Orthogonal Polynomials and Their Properties

Republic of the Sudan Ministry of High Education and Scientific Research Nile Valley University

College of Post Graduate Mathematics Department

A thesis submitted partially for M.sc. in Mathematics

Orthogonal Polynomials and their Properties

Rahama Abdalla Elballa

Supervised by:

Dr. Adam Abdella Abakar

Aug 2009

Dedication

To my father and mother

And

To my wife and brother

Acknowledgement

First I am thanks Allah and I am grateful to express my deep thank to my supervisor

Dr/ Adam Abdella Abakar

Also my thanks to all persons who helps me, and thank for all

Nile Valley University staff

Contents:  Significance of the study  Statement of the problem  Hypothesis of the study  Methodology of the study Chapter one: Basic Concepts 1-1 Introduction 1-2 Objectives Chapter two: Orthogonal Polynomials Sequences 2-1 Definition 2-2 Standardization 2-3 General properties of orthogonal polynomials sequences 2- 3 -1 Recurrence relations 2-3-2 Existence of roots 2-3-3 Interlacing of roots 2-4 Differential equations leading to orthogonal polynomials 2-4-1 Rodrigue's formula 2-4- 2 The number 2-4-3 Second form for the differential equation 2-4-4 Third form for the differential equation 2-4-5 Orthogonality Chapter three :The Classical Orthogonal Polynomials 3-1 Jacobi polynomials 3-2 Gegenbauer polynomials 3-3 Legendre polynomials 3-3-1 Associated Legendre polynomial 3-4 Chebyshev polynomials 3-5 Laguerre polynomials 3-6 Hermite polynomials

Chapter four: Constructing orthogonal polynonlials ' using moment 4-1 Significance of the moment 4-2 Constructing orthogonal polynomials 4-3 Table of classical orthogonal polynomials. Chapter five: Some application 5-1 Mathematical applications 5-1-1 Numerical analysis 5-1-2 Theory of random matrices 5-1-3 Computer approximations 5-2 Physics and Engineering applications 5-2-1 Electromagnetism 5-2-3 Classical mechanics 5-2-3 Communition Conclusion & Recommendation References

Abstract:

Orthogonal polynomials are classes of polynomials {Pn(x)} defined over an interval that obey the property of orthogonality. The main contributions of this work concern the description of a small sample of what is known about specific sets of orthogonal polynomials the general basic concepts, and the constructing of orthogonal polynomials. by using the moment, and also the presentation of some uses or applications in the field of Mathematics, Physics, and Engineering.

الخالصة

كثيرات الحدود المتعامدة هي أنواع من كثيرات الحدود يمكن تعريفها عمى أنها

الفترة التي تتفق مع خواص التعامد. إن اإلسهامات الرئيسية في هذا العمل تتضمن

وصف نموذج صغير فيما يعرف بمجموعات محددة من كثيرات الحدود المتعامدة،

المفاهيم األساسية العامة، وتكوين كثي ارت الحدود المتعامدة باستخدام العزوم وأيضاً

تقديم بعض االستخدامات أو التطبيقات في مجال الرياضيات والفيزياء والهندسة.

Chapter One Basic Concepts

1-Introduction: Orthogonal polynomials seelned to have first been developed by Legendre and Laplace in their work of celestial mechanics. One of the developments was work of Laplace on probability theory. Both of these dealt with specific sets of orthogonal polynomials, first polynomials orthogonal with respect to a symmetric beta distribution on (-1,1) and then the normal distribution on the whole real line. The general theory starts with Chebyshev in the 1850s. Orthogonal polynomials are classes of polynomials {Pn(x)} defined over an interval that obey the property of orthogonality. In the present chapter we shall introduce the corresponding concepts and notations. Two polynomials Pm(x) and Pn(x) which are real-valued continuous functions defined on the interval a < x < b and are such that the integral of the product Pm(x) Pn(x) over that interval exists. We shall denote this integral by ( P m,P n) such that:

(Pm, pn) = ( 1-1) where ( Pm, Pn) means the inner product space of continuous functions Pm, Pn with the following properties: 1- For any scalars and and any polynomials Pm(x), Pn(x),Ps(x) in polynomial space V, then ( Pm (x) + Pn (x),Ps (x)) = (Pm (x ),Ps (x ))+ (Pn (x ),Ps(x) (linearity)

2- For any polynomials Pm(x) and Pn(x) in V, (Pm, Pn) = (Pn, Pm) (symmetry) 3- For any Pm(x) in V, ( Pm, Pn) > 0 and (positive-defmiteness) (Pm, Pn) = 0 if and only if Pm =0 The functions are said to be orthogonal on the interval a

if: (Pm,Pn)= ,(m n) (1-2) [1][4]

A set of real-valued functions P1(x), P2(x), ...... is called an orthogonal set of functions on an interval a

(Pm,Pn) = = (1-4) Such a set is called an orthonormal set of functions on the interval a < x

(Pm, Pn) = = (1-5)

The norm ||Pm|| equals , because

2 (Pm, Pn) = ||Pm|| = , (m=1,2,…..)

So the corresponding orthonormal set consists of the functions:

Then the orthogonal sets yield important types of series development. In fact ,let P1(x), P2(x), .... be any orthogonal set of functions on an interval a < x

This series is called the generalized Fourier series of f(x) and its coefficients cl,c2, .. : ... are called the Fourier constants of f(x) with respect to that orthogonal set of functions. In fact, multiplying both sides of (1-6) by Pm(x) (m fixed), integrating over a

(f,Pm)= = Integral for which n=m equals (Pm ,Pm) = || Pm||2,wehere as all the other integrals on the right are zero, because of the 2 othogonality. Hence (f, Pm) = cm|| Pm|| , the desired formula for the Fourier constant is: (1-7) Some important sets of real functions Pl,P2, ..... occurring in applications are not orthogonal but have the property that for some function W(x), (1-8) Such a set is then said to be orthogonal with respect to the weight function W(x) on the interval a

(1-9)

And if the norm of each function Pm is 1 , the set is said to be orthonormal on that interval with respect to W(x).If we sethm= Pm then (1-8) becomes: (1-10)

That is the functions hm form an orthogonal set in the usual sense. Various important orthogonal sets of functions arise as solutions of second-order differential equations of the form in some interval a

(a) K1y+K2Y'=0 at x=a (b) L1y+L2y'=0 at x=b; (1-12) [2][3][5]

Where is a parameter , and K1, K2,L1, L2 are given are given real constants, at least one in each conditions (1-12) being different from zero. The equation (1-11) is known as the Sturm-Liouville equation, and we shall see that Legendre's equation, and other important equations can be written in the form (1-11). The conditions (1-12)refer to the boundary points x=a and x=b of that interval and, therefore, are called boundary conditions. A differential equation(1-11) with respect to the boundary conditions (1-12) is called Liouvilleproblem. This problem has the trivial solution y=0 for any value of the parameter . Solution y 0 are called characteristic functions or eigenfunctions of the problem, and thy values of for which such solutions exist. For example: if we need to find the eigenvalues and eigenfunctions of the Sturm-Liouville problem: (a) y"+ y=0 (b) y(o) = 0, y( ) =0 (1-13) For negative the general solution of the equation is: vx -vx y(x) = c1e + c2e From (l-13b) we obtain cl=c2=0 and q=0, which is not an eigenfunction. For = 0 the situation is similar. For positive = v2_ the general solution is y(x) =Acosvx+B sin vx. From the first boundary condition we obtain y(0)=A=0. The second boundary condition then yields: y( ) = B sin v or v = 0,±1,±2, for v=O we have y=0. For =v2 =1,4,9,16, .. , taking B=I,we obtain y(x) = sin vx v = 1,2, . . These are the eigenfunctions of the problem, and the eigenvalues are = v2 where v = 1,2, .. . It can be shown that under rather general conditions on the functions p, q , and r in (1-11) the Sturm-Liouville problem (1-11), (1-12) has infinitely many eigenvalues. Furthermore, theeigenfunctions are orthogonal, as follows: Theorem 1: (orthogonality).Let the functions p, q and r in the strum-Liouville equation (1-11) be real-value and continuous

4 on he interval a

[r(ry'nym) - ry'myn (1-15) because the function in the brackets is continuous on a

(1-16) is zero, and from (1-15) we obtain (1-17). If k2= 0 then by assumption kl 0, .and the argument of proof is similar. Case3. Ifr(a)=O, but r(b) * 0 ,the proof is similar to that in case2, but instead of(I-12a) we now have to use (1-12b). Case4; If r( a) 0 and r(b) 0 , we have to use both boundary conditions (1-12) and prbceed as in case 2 and 3. Case5. Let r(a) = r(b). Then (1-16)takes the form: r(b)[y'n (b )ym (b)- y'm(b )yn (b)- y'n (a )ym (a) + y'm (a )yn (a)] We may use {1-12) as before and conclude that the expression in bracket is zero. However, we immediately see that this would also follow from (1-14), so that we may replace (1-12) by (1-14). Hence (1-15) yields (1-17) as before. This completes the proof of theorem 1. The eigenvalues of Sturm-Liouville problems have the following interesting property. Thorem2 (Real eigenvalues): If the Sturm-Liouville problem (1-11), (1-12) satisfies the condition stated in theorem 1 and p is positive in the whole interval a 0 (or p < 0) for all x between a and b, the integral on the left is not zero. Hence, = 0 , which mean that = is real. This completes the proof. [5][6]

Orthogonal polynomials have very useful properties in the solution of mathematical physics problems. Chapter two covers general properties of orthogonal polynomials and chapter three gives some details of classical orthogonal polynomials. Chapter four mentions the constructing orthogonal polynomials by using moment. . The uses or the applications of these orthogonal polynomials in Mathematics, Physics, and Engineering are described in chapter five, and conclusion and recommendation are contained in the end.

