Open access peer-reviewed chapter

# Simple Approach to Special Polynomials: Laguerre, Hermite, Legendre, Tchebycheff, and Gegenbauer

By Vicente Aboites and Miguel Ramírez

Submitted: September 5th 2018Reviewed: December 4th 2018Published: January 24th 2019

DOI: 10.5772/intechopen.83029

## Abstract

Special polynomials: Laguerre, Hermite, Legendre, Tchebycheff and Gegenbauer are obtained through well-known linear algebra methods based on Sturm-Liouville theory. A matrix corresponding to the differential operator is found and its eigenvalues are obtained. The elements of the eigenvectors obtained correspond to each mentioned polynomial. This method contrasts in simplicity with standard methods based on solving the differential equation by means of power series, obtaining them through a generating function, using the Rodrigues formula for each polynomial, or by means of a contour integral.

### Keywords

• special polynomials
• special functions
• linear algebra
• eigenvalues
• eigenvectors

## 1. Introduction

The polynomials covered in this chapter are solutions to an ordinary differential equation (ODE), the hypergeometric equation. In general, the hypergeometric equation may be written as:

sxFx+txFx+λFx=0,E1

where Fxis a real function of a real variable F:UR, where URis an open subset of the real line, and λRa corresponding eigenvalue, and the functions sxand txare real polynomials of at most second order and first order, respectively.

There are different cases obtained, depending on the kind of the sxfunction in Eq. (1). When sxis a constant, Eq. (1) takes the form Fx2αxFx+λFx=0, and if α=1one obtains the Hermite polynomials. When sxis a polynomial of the first degree, Eq. (1) takes the form xFx+αx+β+1Fx+λFx=0, and when α=1and β=0, one obtains the Laguerre polynomials. There are three different cases when sxis a polynomial of the second degree. When the second degree polynomial has two different real roots, Eq. (1) takes the form 1x2Fx+βαα+β+2xFx+λFx=0; this is the Jacobi equation, and for different values of αand β, one obtains particular cases of polynomials: Gegenbauer polynomials if α=β, Tchebycheff I and II if α=β=±1/2, and Legendre polynomials if α=β=0. When the second degree polynomial has one double root, Eq. (1) takes the form x2Fx+α+2x+βFx+λFx=0, and when α=1and β=0, one obtains the Bessel polynomials. Finally, when the second degree polynomial has two complex roots, Eq. (1) takes the form 1+x2Fx+2βx+αFx+λFx=0, which is the Romanovski equation . These results are summarized in Table 1.

sxCanonical form and weight functionExample
ConstantFx2αxFx+λFx=0(2)
wx=eαx2(3)
When α=1one obtains the Hermite equation, Fx=Hx; this produces the Hermite polynomials, denoted as Hnα.
First degreexFx+αx+β+1Fx+λFx=0(4)
wx=xβeαx(5)
When α=1and β=0, one obtains the Laguerre equation, Fx=Lx; this produces the Laguerre polynomials, denoted as Lnαβ.
Second degree: with two different real roots1x2Fx+βαα+β+2xFx+ λFx=06
wαβx=1xα1+xβ(7)
Eq. (6) is the Jacobi equation, considering Fx=Px, and for each pair αβ, one obtains the Jacobi polynomials, denoted as Pnαβ. Particular cases: Gegenbauer polynomials if α=β, Tchebycheff I and II if α=β=±12, and Legendre polynomials if α=β=0.
Second degree: with one double real rootx2Fx+α+2x+βFx+λFx=0(8)
wαβx=xαeβx(9)
When α=1and β=0, one obtains the Bessel equation, Fx=Bx; this produces the Bessel polynomials, denoted as Bnαβ.
Second degree: with two complex roots1+x2Fx+2βx+αFx+λFx=0(10)
wαβx=1+x2β1eαcot1x(11)
Eq. (10) is the Romanovski equation; considering Fx=Rx, then one obtains the Romanovski polynomials, denoted as Rnαβ.

### Table 1.

Polynomials obtained depending on the sxfunction of Eq. (1).

