathsf{\Xi }\right).\end{array}$$

(22)

$$\begin{array}{c}\hfill \left(ii\right)(2h+1){\mathcal{P}}_h\left(\mathsf{\Xi }\right)={\mathcal{P}}_{h+1}^{\prime }\left(\mathsf{\Xi }\right)-{\mathcal{P}}_{h-1}^{\prime }\left(\mathsf{\Xi }\right).\end{array}$$

(23)

Proof. 

To prove (i), we use the generating function Equation (21), we differentiate both sides with respect to $\mathsf{\Omega }$

$$\begin{array}{c}\hfill \begin{array}{cc}& {\displaystyle \frac{-1}{2}}{(1-2\mathsf{\Xi }\otimes \mathsf{\Omega }+{\mathsf{\Omega }}^2)}^{{\displaystyle \frac{-3}{2}}}(-2\mathsf{\Xi }+2\mathsf{\Omega })=\sum _{h=0}^{\infty }h{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathsf{\Omega }}^{h-1}\hfill \\ & (\mathsf{\Xi }-\mathsf{\Omega })\otimes \sum _{h=0}^{\infty }{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathsf{\Omega }}^h=(1-2\mathsf{\Xi }\otimes \mathsf{\Omega }+{\mathsf{\Omega }}^2)\sum _{h=0}^{\infty }h{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathsf{\Omega }}^{h-1}.\hfill \end{array}\end{array}$$

Equating the coefficients of ${\mathsf{\Omega }}^h$

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathsf{\Xi }\otimes {\mathcal{P}}_h\left(\mathsf{\Xi }\right)-{\mathcal{P}}_{h-1}\left(\mathsf{\Xi }\right)=(h+1){\mathcal{P}}_{h+1}\left(\mathsf{\Xi }\right)-2h\mathsf{\Xi }\otimes {\mathcal{P}}_h\left(\mathsf{\Xi }\right)+(h-1){\mathcal{P}}_{h-1}\left(\mathsf{\Xi }\right).\end{array}\end{array}$$

From this, we obtain the desired relation

$$\begin{array}{c}\hfill \begin{array}{c}\hfill (h+1){\mathcal{P}}_{h+1}\left(\mathsf{\Xi }\right)=(2h+1)\mathsf{\Xi }\otimes {\mathcal{P}}_h\left(\mathsf{\Xi }\right)-h{\mathcal{P}}_{h-1}\left(\mathsf{\Xi }\right)\end{array}\end{array}$$

Now, we verify (ii) by considering the generating function Equation (21), we differentiate both sides with respect to $\mathsf{\Xi }$ to obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& \left(-{\displaystyle \frac{1}{2}}\right)(-2\mathsf{\Omega }){\left(1-2\mathsf{\Xi }\otimes \mathsf{\Omega }+{\mathsf{\Omega }}^2\right)}^{-3/2}=\sum _{h=0}^{\infty }{\mathcal{P}}_h^{\prime }\left(\mathsf{\Xi }\right)\otimes {\mathsf{\Omega }}^h.\hfill \\ & \sum _{h=0}^{\infty }{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathsf{\Omega }}^{h+1}=(1-2\mathsf{\Xi }\otimes \mathsf{\Omega }+{\mathsf{\Omega }}^2)\otimes \sum _{h=0}^{\infty }{\mathcal{P}}_h^{\prime }\left(\mathsf{\Xi }\right)\otimes {\mathsf{\Omega }}^h.\hfill \end{array}\end{array}$$

Equating the coefficients of ${\mathsf{\Omega }}^{h+1}$, we get

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{P}}_h\left(\mathsf{\Xi }\right)={\mathcal{P}}_{h+1}^{\prime }\left(\mathsf{\Xi }\right)-2\mathsf{\Xi }{\mathcal{P}}_h^{\prime }\left(\mathsf{\Xi }\right)-{\mathcal{P}}_{h-1}^{\prime }\left(\mathsf{\Xi }\right)\end{array}\end{array}$$

Thus, we arrive at the required relation. □

Theorem 7.

Bicomplex legendre polynomials satisfy the orthogonality theorem

$$\begin{array}{c}\hfill {\int }_{-1}^1{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathcal{P}}_l\left(\mathsf{\Xi }\right)d\mathsf{\Xi }=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}h\ne l\hfill \\ {\displaystyle \frac{2}{(2h+1)}},\hfill & \mathrm{if}h=l\hfill \end{array}\right.\end{array}$$

(24)

where $\mathsf{\Xi }={\mathsf{\Xi }}_1{e}_1+{\mathsf{\Xi }}_2{e}_2,h,l\in N$

Proof. 

