International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 10, October 2018, pp. 1613–1630, Article ID: IJMET_09_10_164 Available online at http://iaeme.com/Home/issue/IJMET?Volume=9&Issue=10 ISSN Print: 0976-6340 and ISSN Online: 0976-6359

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ORTHOGONAL POLYNOMIALS AND CLASSICAL ORTHOGONAL POLYNOMIALS

DUNIA ALAWAI JARWAN Education for Girls College, Al-Anbar University, Ministry of Higher Education and Scientific Research, Iraq

ABSTRACT The focus of this project is to clarify the concept of orthogonal polynomials in the case of continuous internal and discrete points on R and the Gram – Schmidt orthogonalization process of conversion to many orthogonal limits and the characteristics of this method. We have highlighted the classical orthogonal polynomials as an example of orthogonal polynomials because of they are great importance in physical practical applications. In this project, we present 3 types (Hermite – Laguerre – Jacobi) of classical orthogonal polynomials by clarifying the different formulas of each type and how to reach some formulas, especially the form of the orthogonality relation of each. Keywords: Polynomials, Classical Orthogonal, Monic Polynomial, Gram – Schmidt Cite this Article Dunia Alawai Jarwan, Orthogonal Polynomials and Classical Orthogonal Polynomials, International Journal of Mechanical Engineering and Technology, 9(10), 2018, pp. 1613–1630. http://iaeme.com/Home/issue/IJMET?Volume=9&Issue=10 1. INTRODUCTION The mathematics is the branch where the lots of concepts are included. An orthogonality is the one of the concept among them. Here we focuse on the orthogonal polynomial sequence. The orthogonal polynomial are divided in two classes i.e. classical orthogonal polynomials, Discrete orthogonal polynomials and Sieved orthogonal polynomials .There are different types of classical orthogonal polynomials such that Jacobi polynomials, Associated Laguerre polynomials and Hermite polynomials. The Jacobi polynomials are subdivided as Gegerbauer, legendre, and Chebyshev (it has its own two types). The plain Laguerre polynomial is the subclass of the Laguerre polynomials. Discrete orthogonal polynomials are based on the discrete values. In some cases these values are finite rather than an infinite sequence. The Racah polynomial is the example of discrete orthogonal polynomials. It has some special case like Hahn polynomials and dual Hahn polynomials with Meixner polynomials, Krawtchouk polynomials, and Charlier polynomials. The Sieved orthogonal polynomial is the one of orthogonal polynomial. It is classified as are sieved ultra spherical polynomials, sieved Jacobi polynomials, and polynomials. It includes modified recurrence relations.

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1.1. Definition of polynomial For a general polynomial of degree n,

P x = a X n + a X n −1 + ..... + a , a ≠ 0 n ( ) n n −1 0 n [Bickel et al. 2000]

We call an the leading coefficient of the polynomial.

1.2. Definition (Monic polynomial) A polynomial is monic if the coefficient of its leading term is 1.

% Pn ( x ) n a n − a P ( x ) = = X + n −1 X 1 + .....+ o n a a a We denote by n n n [Bickel et al. 2000],

1.3. Definition (Inner product)

Let V is a real vector space. An Inner product is a real – valued function .,. on V ×V such that, for all f , g , h ∈V ,

I. f + g , h = f , h + g,h λ f ,g = λ f ,g II. for λ ∈ III. f , g = g, f f f IV. , g ≥ 0 and , f = 0 iff f =0 see [Milovanovic and Gradimir, 2013; Axler and Sheldon, 2015]

Let us take V is our vector space, the X = (x 1 , x 2 ,...,x n ) and Y = ( y1 , y 2 ,..., y n ). Then the following is an inner product:

n x , y = ∑x i y i [Axler and Sheldon, 2015] i =1 Remark

2 2 The length of vectors X by X = x , x = x 1 + ..... + x n and

If X = 1 , we call X a unit vector and X is said be normalized. See [Axler and Sheldon, 2015] * The set of real – valued functions that defined on the continuous interval [a ,b ], denoted b C [a ,b ] , is a vector space. And f , g = ∫f ( x ) g ( x )dx , is an inner product. See [Milovanovic a and Gradimir, 2013]

1.4. Definition (Orthogonality in general linear spaces) Two elements a ,b of a vector space are orthogonal with respect to a given inner product .,. if a, b = 0 Definition: Orthonormal An orthonormal set is a subsect S of an inner product space, such that for all X ,y ∈S

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0 if x ≠ y x, y =  if X y 1 = [Axler and Sheldon, 2015]

Remark

The set of the (standard basis) vectors {e1 = (1,0,0 ), e 2 = (0,1,0 ),....,en } forms an orthonormal basis of Rn.