The Sturm-Liouville Theory is covered in most advanced physics and engineering courses. In this context, an eigenvalue equation sometimes takes the more general self-adjoint form: Lux+λwxux=0, where Lis a differential operator; Lux=ddxpxduxdx+qxux, λan eigenvalue, and wxis known as a weight or density function. The analysis of this equation and its solutions is called the Sturm-Liouville theory. Specific forms of px, qx, λand wxare given for Legendre, Laguerre, Hermite and other well-known equations in the given references. There, the close analogy of this theory with linear algebra concepts is also shown. For example, functions here take the role of vectors there, and linear operators here take that of matrices there. Finally, the diagonalization of a real symmetric matrix corresponds to the solution of an ordinary differential equation, defined by a self-adjoint operator L, in terms of its eigenfunctions, which are the “continuous” analog of the eigenvectors [2, 3].

The next section shows some of the most important applications of Hermite, Gegenbauer, Tchebycheff, Laguerre and Legendre polynomials in applied Mathematics and Physics. These polynomials are of great importance in mathematical physics, the theory of approximation, the theory of mechanical quadrature, engineering, and so forth.

## 2. Physical applications

### 2.1 Laguerre

Laguerre polynomials were named after Edmond Laguerre (1834–1886). Laguerre studied a special case in 1897, and in 1880, Nikolay Yakovlevich Sonin worked on the general case known as Sonine polynomials, but they were anticipated by Robert Murphy (1833).

The Laguerre differential equation and its solutions, that is, Laguerre polynomials, are found in many important physical problems, such as in the description of the transversal profile of Laguerre-Gaussian laser beams . The practical importance of Laguerre polynomials was enhanced by Schrödinger’s wave mechanics, where they occur in the radial wave functions of the hydrogen atom .

The most important single application of the Laguerre polynomials is in the solution of the Schrödinger wave equation for the hydrogen atom. This equation is

22m2ψZe2rψ=,E12

in which Z=1for hydrogen, 2 for single ionized helium, and so on. Separating variables, we find that the angular dependence of ψis YLMθφ. The radial part, Rr, satisfies the equation

22m1r2ddrr2dRdrZe2rR+LL+1r2R=ER.E13

By use of the abbreviations

ρ=αr,with α2=8mE2,E<0,λ=2mZe2α2,E14

Eq. (14) becomes

1ρ2dρ2ρ+λρ14LL+1ρ2χρ=0,E15

where χρ=Rρ/α. Eq. (15) is satisfied by

ρχρ=eρ2ρL+1LλL12L+1ρ,E16

in which kis replaced by 2L+1and nby λL1, in order to consider the associated Laguerre polynomials Lnkρ.

These polynomials are also used in problems involving the integration of Helmholtz’s equation in parabolic coordinates, in the theory of propagation of electromagnetic waves along transmission lines, in describing the static Wigner functions of oscillator systems in quantum mechanics in phase space , etc.

### 2.2 Hermite

Hermite polynomials were defined into the theory of probability by Pierre-Simon Laplace in 1810, and Charles Hermite extended them to include several variables and named them in 1864 .

Hermite polynomials are used to describe the transversal profile of Hermite-Gaussian laser beams , but mainly to analyze the quantum mechanical simple harmonic oscillator . For a potential energy V=12Kz2=12mω2z2(force F=V=Kz), the Schrödinger wave equation is

22m2Ψz+12Kz2Ψz=EΨz.E17

The oscillating particle has mass mand total energy E. By use of the abbreviations

x=αzwithα4=mK2=m2ω22,λ=2EmK1/2=2Eω,E18

in which ωis the angular frequency of the corresponding classical oscillator, Eq. (17) becomes

d2ψxdx2+λx2ψx=0,E19

where ψx=Ψz=Ψx/α. With λ=2n+1, Eq. (19) is satisfied by

ψnx=2n2π14n!12ex22Hnx,E20

where Hnxcorresponds to Hermite polynomials.

Hermite polynomials also appear in probability as the Edgeworth series, in combinatorics as an example of an Appell sequence, obeying the umbral calculus, in numerical analysis as Gaussian quadrature, etc.

### 2.3 Legendre

Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre. Spherical harmonics are an important class of special functions that are closely related to these polynomials. They arise, for instance, when Laplace’s equation is solved in spherical coordinates. Since continuous solutions of Laplace’s equation are harmonic functions, these solutions are called spherical harmonics.