${\mathcal{P}}_h\left(\mathsf{\Xi }\right)$ and ${\mathcal{P}}_l\left(\mathsf{\Xi }\right)$ are bicomplex Legendre polynomials which satisfy Equation (17)

$$\begin{array}{c}\hfill (1-{\mathsf{\Xi }}^2)\otimes {\displaystyle \frac{d^2}{d{\mathsf{\Xi }}^2}}{\mathcal{P}}_h\left(\mathsf{\Xi }\right)-2\mathsf{\Xi }\otimes {\displaystyle \frac{d}{d\mathsf{\Xi }}}{\mathcal{P}}_h\left(\mathsf{\Xi }\right)+h(h+1){\mathcal{P}}_h\left(\mathsf{\Xi }\right)=0,\end{array}$$

(25)

and

$$\begin{array}{c}\hfill (1-{\mathsf{\Xi }}^2)\otimes {\displaystyle \frac{d^2}{d{\mathsf{\Xi }}^2}}{\mathcal{P}}_l\left(\mathsf{\Xi }\right)-2\mathsf{\Xi }\otimes {\displaystyle \frac{d}{d\mathsf{\Xi }}}{\mathcal{P}}_l\left(\mathsf{\Xi }\right)+l(l+1){\mathcal{P}}_l\left(\mathsf{\Xi }\right)=0.\end{array}$$

(26)

By multiplying Equation (25) by ${\mathcal{P}}_l\left(\mathsf{\Xi }\right)$ and Equation (26) by ${\mathcal{P}}_h\left(\mathsf{\Xi }\right)$, then subtracting the two equations, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& (1-{\mathsf{\Xi }}^2)\otimes \left({\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\displaystyle \frac{d^2}{d{\mathsf{\Xi }}^2}}{\mathcal{P}}_l\left(\mathsf{\Xi }\right)-{\mathcal{P}}_l\left(\mathsf{\Xi }\right)\otimes {\displaystyle \frac{d^2}{d{\mathsf{\Xi }}^2}}{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\right)\hfill \\ & -2\mathsf{\Xi }\otimes \left({\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\displaystyle \frac{d}{d\mathsf{\Xi }}}{\mathcal{P}}_l\left(\mathsf{\Xi }\right)-{\mathcal{P}}_l\left(\mathsf{\Xi }\right)\otimes {\displaystyle \frac{d}{d\mathsf{\Xi }}}{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\right)\hfill \\ & =(h(h+1)-l(l+1)){\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathcal{P}}_l\left(\mathsf{\Xi }\right).\hfill \end{array}\end{array}$$

Then

$$\begin{array}{c}\hfill \begin{array}{cc}& {\displaystyle \frac{d^2}{d{\mathsf{\Xi }}^2}}\left[(1-{\mathsf{\Xi }}^2)\otimes \left({\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\displaystyle \frac{d}{d\mathsf{\Xi }}}{\mathcal{P}}_l\left(\mathsf{\Xi }\right)-{\mathcal{P}}_l\left(\mathsf{\Xi }\right)\otimes {\displaystyle \frac{d}{d\mathsf{\Xi }}}{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\right)\right]\hfill \\ & -2\mathsf{\Xi }\otimes \left({\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\displaystyle \frac{d}{d\mathsf{\Xi }}}{\mathcal{P}}_l\left(\mathsf{\Xi }\right)-{\mathcal{P}}_l\left(\mathsf{\Xi }\right)\otimes {\displaystyle \frac{d}{d\mathsf{\Xi }}}{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\right)\hfill \\ & =(h(h+1)-l(l+1)){\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathcal{P}}_l\left(\mathsf{\Xi }\right).\hfill \end{array}\end{array}$$

We can write the last equation as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\displaystyle \frac{d}{d\mathsf{\Xi }}}\left[(1-{\mathsf{\Xi }}^2)\otimes \left({\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\displaystyle \frac{d}{d\mathsf{\Xi }}}{\mathcal{P}}_l\left(\mathsf{\Xi }\right)-{\mathcal{P}}_l\left(\mathsf{\Xi }\right)\otimes {\displaystyle \frac{d}{d\mathsf{\Xi }}}{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\right)\right]=\left[h(h+1)-l(l+1)\right]{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathcal{P}}_l\left(\mathsf{\Xi }\right).\end{array}\end{array}$$