Let υ1 , υ 2 ,...,υ k be mutually orthogonal vectors, then

2 2 2 υ + υ + ... + υ = υ + ....+ υ 1 2 k 1 k see [Axler and Sheldon, 2015] 2. ORTHOGONAL POLYNOMIAL

2.1. Orthogonality polynomial on Intervals A set of orthogonal polynomials is an (infinite of finite) sequence of polynomials,

P0 (x ) , P1 (x ), P2 ( x ) ,...., where Pn (x ) has degree n and any two polynomials in the set are orthogonal to each other on the C [a ,b ] , that is

b P i , P j = ∫ P i ( x ) P j ( x ) a dx =0 for i ≠ j see [Agarwal and Gradimir, 1991; Diekema and Tom, 2012; Jia and Yan-Bin, 2016] Otherwise, ∞ A sequence of polynomials P x with degree  P (x ) = n for each n is called { n ( )}n= 0  n  orthogonal with respect to the weight function W (x ) on the C [a, b ] if the inner product of the polynomials Pi , Pj

P , P = P ( x )P ( x )w ( x )dx = h δ , h ≥ 0 n m ∫ n m n n ,m n

0 , n ≠ m δnm =  1 , n = m With where δn ,m is the Kronecker delta function The weight function w (x ) should be continuous and positive on C [a ,b ] such that the b n moments µn = ∫w ( x )X dx , n = 0,1, 2,3,... exist. a Remark: Note that the range of the integral may be infinite. See [Bickel et al, 2000; Everitta et al, 2001; Antonia et al, 2009; Stange, 2010; Diekema and Tom, 2012; Duenas et al, 2013; Sadjang et al, 2015]

2.2. Orthogonality polynomial on Finite point sets Let X be a finite set of discrete points on R, or a countable infinite set of discrete points on R and w x , x ∈X be a set of positive constant then a system of polynomials {Pn (x )}, n = 0,1,2,.... is said to be orthogonal on X with respect the weights w x if

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P n ,P m = ∑ P n (x )P m (x w) x = 0 , n ≠ m x ∈X when X is infinite

∑ Pn (x ) Pm ( x W) x = 0 , n , m = 0,1, 2,.... N , n ≠ m , or x ∈X When X is a finite set of N +1 discrete points.

n In the former case we also require ∑ X w x < ∞ see ([Area et al, 1998; x ∈X Everitta et al., 2001; Arvesu et al, 2003; Clarkson, 2013; Diekema and Tom 2012; Duenas et al., 2013; Eisinberg and Giuseppe, 2007])

2.3. Properties of orthogonal polynomials (1) All sets of orthogonal polynomials have a number of fascinating properties:

Any polynomial f ( x ) of degree n can be expanded in terms of P0 , P1 ,....,Pn , that is, there exist coefficients ai such that

n f ( x ) = ∑ai Pi ( x ) i =0 see [Area et al, 1998]

(2) Given an orthogonal set of polynomials {P0 (x ) , P1 (x ),....} each polynomial, Pk (x ) is orthogonal to any polynomial of degree < k .

(3) Any orthogonal set of polynomials {P0 ( x ), P1 ( x ) ,....} has a recurrence formula that relates any three consecutive polynomials in the sequence, that is, the relation

P = a X + b P −C P n +1 ( n n ) n n n −1 exits, Where a , b ,c coefficients depend on n . Such a recurrence formula is often used to generate higher order members in the set.