2- Objectives: The goals or the objectives of our study orthogonal polynomials is that the orthogonal polynomials are widely used for their importance in various divisions of Mathematics, Physics, and Engineering and so on there is:  In approximation theory the orthogonal polynomials are used to approximate functions and that by making the error minimum.  Some Computer techniques use a recursive method then orthogonal polynomials are desirable for it. For example computing the function by using Chebyshev polynomials.  The orthogonal polynomials are preferable in solving problem of propagation of electromagnetic waves along a transmission line of length L.  The time-varying motion estimation also need the use of orthogonal polynomial; and so on ...... So basing on that it seemed to be the great important the study of orthogonal polynomials. [1]

CHAPTER TWO Orthogonal Polynomials Sequences:

2-1 Introduction: An orthogonal polynomial sequence is an infinite sequence of

real polynomials Po, PI, P2, …. of one variable x, in which each Pn has degree n, and such that any two different polynomials in the sequence are orthogonal to each other under a particular version of the inner product space. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by P.L.Chebyshev and was kept on by A.A.Markov and T.J.Stieljes and by a few other mathematicians. Since then, applications have been developed in many areas of mathematics and Physics.

2-2 Definition: The definition of orthogonal polynomials hinges on an inner product, defined as follows. Let [x1,x2] be an interval in the real line (where xl= and x2= are allowed). This is called the interval orthogonality. Let w:[xl,x2] be a function on the interval, that is strictly positive on the interior (x1,x2), but which may be zero or go to infinity at the end points. . Additionally, W must satisfy the requirement that, for any polynomial f , the integral is finite. Such a W is called a weight function. Given any xl, x2, and W as above, define an operation on pairs of polynomials f and g by :

(2-2-1)

This operation is an inner product on the space of all polynomials. It induces a notion of ortogonality in the usual way, namely that two polynomials are orthogonal if their inner product is zero. A sequence of orthogonal polynomials, then, is a sequence of polynomials Po, P1, P2, ... such that Pn has degree n and all members of the sequence are orthogonal to each other, forallm n,

(Pm,Pn)= 0. [1][5]

In other words, a sequence of orthogonal polynomials is an orthogonal basis for the (infinite dimensional ) space of all polynomials, with the extra requirement that Pn has degree n.

2- 3 Standardization: The chosen inner product induces a norm on polynomials in the usual way: = (2-3-1)

When making an orthogonal basis, one may be tempted to make an orthogonal basis , that is , one in which al basis elements have norm 1. Instead, polynomials are often scaled in a way that mathematicians agree on, that makes the coefficients and other formulas simpler. This is called Standardization. This standardization has no mathematical significance; it is just a convention. Standardization also involves· scaling the weight function in an agreed-upon way. Denoting by hn the square of the norm of Pn : hn = ( Pm, Pn), the values of hn for the standardized classical polynomials are listed in the table in chapter four. In this notation, (2-3-2) where is the kronecker delta .

2- 4 General properties of orthogonal polynomial sequences: All orthogonal polynomials sequences have a number of elegant and fascinating properties. Before proceeding with them: Lemma 1: Given an orthogonal polynomial sequence Pi(x), any. nth-degree polynomial S(x) can be expended in terms of P0, ..... ,Pn. That is , there are coefficients , ...... , such that (2-4-1)

Proof: We prove this by using mathematical induction. Choose so that the xn term of S(x) matches that of

.Then S(x)- anP n(x) )s aI!(p.-))th-degreepolynomial. And we continue downward. Lemma 2: Given an orthogonal polynomial sequence, each of its polynomials is orthogonal to any polynomial of strictly lower degree. Proof: Given Pn, any polynomial of degree n-l or lower can be expended in terms of P0,…., Pn-l .. Pn is orthogonal to each of them . [5][6] 2-4-1 Recurrence relations : Any orthogonal sequence has a recurrence formula relating any three consecutive polynomials in the sequence:

Pn+1=(anx+bn)Pn-cnPn-1 (2-4-1-1) Where the coefficients a, b, and c depend on n, as well as the standardization. Proof: the values of an, bn and cn can be worked out directly. Let

Kj and k'j be the first and second coefficients of Pj : i i-1 Pj(x) = kjx + k'jx +……. And hj be the inner product of Pj with itself: hj=( Pj, Pj)we have

2-4-1Existence of real roots: Each polynomial in an orthogonal sequence has all n of its roo s real, distinct, and strictly inside the interval of orthogonality. Proof: Let m be the number of places where the sign of Pn changes inside the interval of orthogonality , and let x1, x2,…. , xn be those points . Each of points is a root of Pn. By the fundamental theorem of algebra, m

2-4-2 Interlacing roots: The roots of each polynomial lie strictly between the roots of the next higher polynomial in the sequence. Proof: First, standardize all of the polynomials so that their leading terms are positive. This will not affect the roots. Next ,a lemma: For all n and all x, P'n+l(X)P n(X) > P n+l(X)P'n(x) Proof by induction. For n=0, P'l(x) > 0, Po(x) > 0, and P'0(x) = 0. Otherwise, the recurrence formula has :

Pn+1 (x) = (ax + b)Pn (x) - cPn-1 (x) with

So P'n+l = aPn +(ax+b)P'n -cP'n-l. So P'n+lPn - Pn+1P'n = [aPn+ (ax +b)P'n cP'n-l]Pn -[(ax+b)P n-cP'n-1] = [aPn – cP'n-l]Pn + cP n-l P'n 2 = aP n + c(P'nPn-l - PnP'n-l) ≥c(P'nP n-l - PnP'n-l) But P'nPn-l - PnP'n-l > 0 by the induction step. Now if x is a root of Pn+1 , the lemma tells us that : P'n+1 (x)Pn(x) > Pn+ 1 (x)P'n(x) = 0. So P'n+1 (x) "and Pn(x) have the same sign. But P'n+1(x) must change sign from any root of Pn+ 1 to the next. Therefore Pn must change sign also, so Pn must have a root in the interval. [1][4] 2-5 Differential equations leading to orthogonal polynomials: A very important class of orthogonal polynomials arises from a differential equation of the form : Q(x)f" + L(x)f' + Af=0 (2-5-1) Where Q is a given quadratic (at most) polynomial, and L is a given linear polynomial. The function f, and the constant A, are to be found. (We note that it makes sense for such an equation to have a. polynomial solution. Each term in the equation is a polynomial, and the degrees are consistent. ) This is a Sturm-Liouville type of equation. Such equations generally have singularities in their solution functions f except for particular values of . They can be thought of a eigenvector- eigenvalue problems: Letting D be the differential operator, D(f) = Qf" + Lf' , and changing the sign of ,the problem is to

11 find the eigenvectors ( eigenfunctions) f, and the· corresponding eigenvalues , such that f does not have singularities and D(f) = f. The solutions of this differential equation have singularities unless A takes on specific values. There is a series of numbers O, 1, 2, ..... that lead to a series of polynomial solutions PO, PI, P2, if one of the following sets of conditions are met: 1- Q is actually quadratic, L is linear, Q has two distinct real roots, the root of L lies strictly between the roots of Q, and the leading terms of Q and L have the same sign. 2- Q is not actually quadratic, but is linear, L is linear, the roots of Q and L are different, and the leading terms of Q and L have the same sign if the root of L is less than the root of Q, or Vice-versa. 3- Q is just a nonzero constant, L is linear, and the leading term of L has the. opposite sign ofQ. These three cases lead to Jacobi-like, Laguerre-like, and Hermitelike polynomials, respectively. In each of these three cases, we have the following:  The solutions are 'a series of polynomials P0, P1, P2, ..... , each Pn having degree n, and corresponding to a number n.  The interval of orthogonality is bounded by whatever roots Q has.  The root of L is inside the interval of orthogonality.

 Letting, the polynomials are orthogonal under the weight function w(x) = R(x ) . Q(x)  W(x) has no zeros or infinities inside the interval, though it may have zeros or infinities at the end points.  W(x) gives a finite inner product to any polynomials. .  W (x) can be made. to be greater than zero in the interval. (Negate the entire differential equation if necessary so that Q(x)>0 inside the interval). Because of the constant of integration, the quantity R(x) is determined only up to an arbitrary positive multiplicative

constant. It will be used only in homogeneous differential equations (where this doesn't matter) and in the definition of the weight function (which can also be indeterminate ). The tables in chapter four will give the "official" values ofR (x) and W(x). [5][7][8]

2-5-1 Rodrigue's formula: Under the assumptions of the preceding section, Pn(x) is proportional to : (2-5-1-1) This is known as Rodrigue's formula. It is often written : (2-5-1-2) where the numbers en depend on the' standardization. The standard values of en will be given in the tables in chapter four.[1][4]

2-5-2 The Numbers n : Under the assumptions of the preceding section, we have: .

n = -n( n -1 Q" + L'). (2-5-2-1) 2 Since Q is quadratic and L is linear, Q" and L' are constant, so these are just numbers.[1]

2-5-3 Second form of the differential equation: Let . Then (Ry')' = Ry''+R'y'= Ry" + Now multiplying the differential equation Qy" + Ly' + y =0 by R/Q, getting: (2-5-3-1) Or (2-5-3-2) This is the standard Sturm-Liouville form for the equation. [3][6]

2-5-4 Third form for the differential equation:

Let (2-5-4-1)

Then (2-5-4-2) Now multiplying the differential equation Qy" + Ly' + y =0 by S/Q, getting: (2-5-4-3) (2-5-4-4) But (Sy)"=Sy"+2S'y'+S"y, so: (Sy)"+ ( -S")y=0 (2-5-4-5) Or letting u –Sy, implies that: u"+ (2-5-4-6) [1][3][6] 2-5-5 Orthogonality: The differential equation for particular may be written (omitting explicit dependence on x): (2-5-5-1) Multiplying by (R/Q)fm yields: (2-5-5-2) And reversing the subscripts yields: (2-5-5-3) Subtracting and integrating: ] + (2-5-5-4) But it can be seen that: (2-5-5-5) So that : if the polynomials f are such that the term on the left is zero, and then the othogonality relationship will hold. for m [6][8]

CHAPTER THREE The Classical orthogonal polynomials The class of polynomials arising from the differential equation described in chapter two have many important applications in such areas as mathematical physics, interpolation theory, the theory of random matrices, computer approximations and many others. All of these polynomial sequences are equivalent under scaling and or shifting of the domain, and standardizing of the polynomials, to more restricted classes. Those restricted classes are the "classical orthogonal polynomials".