By integrating from $-1$ to 1, we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left[(1-{\mathsf{\Xi }}^2)\otimes \left({\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\displaystyle \frac{d}{d\mathsf{\Xi }}}{\mathcal{P}}_l\left(\mathsf{\Xi }\right)-{\mathcal{P}}_l\left(\mathsf{\Xi }\right)\otimes {\displaystyle \frac{d}{d\mathsf{\Xi }}}{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\right)\right]}_{-1}^1=& (h(h+1)-l(l+1)){\int }_{-1}^1{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathcal{P}}_l\left(\mathsf{\Xi }\right)d\mathsf{\Xi }\hfill \\ \hfill =& 0.\hfill \end{array}\end{array}$$

So, we get the first case when $h\ne l$

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int }_{-1}^1{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathcal{P}}_l\left(\mathsf{\Xi }\right)d\mathsf{\Xi }=0.\end{array}\end{array}$$

To prove the second case when $h=l$, we square both sides of Equation (21),

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {(1-2\mathsf{\Xi }\otimes \mathsf{\Omega }+{\mathsf{\Omega }}^2)}^{-1}=\sum _{h=0}^{\infty }\sum _{l=0}^{\infty }{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathcal{P}}_l\left(\mathsf{\Xi }\right)\otimes {\mathsf{\Omega }}^{h+l}.\end{array}\end{array}$$

By integrating from $-1$ to 1, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int }_{-1}^1{\displaystyle \frac{d\mathsf{\Xi }}{(1-2\mathsf{\Xi }\otimes \mathsf{\Omega }+{\mathsf{\Omega }}^2)}}=\sum _{h=0}^{\infty }\sum _{l=0}^{\infty }\left[{\int }_{-1}^1{\mathcal{P}}_h\left(\mathsf{\Xi }\right)\otimes {\mathcal{P}}_l\left(\mathsf{\Xi }\right)d\mathsf{\Xi }\right]\otimes {\mathsf{\Omega }}^{h+l}.\end{array}\end{array}$$

By applying the orthogonality theorem and integrating the left side

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left[{\displaystyle \frac{-1}{2\mathsf{\Omega }}}\otimes ln(1-2\mathsf{\Omega }\otimes \mathsf{\Xi }+{\mathsf{\Omega }}^2)\right]}_{-1}^1& =\sum _{h=0}^{\infty }\left[{\int }_{-1}^1{\mathcal{P}}_h^2\left(\mathsf{\Xi }\right)d\mathsf{\Xi }\right]\otimes {\mathsf{\Omega }}^{2h}\hfill \\ \hfill {\displaystyle \frac{1}{\mathsf{\Omega }}}\otimes ln\left[{\displaystyle \frac{1+\mathsf{\Omega }}{1-\mathsf{\Omega }}}\right]& =\sum _{h=0}^{\infty }\left[{\int }_{-1}^1{\mathcal{P}}_h^2\left(\mathsf{\Xi }\right)d\mathsf{\Xi }\right]\otimes {\mathsf{\Omega }}^{2h}\hfill \\ \hfill \sum _{h=0}^{\infty }{\displaystyle \frac{2{\mathsf{\Omega }}^{2h}}{2h+1}}& =\sum _{h=0}^{\infty }\left[{\int }_{-1}^1{\mathcal{P}}_h^2\left(\mathsf{\Xi }\right)d\mathsf{\Xi }\right]\otimes {\mathsf{\Omega }}^{2h}.\hfill \end{array}\end{array}$$

Equating the coefficients of ${\mathsf{\Omega }}^{2h}$, then

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int }_{-1}^1{\mathcal{P}}_h^2\left(\mathsf{\Xi }\right)d\mathsf{\Xi }={\displaystyle \frac{2}{2h+1}}.\end{array}\end{array}$$

as required. □

4. Bicomplex Fractional Calculus

This section, we introduce the bicomplex Riemann Liouville fractional derivative and integration of the bicomplex Legendre polynomials. Mahesh and Kumar were introduce the concept of bicomplex fractional Riemann Liouville operators [11,12].