In the separation of variables of Laplace’s equation, Helmholtz’s or the space-dependence of the classical wave equation, and the Schrödinger wave equation for central force fields,

2ψ+k2frψ=0,E21

the angular dependence, coming entirely from the Laplacian operator, is

ΦϕsinθdsinθdΘ+Θθsin2θd2Φϕdϕ2+nn+1ΘθΦϕ=0.E22

The separated azimuthal equation is

1Φϕd2Φϕdϕ2=m2,E23

with an orthogonal and normalized solution,

Φm=12πeimϕ.E24

Splitting off the azimuthal dependence, the polar angle dependence (θ) leads to the associated Legendre equation, which is satisfied by the associated Legendre functions; that is, Θθ=Pnmcosθ. Normalizing the associated Legendre function, one obtains the orthonormal function

nmcosθ=2n+12nm!n+m!Pnmcosθ.E25

Taking the product of Eqs. (24) and (25) to define,

Ynmθϕ1m2n+14πnm!n+m!Pnmcosθeimϕ.E26

These Ynmθϕare the spherical harmonics .

Legendre polynomials are frequently encountered in physics and other technical fields. Some examples are the coefficients in the expansion of the Newtonian potential that gives the gravitational potential associated to a point mass or the Coulomb potential associated to a point charge, the gravitational and electrostatic potential inside a spherical shell, steady-state heat conduction problems in spherical problems inside a homogeneous solid sphere, and so forth .

### 2.4 Tchebycheff

Tchebycheff polynomials, named after Pafnuty Tchebycheff (also written as Chebyshev, Tchebyshev or Tschebyschow), are important in approximation theory because the roots of the Tchebycheff polynomials of the first kind, which are also called Tchebycheff nodes, are used as nodes in polynomial interpolation. Approximation theory is concerned with how functions can best be approximated with simpler functions, and through quantitatively characterizing the errors introduced thereby.

One can obtain polynomials very close to the optimal one by expanding the given function in terms of Tchebycheff polynomials, and then cutting off the expansion at the desired degree. This is similar to the Fourier analysis of the function, using the Tchebycheff polynomials instead of the usual trigonometric functions.

If one calculates the coefficients in the Tchebycheff expansion for a function,

fxi=0ciTix,E27

and then cuts off the series after the TNterm, one gets an Nth-degree polynomial approximating f(x).

Tchebycheff polynomials are also found in many important physics, mathematics and engineering problems. A capacitor whose plates are two eccentric spheres is an interesting example , another one can be found in aircraft aerodynamics , etc.

### 2.5 Gegenbauer

Gegenbauer polynomials, named after Leopold Gegenbauer, and often called ultraspherical polynomials, include Legendre and Tchebycheff polynomials as special or limiting cases, and at the same time, Gegenbauer polynomials are a special case of Jacobi polynomials (see Table 1).

Gegenbauer polynomials appear naturally as extensions of Legendre polynomials in the context of potential theory and harmonic analysis. They also appear in the theory of Positive-definite functions .

Since Gegenbauer polynomials are a general case of Legendre and Tchebycheff polynomials, more applications are shown in Section 2.3 and 2.4.

The most common methods to obtain the special polynomials are described in the next section.

## 3. Special polynomials

To obtain the polynomials described in the previous section, one can use different methods, some tougher than others. These polynomials are typically obtained as a result of the solution of each specific differential equation by means of the power series method. Usually, it is also shown that they can be obtained through a generating function and also by using the Rodrigues formula for each special polynomial, or finally, through a contour integral. Most Mathematical Methods courses also include a study of the properties of these polynomials, such as orthogonality, completeness, recursion relations, special values, asymptotic expansions and their relation to other functions, such as polynomials and hypergeometric functions. There is no doubt that this is a challenging and demanding subject that requires a great deal of attention from most students.