(4) Each polynomial in {P0 ( x ), P1 (x ),....} has all n of its roots real, distinct and strictly with in the internal of orthogonality (i.e. not on its ends).

th th (5) Furthermore the roots of the n degree polynomial, Pn lie strictly inside the roots of the ( n +1)

degree polynomial Pn+1 . (see [Area et al, 1998; Antonia, 2009; Masjed-jamei, 2006])

2.4. Gram - Schmidt orthogonalization process

Any infinite sequence of polynomials {Pn }with Pn having degree n forms a basis for the infinite dimensional vector space of all polynomials. Such a sequence can be turned into an orthogonal basis using the Gram – Schmidt orthogonalization process, by projecting out the components of each polynomial that are orthogonal to the polynomials already chosen. Let assume V as a vector space along with an inner product.

Let B = {X 1, X 2,...., X n } as a basis for V , and not necessarily orthogonal. An orthogonal basis \ B = {ν1 ,ν 2 , ν3 ,....,ν n } may be constructed from B as follows:

Then V 1 = X 2

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x 2 ,ν 1 V = X − projν X = X − ν 2 2 1 2 2 ν ,ν 1 1 1

x 3 ,ν 2 x3 ,ν1 V = X − projν X − projν X = X − ν − ν 3 3 2 3 1 3 3 ν ,ν 2 ν ,ν 1 2 2 1 1 .

n − 1 n −1 x n ,νi V n = X n − ∑ projν X n = X n − ∑ νi i ν ,ν i =1 i =1 i i \ Then B ={ν 1, ν 2, ν 3,....,ν n} is an orthogonal basis for V see ([Axler and Sheldon, 2015; George and Spencer, 2016]) Proof: "By Mathematical Induction"

i) if n =2, let B = { x 1, X 2} then by Gran – Schmidt

x ,υ υ = x , υ = x − 2 1 υ 1 1 2 2 υ ,υ 1 2 1 x , υ x ,υ υ ,υ = υ , x − 2 1 υ = υ , x − υ , 2 1 υ 1 2 1 2 υ , υ 1 1 2 1 υ ,υ 1 Then 1 1 1 1

x ,υ υ ,x = 2 1 υ ,υ 1 2 υ ,υ 1 1 1 1 = , x − x , υ 1 2 2 υ1 =0

Therefore υ 1, υ 2 is an orthogonal set. \ Suppose that if n −1 then B = {υ1 , υ2 ,...,υn −1} is an orthogonal sets, then we show that if n = {υ1 , υ2 ,..., υk } is orthogonal.

We show that υ n , υn− 1 = 0

x ,υ υ ,υ = υ ,x − n n−1 υ n −1 n n −1 n V ,υ k −1 n −1 n −1 x ,υ = υ , x − υ , n n−1 υ n −1 n n −1 υ ,υ k −1 k −1 k −1 x ,υ = υ , x − υ ,υ n n−1 n− 1 n n− 1 n−1 υ ,υ n −1 n−1 υ , ,υ = n − 1 x n − x n n−1 υ , υ , 0 = n− 1 x n − n− 1 x n = υ ,υ 0 n −1 n =

Therefore {υ 1 ,υ 2,...,υ k } is orthogonal sets.

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 υ υ υ  Remark: B ′′ =  1 , 2 ,..., n  is orthonormal basis for V .  υ1 υ 2 υn 

Properties of the Gram – Schmidt process α α • V k = X K ( 1 X 1 + ..... + k −1 X k −1 ) , 1 ≤ k ≤ n

• The Span of V 1 ,.....,V k is the same as the Span of X 1,....., X k

• V K = X K −Pk , where Pk is the orthogonal projection (proj) of the vector X K on the

subspace Spanned by X 1 ,....., X k −1

V k • is the distance from X K to the subspace Spanned by X 1 ,....., X k −1 and

V k = υk ,υk • (see [Axler and Sheldon,2015; George and Spencer, 2016]) 3. THE CLASSICAL ORTHOGONAL POLYNOMIALS The classical orthogonal polynomials arise in a number of practical situations and models, often as solutions to differential equations arising from boundary value problems. The classical orthogonal polynomials can then be separated into three distinct groups.

I. The first groups, the Hemite polynomials H n (x ) , are defined for the domain −x −∞ < x < ∞ with a weight function w (x ) =e .