The bicomplex fractional Riemann Liouville integration can be formulated as

$$\begin{array}{c}\hfill {}_{0}I_{\delta }^{\lambda }F\left(\delta \right)={\displaystyle \frac{1}{{\mathsf{\Gamma }}_2\left(\lambda \right)}}\otimes {\int }_0^{\delta }{(\delta -\nu )}^{\lambda -1}\otimes F\left(\nu \right)\otimes d\nu ,\end{array}$$

(27)

where $\nu ,\delta ,\lambda \in \mathbb{BC}$, $\nu =x_1+j{x}_2={\nu }_1{e}_1+{\nu }_2{e}_2,\delta =y_1+j{y}_2={\delta }_1{e}_1+{\delta }_2{e}_2,\lambda =z_1+j{z}_2={\lambda }_1{e}_1+{\lambda }_2{e}_2$ with $Re\left(y_1\right)>\left|Im\left(y_2\right)\right|.$

The bicomplex fractional Riemann Liouville derivative can be formulated as

$$\begin{array}{c}\hfill {}_{0}D_{\delta }^{\lambda }F\left(\delta \right)={\displaystyle \frac{1}{{\mathsf{\Gamma }}_2(r-\lambda )}}\otimes {\displaystyle \frac{d^r}{d{\delta }^r}}\otimes {\int }_0^{\delta }{(\delta -\nu )}^{r-\lambda -1}\otimes F\left(\nu \right)\otimes d\nu ,\end{array}$$

(28)

where $\nu ,\delta ,\lambda \in \mathbb{BC}$, $\nu =x_1+j{x}_2={\nu }_1{e}_1+{\nu }_2{e}_2,\delta =y_1+j{y}_2={\delta }_1{e}_1+{\delta }_2{e}_2,\lambda =z_1+j{z}_2={\lambda }_1{e}_1+{\lambda }_2{e}_2$ with $Re\left(y_1\right)>\left|Im\left(y_2\right)\right|$, $Re\left({\lambda }_1\right)>0,Re\left({\lambda }_2\right)>0$ and $r=\left[Re\left(z_1\right)\right]+1$.

Theorem 8.

Let ${\mathcal{P}}_h\left(\mathsf{\Xi }\right)$ be the bicomplex Legendre polynomials, piecewise continuous on $M^{\prime }=(0,\infty )$ and integrable on any finite sub-interval of $M=[0,\infty )$, where $\nu ,\mathsf{\Xi },\lambda \in \mathbb{BC}$, $\nu =x_1+j{x}_2={\nu }_1{e}_1+{\nu }_2{e}_2,\mathsf{\Xi }=a_1+j{a}_2={\mathsf{\Xi }}_1{e}_1+{\mathsf{\Xi }}_2{e}_2,\lambda =z_1+j{z}_2={\lambda }_1{e}_1+{\lambda }_2{e}_2$, with $Re\left(a_1\right)>\left|Im\left(a_2\right)\right|$; then, we get

$$\begin{array}{c}\hfill {}_{0}I_{\mathsf{\Xi }}^{\lambda }{\mathcal{P}}_h\left(\mathsf{\Xi }\right)=\sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r}{2^h(h-r)!r!}}\otimes {\mathsf{\Xi }}^{\lambda +h-2r}\otimes {\displaystyle \frac{(2h-2r)!}{{\mathsf{\Gamma }}_2(\lambda +h-2r+1)}}.\end{array}$$

(29)

Proof. 

Definition of the bicomplex Riemann–Liouville integral operator Equation (27) on the expansion of the bicomplex Legendre polynomials Equation (14), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_{0}I_{\mathsf{\Xi }}^{\lambda }{\mathcal{P}}_h\left(\mathsf{\Xi }\right)& ={\displaystyle \frac{1}{{\mathsf{\Gamma }}_2\left(\lambda \right)}}\otimes {\int }_0^{\mathsf{\Xi }}{(\mathsf{\Xi }-\nu )}^{\lambda -1}\otimes {\mathcal{P}}_h\left(\nu \right)\otimes d\nu \hfill \\ & ={\displaystyle \frac{1}{{\mathsf{\Gamma }}_2\left(\lambda \right)}}\otimes \sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}\otimes {\int }_0^{\mathsf{\Xi }}{(\mathsf{\Xi }-\nu )}^{\lambda -1}\otimes {\left(\nu \right)}^{(h-2r)}\otimes d\nu \hfill \\ & ={\displaystyle \frac{1}{{\mathsf{\Gamma }}_2\left(\lambda \right)}}\otimes \sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}\otimes {\int }_0^{\mathsf{\Xi }}{(1-{\displaystyle \frac{\nu }{\mathsf{\Xi }}})}^{\lambda -1}\otimes {\mathsf{\Xi }}^{\lambda -1}\otimes {\left(\nu \right)}^{(h-2r)}\otimes d\nu .\hfill \end{array}\end{array}$$