3-1 Jacobi polynomials: The Jacobi-like polynomials, once the have their domain shifted and scaled so that the interval of orthogonality is [-1,1], still have two parameters to be determined. They are α and β in the Jacobi  ,  2 polynomials, written Pn . We have Q(x)= 1-x and L(x)         2x . Both α and β are required to be greater than - 1.(this puts the root of L inside the interval of orthogonality). When α and β are not equal, these polynomials are not symmetrical about x= 0. The differential equation: 1 x2 y        2xy  y  0with  nn 1    (3-1-1) is Jacobi's equation. Jacobi polynomials are obtained from hypergeometric series in cases where the series is in fact finite :

 ,   1n  1 z  Pn z  2 F1 n,1    n; 1, , (3-1-2) n!  2  where (α+1)n is Pochhammer's symbol (for rising factorial), and thus have the explicit expression : n m  ,    n 1 n      n  m 1 z 1 Pn z      (3-1-3) n!    n 1 m0 m   m 1  2 

 ,  n   from which the terminal value follows : Pn 1    . Here for n   z  z 1 integer n,    , and z is the usual Gamma function, n n 1z  n 1 1  z  which has the property  0,n  1,2,.....Thus   0 for n  0 . [1][3] n 1 n

They satisfy the orthogonality condition : 1    1  2 n  1n   1 1 x 1 x P  ,  xP  ,  xdx  nm,  m n (3-1-4) 1 2n    1 n    1n! for  1and  1 ,  n ,  The polynomials have the symmetry relation : Pn  z  1 Pn  z;

 ,  n n    Thus the other terminal value is Pn 1  1    . n  For real x the Jacobi polynomial can alternatively be written as: ns s  ,  1 x 1  x 1 Pn x  n  !n   !s!n   s!  s!n  s!     . (3-1-5) s  2   2  The sum on s extends over all integer values for which the arguments of the factorials are nonnegative. This form allows the expression of the Wigner d-matrix j dm'm , 0    4  in terms of Jacobi polynomials : 1 mm' mm' 2 j   j  m! j  m!       mm',mm' d m'm     sin  cos  Pjm cos . (3-1-6)  j  m'! j  m'  2   2  The k+h derivative of the explicit expression leads to : d k     n 1 k P ,  z  P k, k  z. (3-1-7) dz k n 2k     n 1 nk  ,  the Jacobi polynomials Pn are solution of : 1 x2 y         2xy  nn    1y  0 (3-1-8) There are several important subclasses of Jacobi polynomials: Gegenbauer Legendre,and two types of Chebyshev.

3-2 Gegenbauer polynomials: When one sets the parameters α and β in the Jacobi polynomials equal to each other, then one obtains the Gegenbauer or ultraspherical polynomials. "The Gegenbauer polynomials are named for Leopold Gegenbauer (1849-1903)". They are written, and defined as:  1  2  n    1 1  2   ,   G   x    P 2 2  . (3-2-1) n  1  n 2   n    2  They are also obtained from hypergeometric series in cases where

16 the series is in fact finite: [2][5] n   2   1 1 z  Gn z  2 F1  n,2  n;  ,  (3-2-2)  2 2 

where n is the falling factorial. We have Q(x)=1+x2 and L(x)= -(2α+1)x. α is required to be greater than -1/2. (Incidentally, the standardization given would make no sense for α=0 and n#0, because it would set the polynomials to zero. 2 In that case, the accepted standardization sets G 0 1  instead of the n n value given in the table in chapter four ). Ignoring the above considerations, the parameter α is closely related   to the derivatives of Gn : 1 d G  1 x  G   x (3-2-3) n 2 dx n   or more generally : G  m x  G  m x (3-2-4) n 2m   m nm Gegenbauer polynomials appear from solving the Gegenbauer differential equation : 1 x2 y  2  3xy  ny  0 . They are closely related to ultraspherical polynomials and can be viewed as an extension of the Legendre polynomials, since they can be obtained  1   n from the generating function :  Cn xt (3-2-5) 2   1 2xt  t  n0 They are orthogonal with respect to the weighting function : 1   Wz  1 z 2  2 (3-2-6)

3-3 Legendre polynomials: The Legendre polynomials are named after Adrien-Marie Legendre. Legendre functions (polynomials) are solutions to Legendre's differential equation : 1 x2 y  2xy  y  0 with   nn 1. (3-3-1)

d  2 d  or 1 x  Pn x  nn 1Pn x  0. (3-3-2) dx  dx  The second form of the differential equation (3-3-1) is :  1 x 2 y  y  0 (3-3-3)

17 and the recurrence relation is :

n 1Pn1 x  2n 1xPn x nPn1 x (3-3-4) A mixed recurrence is : r1 r1 r  Pn1 x  Pn1 x 2n 1Pn x (3-3-5) The ordinary Legendre differential equation is frequently encountered in physics and other technical fields. In particular , it occurs when solving Laplace's equation(and related partial differential equations) in spherical coordinates. The Legendre differential equation may be solved using the standard power series method. The equation has regular points at x = ±1 so in general, a series solution about the origin will only converge for │x│< 1. When n is an integer, the solution Pn(x) that is regular at x=1 is also regular at x= -1, and the series for this solution terminates (i.e. is a polynomial). These solutions for n= 0,1,2,….. (with the normalization Pn(1)= 1) form a polynomial sequence of orthogonal polynomials called the Legendre polynomials . Each Legendre polynomial Pn(x) is an nth- degree polynomial. It may be expressed using Rodrigue's formula : n 1 d 2 n Pn x  x 1  (3-3-6) 2n n! dx n 3-3-1 Associated Legendre polynomials: m The associated Legendre polynomials, denoted Pl x where l and m are integers with 0≤ m ≤l are defined as : m m m 2 2 m Pl x  1 1 x  Pl x. (3-3-1-1) the m in parentheses (to avoid confusion with an exponent) is a parameter. The m in brackets denotes the mth derivative of the Legendre polynomial. These "polynomials are misnamed.. they are not polynomials when m is odd. They have a recurrence relation : m m m l 1 mPl1 x  2l 1xPl x l  mPl1 x (3-3-1-2) m m m For fixed m, the sequence Pm , Pm1 , Pm2 ,...... are orthogonal over [-1,1] m with weight 1. For given m, Pl x are the solutions of:  m2  2   1 x y  2xy    2  y  0 with   ll 1. [1][4][5]  1 x 

3-3-2 The orthogonality property : An important property of Legendre polynomials is that they are orthogonal with respect to the L2 inner product on the interval [-1,1]: 1 2 P x P x dx  mn  m   n   (3-3-2-1) 1 2n 1

( where δmn denotes the kronecker delta, equal to 1 if m=n and to 0 otherwise ). In fact , an alternative derivation of the Legendre polynomials is by carrying out the Gram-Schmidt process on the polynomials {1,x,x2,……} with respect to the inner product . The reason for this orthogonality property is that the Legendre differential equation can be viewed as a Sturm-Liouville problem : d  d  1 x 2  Px  Px (3-3-2-2) dx  dx  where the eigenvalue λ corresponds to n(n+1). As an example , the first few Legendre polynomials are : n Pn(x) 0 1 1 X 1 2 3x 2 1 2 1 3 5x3  3x 2 1 4 35x 4  3x 2  3 8 1 5 63x5  70x3 15x 8 1 6 231x6  315x 4 105x 2  5 16 1 7 429x7  693x5  315x3  35x 16 1 6435x8 12012x6  6930x 4 1260x 2  8     128  35  [1][4] 3-3-3 Additional properties of Legendre polynomials Legendre polynomials are symmetric or antisymmetric, that is k Pk(x) = (-1) Pk(x) (3-3-3-1)

Since the differential equation and the orthogonality property are independent of scaling , the Legendre polynomials ' definitions are "standardized" (some times called "normalization", but the actual norm is not unity) by being scaled so that Pk(1) = 1. kk 1 The derivative at the end point is given by P ' 1  . Legendre k 2 polynomials can be constructed using the three terms recurrence x relations: n 1Pn1  2n 1xPn x nPn1 x (3-3-3-2) x 2 1 d And : P x  xP x P x (3-3-3-3) n dx n n n1 Useful for the integration of Legendre polynomials is : d 2n 1P x  P x P x ] (3-3-3-4) n dx n1 n1 [1][3] 3-3-4 Shifted Legendre polynomials: The shifted Legendre polynomials are defined as :