II. The second group, the Laguerre polynomials L n (x ) , are defined for 0≤ x < ∞ with a weight function w (x ) =e −x

( α ,β ) III. The third group the Jacobi polynomials Pn ( x ) and all of its special cases, are defined for the domain −1 < x <1 with a weight function w ( x ) = (1 − x )α (1 + x )β α , β > −1 See [Agarwal and Gradimir, 1991; Ben-Cheikh and Douak, 2000; Everitta et al, 2001; Diekema and Tom, 2012; Clarkson, 2013; Milovanovic, 2013; Marcellan and Xu, 2014; Barry and Arnauld, 2018] These classical orthogonal polynomials satisfy an orthogonality relation, a three term recurrence relation. a second order liner differential equation and a So – Rodrigues formula. Furthermore: We have a generating function for each family of classical orthogonal polynomials. Remark:

The orthogonal polynomial P n (x ) can be represented by a Rodrigues formula n C n d    n  Pn ( x ) =  A ( x )  w ( x ) for a given constant C n where w (x ) is the weight function w ( x ) dx n   defining the inner product and A (x ) is a polynomial of degree at mots 2. See ([Richard, 1981; Wolfram and Schmersau, 1996; Bickel et al, 2000; Bustoz et al., 2000, Miranian, 2005; Walter, 2006; Milovanovic, 2013]) We wish to differentiate this n times by use of Leibniz's rule

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d n n  n  d k d n −k ( A ( x )) (B (x )) = ∑  A ( x ) B ( x ) dx n  k  dx k dx n −k k =0 see [Bustoz et al., 2000; Daniel, 2014]

Classical orthogonal polynomials

Hermite polynomials

Laguerre polynomials

Jacobi polynomials

Legendre polynomials Chebyshev polynomials Chebyshev polynomials first kinds second kinds

3.1. Hermit polynomials The classical Hermite polynomials have two important properties: I. They are included in a family of orthogonal polynomials II. Intimately connected with the commutation properties between the multiplication operator x and the differentiation operator.

Hermite polynomials H n (x ) are an orthogonal on the interval ( −∞ ,∞) regarding the weight 2 function w ( x ) = e −x . See [Johann and Zeng, 2010] They can be defined by means of their Rodrigues formula:

n (− 1) d n H n ( x ) = (w ( x )) w x dx n ( )

n n d H x w x w x (− 1 ) n ( ) ( ) = n ( ) dx

n n x 2 d −x 2 ( −1) e n e dx n =0,1,2,.... The creating function for Hermite polynomials is depicted by

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∞ n t 2xt −t 2 ∑H n ( x ) =e n! n=0 see ([Kelly, 2010; Varma and Tasdelen, 2013]) \ The recurrence relation is: H n +1 = 2xH n (x ) −H n (x )

n +1 ( −1) d n +1 H = w ( x ) n+1 w x n +1 ( ) Proof: ( ) dx

n +1 ( −1) d  d n  H n =  w ( x ) +1 w x dx  dx n  ( )  

n+ 1− n (− 1) d = H ( x ) w ( x ) w x dx n ( )

(−1) 2 2 =  −2 e− x H ( x ) + e −x H \ ( x ) w x  n n  ( ) n =0,1,2,....

(−1) = w \ ( x ) H ( x ) +w (x ) H \ ( x ) w x  n n  ( ) (−1) = w ( x )  −2 xH (x ) + H \ ( x ) w x  n n  ( ) = (− 1 ) − 2 xH (x ) − H \ (x )  n n 

\ H n +1 = −2 xH n (x ) −H n (x ) see [Kelly, 2010]

The definition of H n ( x ) implies that

0 2 d 2 = ⇒ = − 0 x − x n 0 H 0 (x ) ( 1) e 0e If dx

0 2 2 = ( − 1) e x .e −x = (1 ) (1 ) = 1

2 d 2 If n = 1⇒ H ( x ) = ( −1)e x e −x 1 dx

2 2 = ( − 1) e x . − 2x e − x = 2x

2 2 x 2 d −x 2 n = 2 ⇒H 2 ( x ) = ( −1) e e If dx 2

x 2  d −x 2  = −2e  xe   dx 

2 2 2 = −2 e x  −2 x 2e − x +e − x   

= 4x 2 −2

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3 2 d 2 n = ⇒ H x = − e x e − x 3 3 ( ) ( 1) 3 If dx