We separate the integral and denote it by I as follows

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill I& ={\int }_0^{\mathsf{\Xi }}{(1-{\displaystyle \frac{\nu }{\mathsf{\Xi }}})}^{\lambda -1}\otimes {\mathsf{\Xi }}^{\lambda -1}\otimes {\left(\nu \right)}^{(h-2r)}\otimes d\nu \hfill \\ & =\left({\int }_0^{{\mathsf{\Xi }}_1}{(1-{\displaystyle \frac{{\nu }_1}{{\mathsf{\Xi }}_1}})}^{{\lambda }_1-1}{\mathsf{\Xi }}_1^{{\lambda }_1-1}{\left({\nu }_1\right)}^{(h-2r)}d{\nu }_1\right)e_1\hfill \\ & +\left({\int }_0^{{\mathsf{\Xi }}_2}{(1-{\displaystyle \frac{{\nu }_2}{{\mathsf{\Xi }}_2}})}^{{\lambda }_2-1}{\mathsf{\Xi }}_2^{{\lambda }_2-1}{\left({\nu }_2\right)}^{(h-2r)}d{\nu }_2\right)e_2\hfill \\ & =I_1{e}_1+I_2{e}_2\hfill \end{array}\end{array}$$

where $q_1={\displaystyle \frac{{\nu }_1}{{\mathsf{\Xi }}_1}}$ and $q_2={\displaystyle \frac{{\nu }_2}{{\mathsf{\Xi }}_2}}$. Note that when ${\nu }_1=0\Rightarrow q_1=0,$ when ${\nu }_1={\mathsf{\Xi }}_1\Rightarrow q_1=1$ and similarly for $q_2$, we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill I_1& ={\mathsf{\Xi }}_1^{{\lambda }_1+h-2r}{\int }_0^1{(1-q_1)}^{{\lambda }_1-1}{\left(q_1\right)}^{h-2r}d{q}_1={\mathsf{\Xi }}_1^{{\lambda }_1+h-2r}{\displaystyle \frac{\mathsf{\Gamma }({\lambda }_1\left)\mathsf{\Gamma }(h-2r+1\right)}{\mathsf{\Gamma }({\lambda }_1+h-2r+1)}}\hfill \\ \hfill I_2& ={\mathsf{\Xi }}_2^{{\lambda }_2+h-2r}{\int }_0^1{(1-q_2)}^{{\lambda }_2-1}{\left(q_2\right)}^{h-2r}d{q}_2={\mathsf{\Xi }}_2^{{\lambda }_2+h-2r}{\displaystyle \frac{\mathsf{\Gamma }({\lambda }_2\left)\mathsf{\Gamma }(h-2r+1\right)}{\mathsf{\Gamma }({\lambda }_2+h-2r+1)}}.\hfill \end{array}\end{array}$$

So, we get

$$\begin{array}{c}\hfill I={\mathsf{\Xi }}^{\lambda +h-2r}\otimes {\displaystyle \frac{{\mathsf{\Gamma }}_2(\lambda \left)\mathsf{\Gamma }(h-2r+1\right)}{{\mathsf{\Gamma }}_2(\lambda +h-2r+1)}}={\mathsf{\Xi }}^{\lambda +h-2r}\otimes {\displaystyle \frac{{\mathsf{\Gamma }}_2(\lambda \left)(h-2r\right)!}{{\mathsf{\Gamma }}_2(\lambda +h-2r+1)}}.\end{array}$$

Combining the partial results to obtain the bicomplex form, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_{0}I_{\mathsf{\Xi }}^{\lambda }{\mathcal{P}}_h\left(\mathsf{\Xi }\right)=& {\displaystyle \frac{1}{{\mathsf{\Gamma }}_2\left(\lambda \right)}}\otimes \sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}\otimes {\mathsf{\Xi }}^{\lambda +h-2r}\otimes {\displaystyle \frac{{\mathsf{\Gamma }}_2(\lambda \left)(h-2r\right)!}{{\mathsf{\Gamma }}_2(\lambda +h-2r+1)}}\hfill \\ \hfill =& \sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r}{2^h(h-r)!r!}}\otimes {\mathsf{\Xi }}^{\lambda +h-2r}\otimes {\displaystyle \frac{(2h-2r)!}{{\mathsf{\Gamma }}_2(\lambda +h-2r+1)}}.\hfill \end{array}\end{array}$$

□

Theorem 9.