Pn x  Pn 2x 1 (3-3-4-1) Here the "shifting " function x → 2x-1 (in fact, it is an affine transformation) is choosen such that it bijectively maps the interval

[0,1] to the interval [-1,1], implying that the polynomials Pn x are 1 1 P x P x dx  mn orthogonal on [0,1] :  m   n   (3-3-4-2) 0 2n 1 An explicit expression for the shifted Legendre polynomials is given n n nn  k  k by : Pn x  1    x (3-3-4-3) k0 k k  The analogue of rodrigue's formula for the shifted Legendre n 1 d 2 n polynomials is : Pn x  n! x  x  (3-3-4-4) dx n The first few shifted Legendre polynomials are :

N Pn x 0 1 1 2x-1 2 6x2-6x+1 3 20x3-30x2+12x-1 [1][4] 3-3-5 Legendre polynomials of fractional order : Legendre polynomials of fractional order exist and follow from

insertion of fractional derivatives as defined by fractional calculus and non-integer factorials (defined by the Gamma function) into the Rodrigue's formula. The exponents of course become fractional exponents which represent roots.[1]

3-4 Chebyshev polynomials : The Chebyshev polynomials , named after "Pafnuty Chebyshev", are a sequence of orthogonal polynomials which are related to "de Moivre's" formula and which are easily defined recursively , like Fibonacci or Lucas numbers. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted by Tn and Chebyshev polynomials of the second kind which are denoted by Un. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebyshef or Tshebysheff. The Chebyshev polynomials Tn or Un are polynomialsof degree n and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence. The Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials of the first kind , which are also called Chebyshev nodes ,are used as nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is closed to the polynomial best approximation to a continuous function under the maximum norm. This approximation leads directly to the method of Clenshaw-Curtis quadrature. In the study of differential equations they arise as the solution to the Chebyshev differential equation : 1 x2 y  xy  n2 y  0 with   n2 and 1 x2 y  3xy  nn  2y  0, with   nn  2. For the polynomials of the first kind , respectively. These equations are special cases of the Sturm-Liouville differential equation. [1][8]

3-4-1 Definitions : The Chebyshev polynomials of the first kind are defined by the recurrence relation:

T0(x)=1, T1(x)=x, Tn+1(x)=2xTn(x)-Tn-1(x). (3-4-1-1) One example of a generating function for Tn is :  1 tx T x t n   n   2 (3-4-1-2) n0 1 2tx  t The Chebyshev polynomials of the second kind are defined by the recurrence relation : U0(x)=1, U1(x)=2x, Un+1(x)=2xUn(x)-Un-1(x). (3-4-1-3) One example of the generating function for Un is:  1 U x t n   n   2 (3-4-1-4) n0 1 2tx  t [1][4] 3-4-1-1 Trigonometric definition: The Chebyshev polynomials of the first kind can be defined as the trigonometric identity : Tn(x) = cos(n arc cosx)= cosh(n arc coshx) (3-4-1-1-1) Whence : Tn(cos(v)) = cos (nv) for n= 0,1,2,3,…., while the sinn 1v polynomials of the second kind satisfy : U cosv  n sin v which is structurally quite similar to the Drichlet kernel. That cos(nv) is an nth-degree polynomial in cos (x) can be seen by observing that cos(nx) is the real part of the one side of the de Moivre's formula, and the real part of the other side is a polynomial in cos(x) and sin(x), in which all the powers of sin(x) are even and thus replaceable via the identity: cos2(x)+sin2(x)=1. This identity is extremely useful in conjunction with the recursive generating formula in as much as it enables one to calculate the cosine of any integral multiple of an angle solety in terms of the cosine of the base angle. Evaluating the first two Chebyshev polynomials: T0(x) =cos(0x) =1 and T1(cos(x)) = cos, one can straight forwardly determine that: cos2v  2cos vcos v  cos(0v)  2cos 2 v 1 (3-4-1-1-2) cos(3v)  2cos vcos(2v)  cos v  4cos 3 v  3cos v and so forth. To trivially check whether the results seen reasonable, sum the coefficients on both side of the equal sign (that is, setting equal to zero, for which the cosine is unity), and one sees that 1=2-1 in the former expression and 1=4-3 in the

latter. An immediate corollary is the composition identity (or the "nexting property"): Tn(Tm(x)) = Tn.m(x) (3-4-1-1-3) [1] 3-4-1-2 Pell equation definition : The Chebyshev polynomials can also be defined as the 2 2 2 solutions to the Pell equation : Ti  x 1Ui1 1 (3-4-1-2-1) In a ring R[x]. Thus , they can be generated by the standard technic for Pell equations of taking powers of a fundamental solution : i 2 2 Ti Ui1 x 1  x  x 1 (3-4-1-2-2) [1][5] 3-4-1-3Relation between Chebyshev polynomials of thefirstkind and the second kind : The Chebyshev polynomials of the first and second kind are closely related by the following equations: d T x  nU x, n  1,2,...... dx n n1 1 T x  U xU x.T x  xT x 1 x 2 U x (3-4-1-3-1) n 2 n n2 n1 n n1

Tn x  U n x xU n1 x The recurrence relationship of the derivative of Chebyshev polynomials can be derived from these relations : 1 d 1 d 2T x  T x T x, n  1,2,...... (3-4-1-3-2) n n 1 dx n1 n 1 dx n1 This relationship is used in the Chebyshev spectral method of solving differential equations. Equivalently, the two sequences can also be defined from a pair of mutual recurrence equations : T0(x) = 1, U-1(x) =0 2 Tn+1(x)=xTn(x)-(1-x )Un-1(x),Un(x)=xUn-1(x)+Tn(x) (3-4-1-3-3) These can be derived from the trigonometric formulae; for example if x=cosv, then :

Tn1 x  Tn1 cosv  cosn 1v  cosnvcosv sinnvsinv 2  Tn cosvcosvU n1 cosvsin v (3-4-1-3-4) 2  xTn x 1 x U n1 x Note that both these equatios and the trigonometric equations take a simpler form if we, like some works, follow the alternate condition

23 of denoting our Un (the polynomial of degree n) with Un+1 instead.[1][7]

3-4-2 Explicit formulas : Different approaches to defining Chebyshev polynomials lead to different explicit formulas such as :

cos(narccos(x)), x [1,1]  Tn x  cosh(narccos h(x)), x  1  n (1) cosh(narccos h(x)), x  1

n  k   n    x 2 1 x n2k 2 n 2 n       (x  x 1)  (x  x 1)  2  2k T (x)      n 2   k 0     n  n1 n1    2 2  2  x  x 1  x  x 1 n 1  2 n2k  (3-4-2-1) U n x    x 1x 2    2 x 1 k1 2k 1    n1   2  x 1  Tnx  1 n x 1 1   k   k1  2sin 2      n   [1][6] 3-4-2 Properties: 3-4-3-1 Orthogonality: Both the Tn and the Un form a sequence of orthogonal polynomials . The polynomials of the first kind are orthogonal with respect to the weight 1 on the interval, i.e we have : 1 x 2  0 : n  m 1 dx  Tn xTm x   : n  m  0 (3-4-3-1-1)  2 1 1 x   : n  m  0  2 This can be proved by letting x=cos(v) and using the identity Tn(cos(v)) =cos(nv). Similarly, the polynomials of the second kind are orthogonal with respect to the weight 1 x 2 on the interval

1 0 : n  m, 2  [-1,1], i.e we have : U n xU m x 1 x dx   (3-4-3-1-2)  : n  m. 1  2 ( Note that the weight 1 x 2 is, to within a normalizing constant, the density of the wigner semicircle distribution). [1][8]

3-4-3-2 Minimal ∞-norm : For any given n ≥ 1, among the polynomials of degree n with 1 leading coefficient 1, f (x)  T x is the one of which the 2n1 n maximal absolute value on the interval [-1,1] is minimal. The 1 maximal absolute value is : , and f x reaches this maximum 2n1 k exactly n+1 times at x  cos for 0  k  n .[1] n 3-4-3-3 Differentiation and Integration : The derivatives of the polynomials can be less than straight forward. By differentiating the polynomials in their trigonometric forms , it's easy to show that :

dTn dU n n 1Tn1  xU n  nU n1 ,  2 dx dx x 1 (3-4-3-3-1) d 2T nT  xU n 1T U n  n n n1  n n n dx 2 x 2 1 x 2 1 The last two formulas can be numerically troublesome due to the division by zero (0/0 indeterminate form, specifically) at x= -1. It can be shown that : d 2T n4  n2 d 2T n4  n2 n  , n  1n (3-4-3-3-2) dx 2 x1 3 dx 2 x1 3 Indeed, the following more general formula holds : d pT p1 n2  k 2 n  1 n p p x1    (3-4-3-3-3) dx k0 2k 1 This latter result is of great use in the numerical solution of eigenvalue problems. Concerning integration , the first derivative of T the Tn implies that : U dx  n1 (3-4-3-3-4)  n n 1 And the recurrence relation for the first kind polynomials involving derivatives establishes that : [1][7]