2 2  d 2  = − 1 e x   −2xe− x  ( ) 2    dx 

x 2  d  2 − x 2 − x 2  = ( + 2 )e  − 2x e +e   dx  

2 2 2 2 = 2e x  − 4x e − x + 4 x 3e − x − 2xe −x   

3 H 3 ( x ) = 8 x −12x

4 2 d 2 n = 4 ⇒H x = −1 4 e x e −x 4 ( ) ( ) 4 If dx

3 2 d 2 = e x  −2xe−x  3   dx  

2 x 2 d  −x 2 2 −x 2  = e − 2 e +4 x e  dx 2  

2 d  −x 2 −x 2 3 −x 2  = ex 4 x e + 8x e − 8x e  dx  

2 2 2 2 2 2 2 = e x  4e −x − 8 x 2e −x + 8e − x −1 6 x 2e − x − 24 x 2e −x +16x 4e −x   

= 16x 4 − 48x 2 +12 see [Kelly, 2010] By using Leibniz rule

n/2 k ( −1) n! n −2k H n ( x ) = ∑ ( 2x ) k ! ( n −2 k )! We obtain k =0

d n H x n n n n ( ) =2 ! Then dx

It is clear that H 2n ( x ) is an even function H 2n +1 is an odd function. See [Kelly, 2010]

The orthogonality of H n (x ) The Hermite polynomials satisfy the orthogonality relation ∞ 2 −x n m n ∫ e H m ( x ) H n ( x )dx = 2 n! π δmn , ∈ {0,1,3,...} see ([Mourad and −∞ Stanton, 1997; Kelly, 2010; Sadjang et al., 2015])

Proof:

(i) if m ≠ n

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∞ −x 2 ∫ H n ( x ) H m ( x )e dx −∞

∞ 2 = H x  H x e −x dx ∫ m ( )  n ( )  −∞

∞ n  n 2 2 d 2  = H ( x ) ( −1) e x .e − x e − x dx ∫ m  dx n  −∞  

∞  n 2  n d −x = ( −1) H m ( x ) e dx ∫  dx n  −∞   Now use integration by parts in times Then

∞ −x 2 ∫ H n ( x ) H m ( x )e dx = 0 −∞

(ii) if n = m

∞ − x 2 ∫ e H n ( x ) H n ( x )dx −∞

∞ n n − x 2 x 2 d −x 2 = ( −1) e e. e H n ( x )dx ∫ dx n −∞ We used integration by parts

∞ n n −x 2 d = e H n ( x ) dx ∫ dx n −∞

d n Since H ( x ) = 2 n n! dx n n

∞ 2 = ∫ e −x 2n n !dx −∞

∞ 2 ∫ e −x dx = π And consequently we achieve the result using the elementary fact that −∞

∞ 2 = 2 n n ! ∫ e −x dx = 2 n n! π −∞ see ([Kwon and Littejohn, 1997; Kelly, 2010])

3.2. Laguerre polynomials

The Laguerre polynomials L n (x ) are orthogonal on the interal (0, ∞) with respect to the weight function w (x ) =e −x

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They can be defined by means of their Rodrigues formula:

n 1 1 d n L n ( x )= w ( x ) x n !w x dx n ( ) ( )

1 d n = e x e −x x n [Mourad and Stanton, 1997] n ! dx n ( ) The generating function for Laguerre polynomials is given by

∞ e ( − xtl 1−t ) L ( x )t n = see ([Khalfa, 1999; Bickel et al. 2000; ∑ n 1−t n =0 Diekema and Tom, 2012]) Not that we have the first few polynomials is