Let the bicomplex legendre polynomials ${\mathcal{P}}_h\left(\mathsf{\Xi }\right)$ be piecewise continuous on $M^{\prime }=(0,\infty )$ and integrable on any finite subinterval of $M=[0,\infty )$. Consider $r=Re\left(z_1\right)+1$, with $r-1<\lambda <r$, where $\nu ,\mathsf{\Xi },\lambda \in \mathbb{BC}$, $\nu =x_1+j{x}_2={\nu }_1{e}_1+{\nu }_2{e}_2,\mathsf{\Xi }=a_1+j{a}_2={\mathsf{\Xi }}_1{e}_1+{\mathsf{\Xi }}_2{e}_2,\lambda =z_1+j{z}_2={\lambda }_1{e}_1+{\lambda }_2{e}_2$, $Re\left(a_1\right)>\left|Im\left(a_2\right)\right|$, $Re\left({\lambda }_1\right)>0,Re\left({\lambda }_2\right)>0$; then, we get

$$\begin{array}{c}\hfill {}_{0}D_{\mathsf{\Xi }}^{\lambda }{\mathcal{P}}_h\left(\mathsf{\Xi }\right)=\sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r}{2^h(h-r)!r!}}\otimes {\mathsf{\Xi }}^{h-\lambda -2r}\otimes {\displaystyle \frac{(2h-2r)!}{{\mathsf{\Gamma }}_2(h-\lambda -2r+1)}}.\end{array}$$

(30)

Proof. 

By employing the definition of the bicomplex Riemann–Liouville derivative operator and applying it to the bicomplex Legendre polynomials, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_{0}D_{\mathsf{\Xi }}^{\lambda }{\mathcal{P}}_h\left(\mathsf{\Xi }\right)=& {\displaystyle \frac{1}{{\mathsf{\Gamma }}_2(r-\lambda )}}\otimes {\displaystyle \frac{d^r}{d{\mathsf{\Xi }}^r}}\otimes {\int }_0^{\mathsf{\Xi }}{(\mathsf{\Xi }-\nu )}^{r-\lambda -1}\otimes {\mathcal{P}}_h\left(\nu \right)\otimes d\nu \hfill \\ \hfill =& {\displaystyle \frac{1}{{\mathsf{\Gamma }}_2(r-\lambda )}}\otimes \sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}\otimes {\displaystyle \frac{d^r}{d{\mathsf{\Xi }}^r}}\otimes {\int }_0^{\mathsf{\Xi }}{(\mathsf{\Xi }-\nu )}^{r-\lambda -1}\otimes {\left(\nu \right)}^{(h-2r)}\otimes d\nu \hfill \\ \hfill =& {\displaystyle \frac{1}{{\mathsf{\Gamma }}_2(r-\lambda )}}\otimes \sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}\otimes {\displaystyle \frac{d^r}{d{\mathsf{\Xi }}^r}}\otimes {\int }_0^{\mathsf{\Xi }}{(1-{\displaystyle \frac{\nu }{\mathsf{\Xi }}})}^{r-\lambda -1}\otimes {\mathsf{\Xi }}^{r-\lambda -1}\otimes {\left(\nu \right)}^{(h-2r)}\otimes d\nu .\hfill \end{array}\end{array}$$

We separate the integral and denote it by I as follows

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill I& ={\int }_0^{\mathsf{\Xi }}{(1-{\displaystyle \frac{\nu }{\mathsf{\Xi }}})}^{r-\lambda -1}\otimes {\mathsf{\Xi }}^{r-\lambda -1}\otimes {\left(\nu \right)}^{(h-2r)}\otimes d\nu \hfill \\ & =\left({\int }_0^{{\mathsf{\Xi }}_1}{(1-{\displaystyle \frac{{\nu }_1}{{\mathsf{\Xi }}_1}})}^{r-{\lambda }_1-1}{\mathsf{\Xi }}_1^{r-{\lambda }_1-1}{\left({\nu }_1\right)}^{(h-2r)}d{\nu }_1\right)e_1\hfill \\ & +\left({\int }_0^{{\mathsf{\Xi }}_2}{(1-{\displaystyle \frac{{\nu }_2}{{\mathsf{\Xi }}_2}})}^{r-{\lambda }_2-1}{\mathsf{\Xi }}_2^{r-{\lambda }_2-1}{\left({\nu }_2\right)}^{(h-2r)}d{\nu }_2\right)e_2\hfill \\ & =I_1{e}_1+I_2{e}_2.\hfill \end{array}\end{array}$$