1  Tn1 Tn1  nTn1 xTn Tn dx       (3-4-3-3-5)  2  n 1 n 1 n2 1 n 1

3-4-3-4 Roots and extrema: A Chebyshev polynomials of either kind with degree n has n different simple roots , called Chebyshev roots , in the interval [-1,1]. The roots are some times called Chebyshev nodes because they are used as nodes in polynomial interpolation. Using the trigonometric definition and the fact  that cos( 2k 1)  0 one can easily prove that the roots of Tn 2  2k 1 are : xk  cos , k=1,2,…………,n (3-4-3-4-1)  2 n  Similarly, the roots of Un are :  k  xk  cos , k=1,…,n (3-4-3-4-2)  n 1 One unique property of the Chebyshev polynomials of the first kind is that on the interval [-1,1] all the extrema have values that are either -1 or 1. Thus these polynomials have only two finite critical values , the defining property of Shabat polynomials. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given n n by : Tn(1)=1, Tn(-1)=(-1) , Un(1)=n+1, Un(-1)=(n+1)(-1) . [1][8]

3-4-3-5 Other properties : The Chebyshev polynomials are a special case of the ultra spherical or Gegenbauer polynomials , which themselves are the special case of the Jacobi polynomials. For every nonnegative integer n, Tn(x) and Un(x) are both polynomials of degree n. They are even or odd functions of x as n is even or odd, so when written as polynomials of x ,it only has even or odd degree terms respectively. The leading coefficient of Tn is 2n-1 if 1≤n, but 1 if 0=n. Tn are a special case of Lissajous curves with frequency ratio equal to n. [1]

Examples: The first few Chebyshev polynomials of the first kind are : T0(x) = 1 T1(x) = x 2 T2(x) =2x -1 3 T3(x) = 4x -3x 4 2 T4(x) = 8x -8x +1 5 3 T5(x) = 16x -20x +5 6 4 2 T6(x) = 32x -48x +18x -1 7 5 3 T7(x) = 64x -112x +56x -7x The first few Chebyshev polynomials of the second kind are : U0(x) = 1 U1(x) = 2x 2 U2(x) = 4x -1 3 U3(x) = 8x -4x 4 2 U4(x) = 16x -12x +1 5 3 U5(x) = 32x -32x +6x 6 4 2 U6(x) = 64x -80x +24x -1 7 5 3 U7(x) = 128x -192x +80x -8x [1][4]

3-4-3 As a basis set: In the appropriate Sabolev space , the set of Chebyshev polynomials form a complete basis set, so that a function in the same space can, on [-1,1] be expressed via the expression :  f (x)  anTn x (3-4-5-1) n0 Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which implies that the coefficients an can be determined easily through the application of an inner product. This sum is called a Chebyshev series or a Chebyshev expansion. Since a Chebyshev series isrelated to a Fourier cosine series through a change of variables , all of the theorems , identities etc that apply to Fourier series have a Chebyshev counterpart. These attributes include :

 The Chebyshev polynomials form a complete orthogonal system.

 The Chebyshev series converges to f(x) if the function is piecewise smooth and continuous . The smoothness requirement can be relaxed in most cases as long as there are a finite number of discontinuities in f(x) and its derivatives.  At a discontinuity, the series will converge to the average of the right and left limits. The abundance of the theorems and identities inherited from Fourier series make the Chebyshev polynomials important tools in numerical analysis ;for example they are the most popular general purpose basis functions used in the spectral method, often in favor of trigonometric series due to generally faster convergence for continuous functions.[1][7]

3-4-3-2 Partial sum:  The partial sum of f (x)  anTn x are very useful in the n1 approximation of various functions and in the solution of differential equations . Two common methods for determining the coefficients an are the used of the inner product as Galerkin's method and through the use of collocation which is related to interpolation. As an interpolant the N coefficients of the (N-1)th partial sum are usually obtained on the Chebyshev-Gauss-Lobatto points (or Lobatto grid), which results in minimum error and avoid Runge's phenomenon associated with a uniform grid. This collection of points corresponds to the extrema of the highest order polynomial in the sum, plus the endpoints and is given by :  i  xi  cos  ; i=0,1,2,……..,N-1 (3-4-5-1-1)  N 1 [1][5] 3-4-3-3 Polynomial in Chebyshev form: An arbitrary polynomial of degree N can be written in terms of the Chebyshev polynomials of the first kind .Such a polynomial p(x) N is of the form : P(x)  anTn x . Polynomials in Chebyshev form can n0

be evaluated using the Clenshaw algorithm.

3-4-4 Spread polynomials : The spread polynomials are in a sense equivalent to the Chebyshev polynomials of the first kind , but enable one to avoid square roots and conventional trigonometric functions in certain contexts, notably in rational trigonometry.

3-5 Laguerre polynomials : The Laguerre polynomials , named after Edmond Laguerre (1834-1886), are the canonical solutions of Laguerre's equation : xy  1 xy  ny  0, with  n (3-5-1) which is a second order linear differential equation. This equation has non singular solutions only if n is a non-negative integer. These polynomials usually denoted L0, L1, ….., are a polynomial sequence which may be defined by the Rodrigue's formula: e x d n L x  ex x n  (3-5-2) n n! dx n They are orthogonal to each other with respect to the inner  product given by :  f , g   f xgxex dx (3-5-3) 0 The sequence of Laguerre polynomials is a Sheffer sequence. The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the SchrỜdinger equation for one electron atom.[1][4][5]

These are the first few Laguerre polynomials : N Ln(x) 0 1 1 -x+1 1 2 x 2  4x  2 2 1 3  x3  9x 2 18x  6 6 1 4 x 4 16x 3  72x 2  96x  24 24 1  x5  25x 4  200x3  600x 2  600x 5     120 120  1  x6  36x5  450x 4  2400x3  5400x 2  6     720  4320x  720 

We can also define the Laguerre polynomials recursively, defining the first two polynomials as : L0(x) =1, L1(x) =1-x and then using the recurrence relation for any K ≥ 1: 1 L x  2k 1 xL x kL x [1][4] k1 k 1 k k1

3-5-1 Generalized Laguerre polynomials : The orthogonality property stated above is equivalent to saying that if X is an exponentially distributed random variable with probability ex if x  0, density function : f (x)   (3-5-1-1) 0 if x  0. Then E[Ln(x)Lm(x)] = 0 whenever n≠m. The exponential distribution is not the only gamma distribution. A polynomial sequence orthogonal with respect to the gamma distribution whose probability density function is for α > -1, x ex / 1  , x  0, f (x)   0 , x  0 (3-5-1-2) is given by the defining Rodrigue's equation for the generalized x  e x d n Laguerre polynomials : L  x  exen  (3-5-1-3) n n! dx n [1][4]

These are also sometimes called associated Laguerre polynomials. The simple Laguerre polynomials are recovered from the 0 generalized polynomials by setting α = 0 : Ln x  Ln x.

3-5-1-1 Explicit examples and properties of generalize Laguerre polynomials : The generalized Laguerre polynomials of degree n is (as follows from applying Leibnitz's theorem for differentiation of a product to n i   i n   x the defining Rodrigue's formula) : Ln x  1   (3-5-1-1-1) i0 n  i  i! n  The coefficient of the leading term is 1 . n!  The constant term, which is the value at the origin, is    n   n Ln 0     . n   1    Ln has n real,strictly positive roots which are all in the open interval 0,n   n 1 n  .  Derived from the root test and convergent series below is n   Limsup Ln x1 n  The first few generalized Laguerre polynomials are : 2       x   2 1 L0 x  1, L1 x  x   1, L2 x     2x  , 2 2 x3   3x 2   2  3x  1  2  3 L  x      3 6 2 2 6  The explicit formula gives rise to compute Laguerre's polynomial using Horner's method.

3-5-1-2 Recurrence relations : Laguerre polynomials satisfy the recurrence relations : n   1     Ln x  y   Li xLni y (3-5-1-2-1) i0 n  1   in particular Ln x   Li x, (3-5-1-2-2) i0 n       n  i 1   and Ln x   Li x (3-5-1-2-3) i0 n  i  [4][5] 31

n      n1     n    x n 1 i  1 moreover Ln x      Li x (3-5-1-2-4) n  n i0 1   i     1  1 They can be used to derive Ln x  Ln x Ln1 x (3-5-1-2-5)     1 And nLn x  n Ln1 x xLn1 x (3-5-1-2-6) Combined , they give this additional , popular recurrence relation : 1 L  x  2n 1  xL  x n  L  x (3-5-1-2-7) n1 n 1 n n1 A some what curious identity, valid for integer i and n, is :  xi  xn Lin x  Lni x (3-5-1-2-8) i! n n! i

3-5-1-3 Derivatives of generalized Laguerre polynomials: Differentiating the power series representation of a generalized Laguerre polynomial k times leads to : d k L  x  1k L k  x (3-5-1-3-1) dx k n n1 moreover, this following equation holds : 1 d k n   x L  x   x k L k  x k n     n   (3-5-1-3-2) k! dx k  which generalizes with Cauchy's formula to :     n x t  x  t 1 L   x    L  tdt n     n (3-5-1-3-3)   0 x The generalized associated Laguerre polynomials obey the differential equation :

      xLn x  1 xLn x nLn x  0 (3-5-1-3-4) [5][6] 3-5-1-4 Orthogonality: The associated Laguerre polynomials are orthogonal over [0,∞) with respect to measure with weighting function x ex :  n  1 x ex L  x L  x dx  nm  n   m   (3-5-1-4-1) 0 n!