1 d 0 n = ⇒ L x = e x e − x x 0 = e x e− x = 0 0 ( ) 0 . 1 If 0! dx

1 x d −x n =1 ⇒ L1 ( x ) = e e x If 1! dx

x  − x − x  = (1 ) (e )− xe + e  = − x +1

1 d 2 If n = 2⇒ L (x )= e x e −x x 2 2 2! dx 2 ( )

x 1 x d  2 − x −x  e  − x 2 −x − x −x  1  2  = e − x e + 2 xe  = − 2 xe + x e + 2e − 2 xe  =  x − 4 x + 2 2 dx 2 2

e x d3 If n =3 ⇒L ( x ) = e −x x 3 3 3! dx 3

e x  d 2  =   −x 3e − x +3x 2e−x  2   6  dx 

x e  d  2 − x 3 − x − x 2 −x  =  −3 x e + x e + 6 xe −3x e  6  dx  

e x =  − 6 xe − x + 3 x 2e − x + 3 x 2e − x − x 3e − x + 6e − x − 6 xe − x − 6 xe − x + 3x 2e −x  6   1 = − 6 x − 6 x − 6 x + (9 x 2 )+ 6− x 3  6  

1  3 2  =  −x +9 x −18 x +6  6 By using Leibnize rule we obtain

k n  n  (− 1) L ( x ) =   x k , n =0,1,2,.... see [Kelly, 2010] n ∑ k k ! k =0   The recurrence relation is

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(n + 1 )L n +1 (x ) = ( 2n + 1− x )L n (x ) − nL n −1 (x ) see [Herberta and Marcellán, 2014] Proof: By generating function

∞ e− xt /1−t L ( x )t n = ∑ n 1−t n=0 We differentiate two sides with respect to (t ) to obtain

∞  −x ( 1 − t ) − xt e −x t /1− t ( 1− t ) +e −x t /1−t L x nt n −1 =   ∑ n ( ) 2 2  1 − t  1− t n =0  ( )  ( )

− x + xt − xt 1 e − xt /1−t = e− xt /1−t. + ( 1−t) (1 − t )2 (1 −t )2

− xt /1− t − xt /1− t − xt /1−t e x −x t /1−t 1 xte 1 xte 1 = 2 − e + 2 2 (1 −t ) ( 1 − t ) (1 −t 2) ( 1− t ) (1 −t ) ( 1−t ) ( 1 −t )

1 − xt /1− t 1 − xt /1−t x = 2 e − e 2 (1 − t ) ( 1−t) (1 −t )

1  1   1  x =  e −x t /1 −t  −  e −x t /1−t      2 ( 1 − t )  ( 1 − t )   ( 1− t)  (1 −t )

1 ∞ x ∞ = L ( x )t n − L ( x )t n 1 t ∑ n 2 ∑ n − n =0 (1 −t ) n =0

2 Multiplying both sides by (1 − t ) = 1− 2t +t 2

∞ ∞ ∞ n −1 n n +1 ∑ nLn ( x )t − 2∑ nLn ( x )t + ∑nLn ( x )t n =0 n =0 n =0

∞ ∞ ∞ n n +1 n = ∑ L n ( x ) t − ∑ L n ( x ) t − x ∑L n ( x )t n =0 n =0 n =0

 ∞ ∞   ∞ ∞ ∞  ∞  n + 1 n +1   2 n n n  n−1  ∑ nL n ( x )t + ∑ L n ( x )t  =  + ∑ nL n ( x )t + ∑ L n ( x )t − x ∑ L n ( x )t  −∑nL n ( x )t  n= 0 n =0   n =0 n= 0 n = 0  n=0

∞ ∞ ∞ n +1 n n −1 ∑ ( n + 1) L n+ 1 ( x ) t = ∑ ( 2n + 1− x ) L n ( x ) t − ∑nL n−1 ( x )t n = 0 n =0 n =0

⇒ ( n + 1) L n +1 (x )= (2 n + 1− x )L n ( x )− nL n −1 (x )

Orthogonality of L n (x ) The Laguerre's polynomials satisfy the orthogonality relation

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∞ e −x L x L x dx ∫ n ( ) m ( ) = δ m ,n see ([Herberta and Marcellán, 2014, Sadjang et 0 al., 2015]) Proof:

(i) if m ≠ n ⇒

∞ − x (i) if m ≠ n then ∫e L n ( x ) L m ( x )dx = 0 0 By generating function for Laguerre's polynomial gives