where $q_1={\displaystyle \frac{{\nu }_1}{{\mathsf{\Xi }}_1}}$ and $q_2={\displaystyle \frac{{\nu }_2}{{\mathsf{\Xi }}_2}}$. Note that when ${\nu }_1=0\Rightarrow q_1=0,$ when ${\nu }_1={\mathsf{\Xi }}_1\Rightarrow q_1=1$ and similarly for $q_2$, then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill I_1& ={\int }_0^1{(1-q_1)}^{r-{\lambda }_1-1}{\mathsf{\Xi }}_1^{h-{\lambda }_1-r}{\left(q_1\right)}^{(h-2r)}d{q}_1\hfill \\ & ={\mathsf{\Xi }}_1^{h-{\lambda }_1-r}{\displaystyle \frac{\mathsf{\Gamma }(r-{\lambda }_1)\mathsf{\Gamma }(h-2r+1)}{\mathsf{\Gamma }(h-{\lambda }_1-r+1)}}\hfill \\ \hfill I_2& ={\int }_0^1{(1-q_2)}^{r-{\lambda }_2-1}{\mathsf{\Xi }}_2^{h-{\lambda }_2-r}{\left(q_2\right)}^{(h-2r)}d{q}_2\hfill \\ & ={\mathsf{\Xi }}_2^{h-{\lambda }_2-r}{\displaystyle \frac{\mathsf{\Gamma }(r-{\lambda }_2)\mathsf{\Gamma }(h-2r+1)}{\mathsf{\Gamma }(h-{\lambda }_2-r+1)}}\hfill \end{array}\end{array}$$

So, we get

$$\begin{array}{c}\hfill I={\mathsf{\Xi }}^{h-\lambda -r}\otimes {\displaystyle \frac{{\mathsf{\Gamma }}_2(r-\lambda )\mathsf{\Gamma }(h-2r+1)}{{\mathsf{\Gamma }}_2(h-\lambda -r+1)}}={\mathsf{\Xi }}^{h-\lambda -r}\otimes {\displaystyle \frac{{\mathsf{\Gamma }}_2(r-\lambda )(h-2r)!}{{\mathsf{\Gamma }}_2(h-\lambda -r+1)}}.\end{array}$$

We combine the partial results to obtain the bicomplex form, and then we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_{0}D_{\mathsf{\Xi }}^{\lambda }{\mathcal{P}}_h\left(\mathsf{\Xi }\right)=& {\displaystyle \frac{1}{{\mathsf{\Gamma }}_2(r-\lambda )}}\otimes \sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r(2h-2r)!}{2^h(h-r)!r!(h-2r)!}}\otimes {\displaystyle \frac{d^r}{d{\mathsf{\Xi }}^r}}\otimes {\mathsf{\Xi }}^{h-\lambda -r}\otimes {\displaystyle \frac{{\mathsf{\Gamma }}_2(r-\lambda )(h-2r)!}{{\mathsf{\Gamma }}_2(h-\lambda -r+1)}}\hfill \\ \hfill =& \sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r}{2^h(h-r)!r!}}\otimes {\displaystyle \frac{d^r}{d{\mathsf{\Xi }}^r}}\otimes {\mathsf{\Xi }}^{h-\lambda -r}\otimes {\displaystyle \frac{(2h-2r)!}{{\mathsf{\Gamma }}_2(h-\lambda -r+1)}}\hfill \\ \hfill =& \sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r}{2^h(h-r)!r!}}\otimes {\displaystyle \frac{{\mathsf{\Gamma }}_2(h-\lambda -r+1)}{{\mathsf{\Gamma }}_2(h-\lambda -2r+1)}}\otimes {\mathsf{\Xi }}^{h-\lambda -2r}\otimes {\displaystyle \frac{(2h-2r)!}{{\mathsf{\Gamma }}_2(h-\lambda -r+1)}}\hfill \\ \hfill =& \sum _{r=0}^{\lceil {\displaystyle \frac{h}{2}}\rceil }{\displaystyle \frac{{(-1)}^r}{2^h(h-r)!r!}}\otimes {\mathsf{\Xi }}^{h-\lambda -2r}\otimes {\displaystyle \frac{(2h-2r)!}{{\mathsf{\Gamma }}_2(h-\lambda -2r+1)}}.\hfill \end{array}\end{array}$$

□

5. Numerical Experiments: Bicomplex Legendre Polynomials

In this section, we present numerical investigations of bicomplex Legendre polynomials. The study focuses on polynomial construction, orthogonality verification, spectral projection, coefficient decay, and spectral convergence for bicomplex-valued test functions.