The associated , symmetric kernel polynomial has the representations : 32

n             1 Li xLi y 1 Ln xLn1 y Ln1 x K n x, y     1 i0   i  1 x  y n        i  n 1 n  1 n xi L i xL i1 y   ni ni (3-5-1-4-2)  1 i0 i!   nn    n i  y 1 L 1 xL  y recursively: K   x, y  K  1 x, y n n (3-5-1-4-3) n  1 n1  1   n   n    y   2 Moreover, y e kn , y   y  , in the associated L [0, ∞)-space. The following integral is needed in the quantum mechanical treatment of the hydrogen atom,  n  ! x 1ex [L  ]2 dx  2n  1  n   (3-5-1-4-4) 0 n!

[1][7]

3-5-1-5 Series expansions : n     Let a function have the (formal) series expansion f x   fi Li x, i0  L  x x ex f    i   f (x)dx then i  (3-5-1-5-1) 0 i    1   i  The series converges in the associated Hilbert space L2[ 0, ∞], if   x 2 x e 2 i     2 and only if f 2  f (x) dx    f   (3-5-1-5-2) L    i 0  1 i0 i  A related series expansion is : L  x i i  ex  f x 1   i  in   f       i 1    n (3-5-1-5-3) i0 1   n0 n L  i  ex  L  x 1  i  in   in particular n     i 1   (3-5-1-5-4) i0 1  n  x  1 n n   L      ni  L  x which follows from : n   n    i   (3-5-1-5-5) 1   1   i0 n  i  Secondly,

   i x f (x)     Li x in      n       1    f n (3-5-1-5-6)  1   i0   i n0 i  nn    i  A consequence derived from :       x Ln x        n in   Li x    1   (3-5-1-5-7)  1  n  i0 i  n   i   i  for Re(α +2β) > -1. [1][8]

3-5-2 As Contour integral : The polynomials may be expressed in terms of a contour integral : xt  1t   1 e Ln x  dt (3-5-2-1) 2i  1 t 1t n1 where the contour circles the origin once in a counterclockwise direction.

3-5-3 Relation to Hermite polynomials : The generalized Laguerre polynomials are related to the Hermite  1   1       n 2n  2  2 n 2n1  2  2 polynomials : H 2n x  1 2 n!Ln x  and H 2n1 x  1 2 n!xLn x , where the Hn(x) are the Hermite polynomials based on the weighting function exp(-x2), the so-called "physicist's version". Because of this , the generalized Laguerre polynomials arisein the treatment of the quantum harmonic oscillator.

3-5-4 Relation to hypergeometric functions : The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric function, as:  1n L  x  n  M  n, 1, x  F  n, 1, x. n n! 2 1

3-6 Hermite polynomials: The Hermite polynomials are named in honor of Charles Hermite. They are a classical orthogonal polynomial sequence that arise in probability, such as the Edge worth series ; in combinatory,

34 as example of an Apell sequence, obeying the umbral calculus ; and in physics , where they give rise to the eigenstates of the quantum harmonic oscillator (Schrödinger's equation). They are also eigenfunctions (with eigenvalue (-i)n)) of the continuous Fourier transform. The Hermite polynomials are solutions of the differential equation : Y''-2xy'+λy = 0, with λ = 2n . And the second form of this differential equation is : x2 2 2 2     (ex y)  ex y  0 ; and the third form is: ex / 2 y   1 x 2 e 2 y  0.        Many authors, particularly probabilists,use an alternative definition x2  of the Hermite polynomials, with a weigth function of e 2 instead 2 of ex .The Hermite polynomials are defined either by : 2 2 x n x d  H x  1n e 2 e 2 (3-6-1) n dx n which is the "probabilist's Hermite polynomials" , or sometimes by : n 2 d 2 H x  1n e x ex (3-6-2) n dx n which is the " physicist's Hermite polynomials ". These two definitions are not exactly equivalent , either is a trivial rescaling of n phys 2 prob the other , to wit : H n x  2 H n  2x (3-6-3) These are Hermite polynomial sequences of different variances; below we usually follow the first convention. That convention is x2 1  often preferred by probabilists because : e 2 is the probability 2 density function for the normal distribution with expected value 0 and standard deviation 1. The first few probabilists' Hermite polynomials are : H0(x) = 1 H1(x) = x 2 H2(x) = x -1 3 H3(x) = x -3x 4 2 H4(x) = x -6x +3 5 3 H5(x) = x -10x +15x

6 4 2 H6(x) = x -15x +45x -15

7 5 3 H7(x) = x -21x +105x -105x And the first few physicists' Hermite polynomials are : H0(x) = 1 H1(x) = 2x 2 H2(x) = 4x -2 3 H3(x) = 8x -12x 4 2 H4(x) = 16x -48x +12 5 3 H5(x) = 32x -160x +120x 6 4 2 H6(x) = 64x -480x +720x -120 7 5 3 H7(x) = 128x -1344x +3360x -1680x [1][2][3][4]

3-6-1 Properties: Hn is a polynomial of degree n . The probabilists' version has leading coefficient 1 , while the physicists' version has leading coefficient 2n. [1][8]

3-6-1-1 Orthogonality: Hn(x) is an nth-degree polynomial for n= 0,1,2,3,….. These x2  polynomials are orthogonal with respect to the weight function e 2 2 (probabilistic) or ex (physicist). i.e we have :  x2  H x H x e 2 dx  n! 2  n   m   nm (probabilistic) (3-6-1-1-1)   2 H x H x ex dx  n!2n  or  n   m   nm (physicist) (3-6-1-1-2) 

where  ij is the kronecker delta , which equal unity when n=m and zero otherwise. The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function . They form an orthogonal basis of the Hilbert space of function  x2  satisfying :  f x 2e 2 dx   (3-6-1-1-3)  in which the inner product is given by the integral including a  x2  gaussian function :  f , g   f (x)g(x)e 2 dx (3-6-1-1-4)  [1][7]

3-6-1-5 Hermite's differential equation : The nth Hermite polynomial satisfies Hermite's differential    H n x xH n x nH n x  0 ( probabilist) equation :  (3-6-1-1-5)   H n x 2xH n x 2nH n x  0 ( physicist) [1][4] 3-6-1-6 Recurssion relation : The sequence of Hermite polynomials also satisfies the recursion H (x)  xH (x)  H ' (x) ( probabilist) relation: n1 n n  (3-6-1-3-1) '  H n1 (x)  2xH n (x)  H n x ( physicist)  The Hermite polynomials constitute an Apell sequence, i.e they are a polynomial sequence satisfying the identity : ' H n x  nH n1 x ( probabilist) ' (3-6-1-3-2) H n x  2nH n1 x ( physicist) n n k  H n x  y   x H nk y ( probabilist) k0 k   or equivalently ,  (3-6-1-3-3) n n H x  y   H x 2y nk  ( physicist)  n     k     k0 k   It follows that the Hermite polynomials also satisfy the recurrence H x  xH x nH x ( probabilist) relation : n1 n n1 (3-6-1-3-4) H n1 x  2xH n x 2nH n1 x ( physicist) These last relations, together with the initial polynomials H0(x) and H1(x), can be used in practice to compute the polynomial quickly. [2][3] 3-6-1-7 Generating function: The Hermite polynomials are given by the exponential generating  t 2   t n exp xt    H x ( probabilist)  2   n n! function :   n0 (3-6-1-4-1)  n 2 t exp(2xt  t )   H n x ( physicist) n0 n!

3-6-1-8 Expected Value : If X is a random variable with a normal distribution with standard deviation 1and expected value μ then n E(Hn(X)) = μ (probabilistic) (3-6-1-5-1)

3-7 Relation to other functions : 3-7-1 Laguerre polynomials : The Hermite polynomials can be expressed as a special case of the Laguerre polynomials :  1      H x   4n n!L 2  x 2  ( probabilistic) 2n n (3-7-1-1)  1     n  2  2  H 2n1 x  2 4 n!xLn x  ( physicist) 

3-7-2 Relation to confluent hypergeometric functions: The Hermite polynomials can be expressed as a special case of the parabolic cylinder functions:

n 1 n 3 2  H n x  2 U , ; x  (physicist) (3-7-2-1)  2 2  where U(a,b,z) is Whittaker's confluent hypergeometric function.

n 2n!  1 2  H 2n (x)  1 1 F1  n, ; x  probabilist Similarly, n!  2  (3-7-2-2) n 2n 1!  3 2  H 2n1 x  1 2x1F1  n, ; x  ( physicist) n!  2  where 1F1(a,b;z)=M(a,b;z) is Kummer's confluent hypergeometric function .[4][5][7]

3-8 Differential operator representation: The probabilist's Hermite polynomials satisfy the identity : D2  2 2 H n x  e x (3-8-1) where D represents differentiation with respect to x, and the exponential is interpreted by expanding it as a power series . There are no delicate questions of convergence of this series when it operates on polynomials, since all but finitely many terms vanish. Since the power series coefficients of the exponential are well known , and higher order derivatives of the monomial xn can be written down explicitly, this differential operator representation gives rise to a concrete formula for the coefficients of Hn that can be used to quickly compute these polynomials.

