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Basic Analogue of Legendre Polynomial and its Difference Equation



Javid Ahmad Ganie and Renu Jain
 
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ABSTRACT

The legendre polynomials belong to rich class of orthogonal polynomials which have been extensively investigated because of their applications in various fields. The main objective of this study was to derive the discrete Legendre polynomials as they represent the discrete functions or discrete data. In this study the difference equation of discrete Legendre polynomials was derived. Firstly, introduced the discrete Legendre polynomial of integer order about point s by using Taylor’s formula and some of its properties and later on it was shown that the current system satisfied Rodrigue’s formula.

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  How to cite this article:

Javid Ahmad Ganie and Renu Jain, 2019. Basic Analogue of Legendre Polynomial and its Difference Equation. Asian Journal of Mathematics & Statistics, 12: 1-7.

DOI: 10.3923/ajms.2019.1.7

URL: https://scialert.net/abstract/?doi=ajms.2019.1.7
 
Received: January 25, 2019; Accepted: May 09, 2019; Published: July 23, 2019


Copyright: © 2019. This is an open access article distributed under the terms of the creative commons attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.

INTRODUCTION

Quantum calculus is very interesting field in mathematics especially in physics. Quantum calculus is also known as calculus without limits. There are two types of quantum calculus, the q-calculus and h-calculus1. This paper focused more on the h-calculus. In classical and quantum physics there has been great interest in the discrete models from last few years2. Cosmic strings and blackholes3, confirmed quantum mechanics4 are mainly based on quantum calculus approach. Certain models are solved by using the theory of the classical discrete polynomials5. Discrete oscillators of Charlier, Meixner oscillators6 and Kravchuk oscillators7-11 that are related to the polynomials of Charlier, Meixner and Kravchuk, respectively and the finite radial oscillator12,13 related with the Hahn polynomials are some of the important instances.

A polynomial is defined on (-∞, ∞) but usually an approximation process is used on a finite domain. The polynomial should be segmented in order to utilize the integration or delta function approach for construction. The nth order polynomial is usually expressed as14:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

In almost, all branches of applied sciences researchers encounter some special classes of orthogonal functions such as; Legendre, Jacobi, Chebyshev and Laguerre polynomials15,16. All of these polynomials, but Legendre polynomials in particular have an extensive usage in the areas of physics and engineering. The Legendre polynomials are widely used in the determination of wave functions of electrons in the orbits of an atom17,18 and in the determination of potential functions in the spherically symmetric geometry19, etc. Also, Legendre polynomials have an extraordinary importance in representing a stream of data or a function. The h-Legendre polynomials are a discrete or quantum variant of the classical Legendre polynomials.

The Legendre polynomial of degree n is defined as20:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation
(1)

where, n is positive integer and x is variable.

So, the purpose of this study was to derive the discrete version of the Legendre polynomial. Legendre polynomials also called hypergeometric polynomials are a class of orthogonal polynomials.

Mathematical preliminaries: The h-derivative for a function f: hZ→ ℂ is defined as:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

and this yields the classical derivative if:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

we understand the limit in the sense of Bonita and Ralph21.

The product rule for Δh is given by:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

and the quotient rule for Δh is given by:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

The discrete analogue of the integral Image for - Basic Analogue of Legendre Polynomial and its Difference Equation are the Δh-integral:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

where, a is hm and b is hn.

Discrete polynomials and related functions: The symbol h has two different meanings h alone referred to a number in (0, ∞) and hn will refer to discrete polynomials. Define the weighted hn monomials of hZ centred about s by the function:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

By Cuctha22:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

where, s is hm and t is hn.

It is also known from the previous study23 that:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

The discrete monomials of hZ centered about s is given by:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Lemma 1: For n, m ∈ ℕ0. we have:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Proof: we compute:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

this lemma is also known as shift lemma22.

The formula:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

is known as binomial series24.

Discrete legendre polynomial
Definition: we define discrete analogue of Legendre polynomial of degree n about point s by:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation
(2)

where, t∈ hZ.

Theorem 1: Define y (t) = Pn (t, s: h). Then y(t) satisfies the equation:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Proof: we have for n = 2m:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

we have by using above lemma:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Also, we have:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation
(3)

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

From Eq. 3, we have:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Combining above results, we get:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

which is desired difference equation for Legendre’s differential equation.

Theorem 2: The following formula holds:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Proof: If n = 2m+1, then:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

and so:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

The case n = 2m is essentially the same.

Examples of first few discrete legendre polynomials:

If n = 0, P0 (t, s: h) = 1
If n = 1, P1 (t, s: h) = (t-s)h
If n = 2, P2 (t, s: h) = 3/2 (t-s)2h-1

Theorem 3
Three-term recurrence: The following formula holds:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Proof: If n = 2m, then:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

and:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

so, we compute:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation
(4)

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Hence proof.

Rodrigue's formula and orthogonality
Theorem 4:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Proof: We have for n = 2m and nεℕ0:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

We have:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

we get the result by using binomial theorem.

Corollary 1: Let <.,.> be an inner product to which the sequence Image for - Basic Analogue of Legendre Polynomial and its Difference Equation of polynomials is orthogonal. Then, there exist constants an, bn, cn such that:

Pn+1 (t) = (an,+tbn)Pn(t)+cnPn-1(t)

holds for all nεℕ022.

Moreover, there does not exist an inner product with respect to which all of the Pn functions are orthogonal.

Proof: We have:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Also:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

and:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

By using above lemma, we have:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation
(5)

Now:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation
(6)

Equations 5 and 6 yields the system:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

⇒a-b = 5h

Also:

Image for - Basic Analogue of Legendre Polynomial and its Difference Equation

⇒ a = 0 so b = -5h

Therefore, by corollary 1 there is no inner product for which the polynomials are orthogonal.

CONCLUSION

In this work, some useful properties of discrete Legendre polynomial were derived by using the Taylor series about point s. Authors have derived h-difference equation analogue of the Legendre's differential equation, recurrence relation and Rodrigue's type formula. The expectation is that these results can be generalized further on Stephen Hilger time scale basis. These results were derived first time and are likely to have useful applications in Physical Sciences and Engineering.

SIGNIFICANCE STATEMENT

This study discovered the results which are discrete version or quantum variant and which are under consideration are better in comparison to classical results that can be beneficial for physicists and engineers. This study will help the researcher to uncover the critical areas of polynomial theory that many researchers were not able to explore.

ACKNOWLEDGMENT

Authors are highly thankful to referees for their valuable suggestions and helpful remarks for the improvement of this paper which led to a better presentation.

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