∞ −x t /1−t n e ∑ L n ( x )t = 1−t n=0

∞ e −x s /1 −s L ( x )s m = ∑ m 1−s m =0

 t s  − x +  ∞ ∞  1 −t 1−s  ⇒ n m e ∑∑ L n ( x ) L m ( x )t s = ( 1 − t )( 1− s) n = 0 m=0 We now multiply both sides by e −x

 t s  −x  1+ +  ∞ ∞ e  1− t 1−s  ⇒ − x n m ∑∑e L n (x )L m (x )t s = ( 1 − t )( 1−s ) n =0 m =0 Integrate both sides from 0 to ∞ with respect to x , which gives:

∞ ∞ ∞   ⇒ −x n m ∑∑  ∫ e Ln ( x ) Lm ( x ) dx  t s   n =0 m =0  0 

∞  t s  1 − x  1+ +  = e  1 − t 1−s dx 1 − t 1 −s ∫ ( )( ) 0

  t s    − x  1+ +   1 e  1− t 1−s  ∞ =   ( 1 − t )( 1− s )   t s  0   − 1 + +     1 − t 1 −s  

  − 1 −t 1−s   1  ( )( )  = [ 0− 1]  (1 − t )(1 − s )   (1 − t )(1 − s ) + t (1 − s ) + s (1 − t ) 

1 (1 −t )(1 −s ) = (1 −t )(1 − s ) (1 −t )(1 − s ) + t −t s + s −ts

1 = 1− s −t + st + t −t s + s −ts

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1 = 1− st

− 1 2 = ( 1− st ) = 1+ st + (s t ) + ...+

∞ = ∑s nt n n =0

∞ ∞  ∞  ∞  − x  n m n n ∑∑  ∫e L n ( x ) L m ( x )dx t s = ∑s t   Therefore n =0 m =0  0  n =0 When m ≠n , equating coefficients of t ns m on both sides of gives

∞ − x 0 ∫e L n ( x ) L m ( x )dx = 0 When m = n , equating coefficients of t ns n from both sides of gives

∞ −x  n ∫e L n ( x ) dx =1 0 Combining we get

∞ e −x L x L x dx ∫ n ( ) m ( ) = δ mn 0 0 if m ≠ n Where δ mn = see ([Perez and Pinar, 1996; Kwon and 1 if m = n Littlejohn, 1997; Mourad and Stanton, 1997; Miranian, 2005; Duenas et al, 2013; Herberta and Marcellán, 2014])

3.3. Jacobi polynomials The Jacobi polynomials are an orthogonal on the interval (−1,1) , regarding the weight function w ( x ) = (1 − x )α (1 + x )β They can be defined by means of their Rodrigues formula:

n n α β ( −1) 1 d  n α β  P ( , ) ( x ) = w ( x ) 1 − x 2 (1 − x ) (1 +x ) n n w x n  ( )  2 n ! ( ) dx

n n ( −1) − α − β d  n + α n+β  = (1 − x ) (1 + x ) ( 1 − x ) (1 + x )  for n = 0,1,2,... see [Johann and Zeng, 2n n ! dx n   2010] By using Leibniz's rule we have

n n (α ,β ) ( −1 ) k  n + α  n + β n −k k Pn (x )= ∑( − 1)   ( 1− x ) ( 1+ x ) 2n  k  n − k  k =0 n = 0,1,2,... A creating function for the Jacobi polynomials is depicted by

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α +β ∞ 2 ( α ,β) n α =∑Pn (x )t R 1+ R − t 1+ R + t ( ) ( ) n =0

Where R = 1 − 2xt +t 2 see ([Antonia et al 2009; Paul, 1990; Szego, 1975])

3.3.1. Orthogonality Jacobi polynomials The Jacobi polynomials satisfy the orthogonality relation

1 α β ( 1 − x ) ( 1− x ) P (α ,β ) ( x ) P (α ,β )dx ( x )δ ∫ m n mn 2 α+ β+1 Γ( n + α +1 )Γ( α + β +1) −1 = δ ( 2 n + α + β + 1) Γ ( n + α + β + 1) n! mn

α > −1 , β > −1 m , n ∈ 0,1,2,... for { } see ([Mourad and Stanton, 1997; Sadjang et al, 2014])