2 Since the formal expression for the Weie e D ,rstrass transform W is n  x  n we see that the Weierstrass transform of  2 H n   is x .  2  Essentially the Weierstrass transform thus turns a series of Hermite polynomials into a corresponding Maclaurin series . The existence of some formal power series g(D), with non zero n constant coefficient, such that Hn(x) = g(D)x , is another equivalent to the statement that these polynomials form an Apell sequence. Since they are an Apell sequence they are a fortiori a Sheffer sequence. [1][4]

3-9 Contour integral representation : The Hermite polynomials have a representation in terms of contour t 2 tx  e 2  n! dt ( probabilist) H n x  n1  2i  t  integral as :  (3-9-1)

2  n! e2txt  H n x  n1 dt ( physicist)  2i  t  With the contour encircling the origin. [1][8]

3-10 Generalization : The probabilists' Hermite polynomials defined above are orthogonal with respect to the standard normal probability x2 1  distribution, whose density function is : e 2 which has expected 2 value 0 and variance 1. One may speak of Hermite polynomials   H n x of variance α , where α is any positive number. These are orthogonal with respect to the normal probability distribution whose x2 1   2  density function is 2  2 e . They are given by : n D2     2 1 x  2 n H n x   H n    e x (3-10-1)     1     2  In particular , the physicist's Hermite polynomials are H n x.

n     k If H n x  hn,k x  (3-10-2) k0 Then the polynomial sequence whose nth term is n         H n  H k x  hn,k H k x (3-10-3) k0 Is the umbral composition of the two polynomial sequences, and it        can be shown to satisfy the identities : H n  H x  H n x (3-10-4) n    n     And : H n x  y   H k xH nk y (3-10-5) k0 k  The last identity (3-10-5) is expressed by saying that this parameterized family of polynomial sequence is a cross-sequence. [1][5][6]

3-10-1 Negative variance : Since polynomial sequences form a group under the operation of   umbral composition , one may denote by H n x the sequence that is inverse to the one similarly denoted but without the minus sign, and thus speak of Hermite polynomials of negative variance . For α   >0, the coefficients of H n x are just the absolute values of the   corresponding coefficients of H n x. These arise as moments of normal distribution with expected value μ and variance σ2 is : n  2  EX  H n  (3-10-1-1) where X is a random variable with the specified normal distribution. A special case of the cross-sequence identity then says that n n     0 n  H k xH nk y  H n x  y  x  y . (3-10-1-2) k0 k  [1][7]

CHAPTER FOUR

Constructing Orthogonal Polynomials by Using Moment :

4-1 Significance of moment : The concept of moment in mathematics evolved from the concept of moment in physics . The nth moment of a real-valued function     x  c n f (x)dx f(x) of real variable about a value c is n    .  It is possible to define moments for random variables in a more general fashion than moments for real values. The moments about zero are usually referred to simply as the moments of a function . The function will be a probability density function. The nth moment (about zero) of a probability density function f(x) is the expected value of Xn . The moment about its mean μ are called central moments. These describe the shape of the function. If f is a probability density function , then the value integral above is called the nth moment of the probability distribution function. More generally, if F is a cumulative probability distribution function of any probability distribution, which may not have a density function , then the nth moment of the probability distribution is given by the Rieman-Stieltjes integral    E(X n )  x n dF(x) n  . Where X is a random variable that has his  distribution and E the expectation operator . When  EX n   x n dF(x)  , then the moment is said not to exist. If the nth  moment about any point exists, so does (n-1)th moment, and all lower-order moments, about every point. The first moment about zero , if it exists, is the expectation of X, i.e. the mean of the probability distribution of X, designated μ . In higher orders, the central moments are more interesting than the moments about zero . The nth central moment of the probability n distribution of a random variable X is : μn = E[(x- μ ) ]. The first central moment is thus zero. The second central moment is the [1]

variance, the positive square root of which is the standard deviation σ .

4-1-1 Standardized moment : In probability theory and statistic, the kth standardized moment

 k of the probability distribution is where μk is the kth moment  k about the mean and σ is the standard deviation.

4-2 Constructing Orthogonal Polynomials :   x n d Let n  be the moments of a measure μ. Then the  polynomial sequence defined by :

0 1 2 ...... n     ......    1 2 3 n1 

2 3 4 ...... n2  P x  det  n   ......      ......    n1 n n1 2n1   2 n  1 x x ...... x  is a sequence of orthogonal polynomials with respect to the k measure μ. To see this , consider the inner product of Pn(x) with x for any k < n. We will see that the value of this inner product is zero. [1][7]

0 1  2 ......  n     ......    1 2 3 n1     ......   x k P x d  x k det 2 3 4 n2 d  n        ......      ......    n1 n n1 2n1  2 n 1 x x ...... x 

0 1 2 ...... n     ......    1 2 3 n1     ......   det 2 3 4 n2 d     ......       ......    n1 n n1 2n1  k k 1 k 2 k n x x x ...... x 

    0 1 2 ...... n     ......    1 2 3 n1  2 3 4 ...... n2     det  ......  

n1  n  n1...... 2n1     k k 1 k2 k n   x d  x d  x d......  x d           ......   0 1 2 n    ......    1 2 3 n1     ......    det 2 3 4 n2   0 if K  n,         ......    n1 n n1 2n1   k k 1 k 2 ......  kn  Since the matrix has two identical rows. ((The entry-by-entry integration merely says the integral of a linear combination of functions is the same linear combination because only one row contains non-scalar entries.)) k Thus Pn(x) is orthogonal to x for all k

4-3 Table of classical orthogonal polynomials: Name, and ndChebyshev, Chebyshev LegendreHermite,, conventional Tn (second kind) Pn Hn symbol Un

Limit of - of1,1 -1,1 -1,1 -∞, ∞ orthogonality 1 1 x2 Weight,W(x) 2 2 1 e 1 x 2 1 x 2 n StandardizationTn (1)=1 Un(1)=n+1 Pn(1)=1 Leadterm=2 Square of  : n  0  : n  0 2 2n n!      2n 1 Norm, hn : n  0 : n  0  2  2 Leading term,2 n1 2n 2n! 2n 2n n! 2 kn   Second term, 0 0 0 0 K'n Q 1-x2 1-x2 1-x2 1 L -x -3x -2x -2x L(x) 1 1 2 x2  2 2 1-x e R(x)  e Q(x) 1 x 2 1 x 2 Constant in n2 n(n+2) n(n+1) 2n Diff.equation, λn n  3  Constant in n   (-1) n n 2 n Rodrigues'  2    2 n! 2(2)n n 1  Formula, en   Recurrence 2 2 2n 1 2 n 1 Relation,an Recurrence 0 0 0 0 Relation, bn Recurrence 1 1 n 2n n 1 Relation, cn

[1][4][7]

Name, and Associated Laguerre, Laguerre, Ln   Conventional Ln Symbol Limits of orthogo- 0, ∞ 0, ∞ nality Weight, W(x) x ex e x n n Standardization Lead term= 1 Lead term= 1 n! n! n  1 Square of norm,hn 1 n! n n Leading term, kn 1 1 n! n! n1 n1 Second term, k'n 1 n   1 n (n 1)! n 1! Q X X L a +1-x 1-x L(x) -x  x 1ex Xe R(x)  e Q(x) Constant in diff.equa- n n Tion, λn Constant in Rodrigue n! n! S' formula, en

Recurrence relation, 1 n 1 an Recurrence relation, 2n 1  2n 1 n 1 n 1 bn Recurrence relation, n   n n 1 n 1 cn

[1][4][7]

   ,  Name,and conven Gegenbauer, Cn Jacobi , Pn Tional symbol Limits of orthogona- -1,1 -1,1 lity 1 -x Weight, W(x) 2   e 1 x  2 n  2  n 1  Standardization C   1  if  0 P ,  1  n n!2  n n!1   212 n  2  2  1  n  1  n   1 Square of norm, hn     n!n    2 n!2n    1n    1 Leading term,k  1  2n 1    n 2n  2     n!2n n 1     2   1  n!2n 2 n      2     2n     Second term, k'n 0 n 1!2n n 1    Q 1-x2 1-x2 L -(2α+1)x β-α-(α+β+2)x Lx 1  1  1  2   1 x 1 x Rx  e Qx 1 x  2 Constant in diff. n(n+2α) n(n+1+α+β) equation, λn n n  1  Constant in Rodri-  2 n!2 n     (-2) n! 2 gue's formula,en    1  n  2     2  Recurrence relation , 2n    2n 1   2n  2     n 1 2n 1n 1    an 2 2 Recurrence relation, 0    2n 1    bn 2n 12n    n 1    Recurrence relation, n  2 1 n  2  2 1 n 1 n 1n 1    cn

[1][4][7]

CHAPTER FIVE Some applications: The class of polynomials arising from differential equation described above have many important applications in such areas as mathematics, physics, engineering, and so many others. 5-1 Mathematical applications:- 5-1-1 Approximation theory: In approximation theory ,for example, the function f(x) can be approximated with a Cheyshev series, basing on the following Lemma : If j, k < n and at last j ≠ 0 or k ≠ 0 we have n1 n T j xr Tk xr    ij where  ij    j  k,n   j  k,n (5-1-1) r0 2 1 Now ,if f (x)  C  C T (x)  C T (x)  ......  C T (x)  R(x) to determine 2 0 1 1 2 2 n1 n1 the coefficients Ck , let x=xr . then 1 f (x )  C  C T x  C T (x )  ......  C T (x )  R(x ). Multiplying r 2 0 1 1 r 2 2 r n1 n1 r r this by Tk(xr) , we obtain : 1 f (x )T (x )  C T (x )  C T (x )T (x )  ......  C T (x )T (x ) (5-1-2) r k k 2 0 k r 1 1 r k k n1 n1 r k r Taking the sum of both sides we get: n1 1 n1 n1 n1  f (xr )Tk (xr )  C0 Tk (xr )  C1 T1 (xr )Tk (xr )  ......  Cn1 Tn1 (xr )Tk (xr ) r0 2 r0 r0 r0 but from (5-1-1), 0 if j  k n1  n T j (xr )Tk (xr )   if j  k  0 r0 2 n if j  k  0 2r 1 where r =0,1,2,……., n-1 and x  cos  .Then r 2n n1 1 n1 n1 n1  f (xr )Tk (xr )  Co Tk (xr )  C1 T1 (xr )Tk (xr )  ......  Ck T1 (xr )Tk (xr )  .... r0 2 r0 r0 r0 n1 ......  Cn1 Tn1 (xr )Tk (xr ) r0 n  C 2 k 2 n1 2r 1 which gives : Ck   f (xr )cos(k  ) (5-1-3) n r0 2n [1][8]