5.1. Test Functions and Parameters

We consider bicomplex-valued functions of the form

$$F\left(x\right)=f_1\left(x\right)e_1+f_2\left(x\right)e_2,x\in [-1,1],$$

with components

$$f_1\left(x\right)=e^x,f_2\left(x\right)=sin\left(\pi x\right),$$

and approximate them using Legendre polynomials of maximum degree $N_{max}=20$. The spatial domain is discretized with 3000 points to ensure accurate numerical integration.

5.2. Construction of Bicomplex Legendre Polynomials

Bicomplex Legendre polynomials $P_n^{\left(1\right)}\left(x\right)$ and $P_n^{\left(2\right)}\left(x\right)$ are constructed recursively using

$$P_0\left(x\right)=1,P_1\left(x\right)=x,P_{n+1}\left(x\right)={\displaystyle \frac{(2n+1)x{P}_n\left(x\right)-n{P}_{n-1}\left(x\right)}{n+1}},n\ge 1,$$

for each component independently. Figure 1 shows the orthogonality heatmaps for the first seven polynomials. The diagonal dominance confirms that the polynomials are orthogonal up to numerical tolerance.

5.3. Bicomplex Projection and Approximation

The bicomplex functions are projected onto the polynomial basis using

$$c_n^{\left(1\right)}={\displaystyle \frac{2n+1}{2}}{\int }_{-1}^1{f}_1\left(x\right)P_n^{\left(1\right)}\left(x\right)dx,c_n^{\left(2\right)}={\displaystyle \frac{2n+1}{2}}{\int }_{-1}^1{f}_2\left(x\right)P_n^{\left(2\right)}\left(x\right)dx,$$

for $n=0,1,\dots ,N$. The approximations are then constructed as

$$f_1^{\mathrm{approx}}\left(x\right)=\sum _{n=0}^N{c}_n^{\left(1\right)}{P}_n^{\left(1\right)}\left(x\right),f_2^{\mathrm{approx}}\left(x\right)=\sum _{n=0}^N{c}_n^{\left(2\right)}{P}_n^{\left(2\right)}\left(x\right).$$

Figure 2 compares the exact and approximated functions for both components. The approximations closely follow the exact functions, indicating high accuracy for $N=10$.

Figure 3 shows the spectral coefficients $|c_n^{\left(1\right)}|$ and $|c_n^{\left(2\right)}|$ for the two components. The rapid decay with increasing polynomial degree confirms spectral accuracy and the efficiency of the bicomplex expansion for smooth functions.

The maximum approximation error is computed as

$${\mathrm{Error}}_1\left(N\right)=\underset{x}{max}|f_1\left(x\right)-f_1^{\mathrm{approx}}\left(x\right)|,{\mathrm{Error}}_2\left(N\right)=\underset{x}{max}|f_2\left(x\right)-f_2^{\mathrm{approx}}\left(x\right)|.$$

Figure 4 illustrates the exponential decay of the maximum error with increasing polynomial degree, confirming spectral convergence for both components.

6. Conclusions

Our paper advances the theory of bicomplex Legendre polynomials by extending classical foundations into the bicomplex domain and incorporating modern analytic methodologies. By examining their expansion formulas and orthogonality properties, we establish a coherent framework that facilitates further theoretical exploration and practical applications.

Furthermore, this paper underscores the central role of bicomplex analysis, special functions, and fractional calculus, providing fresh perspectives and powerful tools for research. In doing so, it strengthens the theoretical framework, fosters innovation, and broadens applications across the mathematical sciences. A compelling direction for future research is the development of a novel finite element method for systems of coupled fractional differential equations [14,15,16]. By utilizing the bicomplex Legendre polynomials introduced in this work as spatial basis functions, one can formulate a unified bicomplex-valued weak form for such systems. The key challenges involve deriving the Caputo fractional derivatives [17] of these polynomials and assembling the resulting bicomplex linear system [18], which benefits from the established orthogonality properties. Success in this endeavor would provide a powerful computational tool for problems in fractional viscoelasticity [19] and diffusion, where coupled phenomena are prevalent. Future work can extend this framework to higher-dimensional bicomplex systems, alternative orthogonal polynomials, bicomplex PDEs, adaptive spectral methods, and rigorous error analysis to enhance computational efficiency and theoretical understanding.