### 3.1 Differential equation

The most common way to solve the special polynomials is solving the associated differential equation through power series and the Frobenius method y=n=0anxn. The corresponding polynomials satisfy the following differential equations:

the Laguerre differential equation,

xy+1xy+ny=0,E28

the Hermite differential equation,

y2xy+2ny=0,E29

the Legendre differential equation,

1x2y2xy+nn+1y=0,E30

the Tchebycheff differential equation,

1x2yxy+n2y=0,E31

and the Gegenbauer differential equation,

1x2y2λ+1xy+nn+2λy=0,E32

with n=0,1,2,3,in all the previous cases. Note that if λ=12, Eq. (32) reduces to the Legendre differential equation (Eq. (30)), and if λ=0, Eq. (32) reduces to the Tchebycheff differential equation (Eq. (31)).

### 3.2 Rodrigues formula

For polynomials ψnx, with interval I, weight function wx, and an eigenvalue equation of the form

pxψnx+qxψnx+λnψnx=0,E33

and with qx=pxwxwx, the general formula

ψnx=wx1dndxnpxnwxE34

is known as the Rodrigues formula,useful to obtain the nth-degree polynomial of ψ.

### 3.3 Generating function and contour integral

Let Γbe a curve that encloses xIbut excludes the endpoints of I. Then, considering the Cauchy integral formula  for derivatives of wxpxnto derive an integral formula from Eq. (34), one obtains

ψnxn!=12πiΓwzwxpznzxndzzx.E35

The generating functionfor the orthogonal polynomials ψnxn!is defined as

Gxs=n=0ψnxn!sn.E36

In the following section, Laguerre , Hermite , Legendre, Tchebycheff  and Gegenbauer  polynomials are obtained through a simple method, using basic linear algebra concepts, such as the eigenvalue and the eigenvector of a matrix.

## 4. Simple approach to special polynomials

The general algebraic polynomial of degree n,

a0+a1x+a2x2+a3x3+anxn,E37

with ao, a1,…,an∈ ℜ, is represented by vector

An=a0a1a2a3an.E38

Taking the first derivative of the above polynomial (x), one obtains the polynomial

ddxa0+a1x+a2x2+a3x3+anxn=a1+2a2x+3a3x2+nanxn1,E39

which may be written as

dAndx=a12a23a3nan0.E40

Taking the second derivative of the polynomial (Eq. (37)) one obtains

d2dx2a0+a1x+a2x2+a3x3+anxn=2a2+6a3x+nn1anxn2,E41

which may be written as

d2Andx2=2a26a3nn1an00.E42

Using Eq. (40), Eq. (39) may be written as

0100000200000300000n00000a0a1a2a3an=a12a23a3nan0;E43

therefore, the first derivative operator Anmay be written as

ddx0100000200000300000n00000.E44

Doing the same for Eq. (41),

00200000600000nn10000000000a0a1a2an1an=a12a2nn1an00,E45

the second derivative operator Anmay be written as

d2dx200200000600000nn10000000000.E46

### 4.1 Laguerre

The Laguerre differential operator is given by.

xd2dx2+1xddx;E47

substituting Eqs. (41) and (44) into Eq. (47),

x2a2+6a3x++nn1anxn2+1xa1+2a2x+3a3x2++nanxn1=a1+4a2a1x+9a32a2x2+16a4+3a3x3+nan,E48

which may be written as

010000014000002900000316000000na0a1a2a3an=a14a2a19a32a216a43a3nan.E49

For simplicity, the Laguerre differential operator, as a 4x4matrix, is represented by

xd2dx2+1xddx0100014000290003.E50

The eigenvalues of a matrix Mare the values that satisfy the equation DetMλI=0. However, since the matrix (Eq. (50)) is a triangular matrix, the eigenvalues λiof this matrix are the elements of the diagonal, namely: λ1=0, λ2=1, λ3=2, λ4=3. The corresponding eigenvectors are the solutions of the equation MλiI·v=0, where the eigenvector v=a0a1a2a3T:

0λi10001λi40002λi90003λia0a1a2a3=0000.E51

Substituting eigenvalue λ1=0in Eq. (51), we obtain eigenvector v1:

v1=1000;E52

the elements of this eigenvector correspond to the first Laguerre polynomial, L0x=1.

Substituting eigenvalue λ2=1in Eq. (51), we obtain eigenvector v2:

v2=1100;E53

the elements of this eigenvector correspond to the second Laguerre polynomial, L1x=1x.