5-1-2 Computer approximation: Chebyshev series are also used on Computer approximation, where a recursive technique is desirable. Such a technique has been n devised by Clenshaw for the computation of f (x)  CkTk (x) (5-1-2-1) k0 Where the coefficients Ck as well as the value x are assumed to be known. We form a sequence of numbers : an, an-1, ………., a1, a0 by means of the relation : ak- 2xak+1+ ak+2 = Ck (5-1-2-2) Where an+1 = an+2 = 0. Insertion in f(x) gives : n2 n f (x)  ak  2xa k1  ak2 Tk x ak  2xa k1  ak2 Tk x k0 kn1 n2 (5-1-2-3)  ak  2xa k1  ak2 Tk x an1  2xa n Tn1 x anTn x k0 then

f (x)  a0  2xa1  a2 T0  a1  2xa 2  a3 T1  a2  2xa 3  a4 T2  a3  2xa 4  a5 T3  ......  ......  ......

 an3  2xa n2  an1 Tn3  an2  2xa n1  an Tn2  an1  2xa n  0Tn1  an Tn

 a0T0  2xa1T0  a2T0  a1T1  2xa 2T1  a3T1  a2T2  2xa3T2  a4T2  a3T3  2xa 4T3  a5T3  ......

 an3Tn3  2xa n2Tn3  an1Tn3  an2Tn2  2xa n1Tn2  anTn2  an1Tn1  2xa nTn1  0

 anTn  0  0

 a0T0  a1  2xT0  T1  a2 T0  2xT1  T2  a3 T1  2xT2  T3  a4 T2  2xT3  T4   ......

 an2 Tn4  2xTn3  Tn2  an1 Tn3  2xTn2  Tn1  an Tn2  2xTn1  Tn  n   ak 2 Tk  2xTk 1  T  T0 a0  a1  2xT0  T1  k 0 but Tk – 2xTk+1+ +Tk+2 = 0, and T0 = 1, T1 = x, then f(x) = a0 – a1x (5-1-2-4)

[1][6]

5-2 Physics and Engineering applications:

5-2-1 Expanding functions: Legendre polynomials are useful in expanding functions like : 1 1  rl   Pl cos  (5-2-1-1) 2 2  l x  x r  r  2rrcos l0 r 1 where r and r' are the lengths of the vectors and respectively and γ is the angle between those two vectors. This expansion holds where r>r'. This expression is used, for example, to obtain the potential of a point charge , felt at point while the charge is located at point. The expansion using Legendre polynomials might be useful when integrating this expression over a continuous charge distribution. Legendre polynomials are also useful in expanding functions of the form (this is the same as before , written a little differently):  1 k   Pk x (5-2-1-2) 2  1  2x k0 which arise naturally in multipole expansions . The left-hand side of the equation is the generating function for the Legendre polynomials. As an example, the electric potential Φ(r,θ) (in spherical coordinates) due to a point charge located on the z-axis at z = a 1 1 varies like : r,    . (5-2-1-3) R r 2  a 2  2ar cos If the radius r of the observation point P is greater than a, the potential may be expanded in the Legendre polynomials: k 1   a  r,     Pk cos  (5-2-1-4) r k0  r  Where we have defined η = a/r < 1 and x = cosθ. This expansion is used to develop the normal multipole expansion. Conversely, if the radius r of the observation point P is smaller than a, the potential may still be expanded in the Legendre polynomials as above, but with a and r exchanged. This expansion is the basis of interior multipole expansion. [1][5] 5-2-2 Laplace equation : Legendre polynomials occur in the solution of Laplace equation of the potential, 2 x, y  0, using the method of separation of variables.

As an example, to solve Laplace equation for heat diffusion in a sphere with unity radius at original point, such that the heat is independent of Φ axis and its boundary condition : u(1,θ) = sinθ. So let u represents the heat in the sphere. Laplace equation 2u  0, such that u independent of Φ in spherical coordinates (r,θ,Φ), gives :  2u 2 u 1   u    sin   0 (5-2-2-1) r 2 r r r 2 sin     put u(r,θ) = R(r)Ψ(θ) (5-2-2-2) substitute (5-2-2-2) in (5-2-2-1) , we obtain 2 R d R  R  sin.  0 (5-2-2-3) r r 2 sin d dividing both sides of (5-2-2-3) over RΨ yields : R 2R 1 d    2 sin     0 R R r sin   d (5-2-2-3) r 2 R 2rR 1 d     sin    2 R R sin   d and from this we obtain : r 2 R  2rR  R2  0 (5-2-2-4) d sin   2sin  0 (5-2-2-5) d B the equation (5-2-2-4) has the solution R(r)  Ar n  (5-2-2-6) r n1 λ2 = -n(n+1) (5-2-2-7) substitution (5-2-2-7) in (5-2-2-5) , yields : d sin   nn 1sin    0 d (5-2-2-8) d 2  d  sin  cos  n(n 1)sin    0 d 2 d the equation (5-2-2-8) represents Legendre equation which solution is : Ψ(θ) = CPn(cosθ) + DQn(cosθ) (5-2-2-9) and we obtain the general solution by substitution (5-2-2-6) and (5-2-2-9) in (5-2-2-2), which is :

 n B  u(r,)  Ar  CPn cos  DQn cos  (5-2-2-10)  r n1  Since The temperature is bounded with r = 0 (at the centre of sphere) , then : B = 0 (5-2-2-11) (If θ =0 or θ = π , then Qn (1) →∞); so D = 0 (5-2-2-12) by substituting (5-2-2-11) and (5-2-2-12) in (5-2-2-10) , we obtain: n u( r , θ) = Er Pn(cosθ) (5-2-2-13)

where E = AC. Applying combination rule on (5-2-2-13) ,we obtain

 n u(r,)   En r Pn cos  (5-2-2-14) n0 Applying the boundary condition : u( 1, θ) = sinθ , then  sin   En Pn (cos) (5-2-2-15) n0 We put u = cosθ ↔ θ = cos-1u. Applying this in (5-2-2-15) :  1 sin(cos u)   En Pn u (5-2-2-16) n0 Equation (5-2-2-16) represents Legendre series of the function sin(cos-1u). And from this : 2n 1 1 E  sin cos 1 u  P u du n    n   2 1  2n 1 0  sin  P (cos )  sind     n   (5-2-2-17)  2   2n 1  sin 2 P (cos )d   n  2  0 we complete the solution by substitution (5-2-2-17) in (5-2-2-14)    2n 1  u(r,)  r n P cos sin 2 P cos d and obtain :  n    n    n0   2  0 

There are multivarious applications of orthogonal polynomials. In our study we are restricted to a few applications. The Hermite polynomials for instance are occurred in interpolation theory ; Laguerre polynomials are applied to the theory of Propagation of Electromagnetic Waves. And son as the applications. [1][2][6]

 Conclusion and Recommendation : Orthogonal systems play an important role in analysis, mainly because functions belonging to very general classes can be expanded in series of orthogonal functions . An important class of orthogonal system consists of orthogonal polynomials Pn(x) , where n is the degree of the polynomial. This class contains many special functions commonly encountered in the applications, for example, Legendre, Hermite, Laguerre, Chebyshev and Jacobi polynomials. In addition to orthogonality property, These functions have many other general properties. For example , they are the integrals of differential equations of a simple form, and can be defined as the coefficients in expansions in power of t of suitable chosen functions w( x, t ), called generating functions . Othogonal polynomials are the great importance in mathematical physics, approximation theory, the theory of mechanical quadratures, etc., and are the subject of an enormous literature, in which the contributions of Russian mathematicians like Adamov, Akhiezer, Bernstein, Chebyshev, Sonine, ….. , play a prominent role. As the orthogonal polynomials have extremely diverse applications, we recommend their maintenance and wide study on them.

 References : 1) http/:www.google.com 2) Erwin Kreyszig , Advanced Engineering Mathematics , Third Edition. Columbus; Ohio, 1972. 3) Courant , R. , and D. Hilbert , Method of mathematical physics. 2 vols. New York: Interscience , 1953, 1962. 4) Rainville , E.D. , Special functions. Nem York: Macmillan, 1960. 5) Abramowitz, M., and I.A. Stegun , Handbook of mathematical fuctions. New York : Dover , 1965. 6) Buck , R.C., Advanced Calculus. 2nd ed. New York: McGraw-Hill, 1965. 7) Magnus , W., F. Oberhettinger , and R.P. Soni , Formulas and Theorems for the Special Functions of mathematical physics. 3rd ed. New York : Springer, 1966. 8) Whittaker , E.T. , and G.N. Watson, A Course of Modern Analysis . 4th ed. Cambridge : Havard University Press, 1927 ( reprinted 1965).