3.3.2. Special cases of Jacobi polynomials

3.3.2.1. Legendre polynomials For α = β = 0, these are called Legendre polynomials for which the interval of orthogonality is [−1,1] and the weight function is simply w (x ) =1 n ( −1) d n n 1 2 The Rodrigues formula is Pn ( x ) = n n ( −x ) 2 n ! dx

The recurrence relation is ( n + 1) Pn +1 ( x ) = (2 n + 1)xPn ( x ) − nPn −1 (x )

n 1  n  n  n −k k Where P ( x ) =   ( x − 1) ( 1+x ) n = 0,1,2,... n n ∑ k n − k 2 k= 0    1 They satisfy P (x ) = 1, P ( x ) = x , P (x )= 3 x 2 −1 0 1 2 2  

1 2 they Rodrigues formula is P m ( x ) P n ( x )dx = δ mn ∫ 2 n + 1 −1

∞ 1 n The orthogonality relation = Pn ( x )t see ([Diekema and Tom, 2012; Jia, 2 ∑ 1 − 2xt + t n= 0 Yan-Bin, 2016; Mourad and Stanton, 1997; Liu, 1998; Kwon and Littlejohn, 1997; Yszard, 2004])

3.3.2.2. Chebyshev polynomials For α = β = ±1/ 2, one obtains the Chebyshev polynomials of the second and the first kind, respectively.

−1  −1   2 2  (a)Chebyshev polynomials of the first kind T n (x ) α = β = , w (x )= (1 − x )  2 

The Rodrigues formula is

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1 d n  n−1/2  T x = 1 − x 2 1− x 2 n ( ) n n ( )  −2 n ! dx   ( ) The recurrence relation is T x = 2xT x −T x n +1 ( ) n ( ) n −1 ( ) n =1, 2,... T x n arc x Where n ( ) = cos ( cos( )) n =1, 2,...

T x = T x = x They satisfy 0 ( ) 1 , 1( ) s ee ([Mourad and Stanton, 1997; Yszard, 2004]) ∞ 1− xt t < 1 n = The generating function ∑T n ( x )t 2 , n =0 1− 2xt + t

1 1 − 2  π  The orthogonality relation is T ( x ) T ( x ) 1 − x 2 dx =  −δ δ see [Szego, 1975] ∫ n m ( )  2 no  mn −1

 1 1  (b)Chebyshev polynomials of the second kind: α = β = , w ( x ) = (1 −x )2   2 

1 1 d n  n+  In Rodrigues formula U x =  1 − x 2 2  n ( ) n n ( ) ( − 2 ) n ! 1− x 2 dx  

The recurrence relation

U (x ) = 2xU ( x ) −U (x ) n +1 n −1 n = 1, 2,...

U ( x ) = s  ( n +1) arc (cos ( x )) / sin (arc cos ( x )) Where n in   U x = 1 , U x = 2x then 0 ( ) 1 ( )

the generating function U n (x ) is

∞ n 1 ∑U n ( x )t = 1 − 2xt + t 2 k = 0 1 1 2 2 π The orthogonality relation is U n ( x )U m ( x ) 1− x = δmn see ([Yszard, 2004; ∫ ( ) 2 −1 Mourad and Stanton, 1997; Szego, 1975; Szego, 1975]) 4. CONCLUSION Here we consider the three types of classical orthogonal polynomials i.e. Hermite, Laguerre and Jacobi. So, It is must to discuss some scenarios related to them. As we know that the Hermite polynomials is classical orthogonal polynomial sequence defined by Pierre-Simon Laplace in 1810. It is denoted by notation He and H and these are standard references. These Hermite polynomials are orthogonal with respect to the measure. Hermite polynomial also satisfies the recursion. The actual mathematics is discussed below. The Laguerre polynomial is implemented using Wolfram Language as Laguerre [n, k, x]. The Laguerre polynomials are orthogonal over 0 to infinity with respect to the weighting function. Here, k is an integer called a Laguerre function.

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Now discuss the Jacobi polynomial. Jacobi polynomial is also called as hyper geometric polynomials. The Jacobi polynomials are invented by Carl Gustav Jacob Jacobi. With the help of the hyper geometric function The Jacobi polynomials are defined. Related mathematical part is explained below. REFERENCES

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