Substituting eigenvalue λ3=2in Eq. (51), we obtain eigenvector v3:

v3=12120;E54

the elements of this eigenvector correspond to the third Laguerre polynomial, L2x=12x+12x2.

Substituting eigenvalue λ4=3in Eq. (51), we obtain eigenvector v4:

v4=133216;E55

the elements of this eigenvector correspond to the fourth Laguerre polynomial, L3x=13x+32x216x3.

### 4.2 Hermite

The Hermite differential operator is given by

d2dx22xddx;E56

substituting Eqs. (41) and (44) into Eq. (56),

2a2+6a3x++nn1anxn22xa1+2a2x+3a3x2++nanxn1=2a2+6a32a1x+12a44a2x2+20a56a3x3+2nan,E57

which may be written as

0020000206000040120000600000002na0a1a2a3an=2a26a32a112a44a220a56a32nan.E58

For simplicity, the Hermite differential operator, as a 4x4matrix, is represented by

d2dx22xddx0020020600400006.E59

The eigenvalues of a matrix Mare the values that satisfy the equation DetMλI=0. However, since the matrix (Eq. (59)) is a triangular matrix, the eigenvalues λiof this matrix are the elements of the diagonal, namely: λ1=0, λ2=2, λ3=4, λ4=6. The corresponding eigenvectors are the solutions of the equation MλiI·v=0, where the eigenvector v=a0a1a2a3T:

0λi02002λi06004λi00006λia0a1a2a3=0000.E60

Substituting eigenvalue λ1=0in Eq. (60), we obtain eigenvector v1:

v1=1000;E61

the elements of this eigenvector correspond to the first Hermite polynomial, H0x=1.

Substituting eigenvalue λ2=2in Eq. (60), we obtain eigenvector v2:

v2=0200;E62

the elements of this eigenvector correspond to the second Hermite polynomial, H1x=2x.

Substituting eigenvalue λ3=4in Eq. (60), we obtain eigenvector v3:

v3=2040;E63

the elements of this eigenvector correspond to the third Hermite polynomial, H2x=4x22.

Substituting eigenvalue λ4=6in Eq. (60), we obtain eigenvector v4:

v4=01208;E64

the elements of this eigenvector correspond to the fourth Hermite polynomial, H3x=8x312x.

### 4.3 Legendre

The Legendre differential operator is given by

1x2d2dx22xddx;E65

substituting Eqs. (41) and (44) into Eq. (65),

1x22a2+6a3x++nn1anxn22xa1+2a2x+3a3x2++nanxn1=2a2+6a32a1x+12a46a2x2+20a512a3x3+n2+nan,E66

which may be written as

0020000206000060120000120000000n2+na0a1a2a3an=2a26a32a112a46a220a512a3n2+nan.E67

For simplicity, the Legendre differential operator, as a 4x4matrix, is represented by

1x2d2dx22xddx00200206006000012.E68

The eigenvalues of a matrix Mare the values that satisfy the equation DetMλI=0. However, since the matrix (Eq. (68)) is a triangular matrix, the eigenvalues λiof this matrix are the elements of the diagonal, namely: λ1=0, λ2=2, λ3=6, λ4=12. The corresponding eigenvectors are the solutions of the equation MλiI·v=0, where the eigenvector v=a0a1a2a3T:

0λi02002λi06006λi000012λia0a1a2a3=0000.E69

Substituting eigenvalue λ1=0in Eq. (69), we obtain eigenvector v1:

v1=1000;E70

the elements of this eigenvector correspond to the first Legendre polynomial, P0x=1.

Substituting eigenvalue λ2=2in Eq. (69), we obtain eigenvector v2:

v2=0100;E71

the elements of this eigenvector correspond to the second Legendre polynomial, P1x=x.

Substituting eigenvalue λ3=6in Eq. (69), we obtain eigenvector v3:

v3=1030;E72

the elements of this eigenvector correspond to the third Legendre polynomial, P2x=32x212.

Substituting eigenvalue λ4=12in Eq. (69), we obtain eigenvector v4:

v4=0305;E73

the elements of this eigenvector correspond to the fourth Legendre polynomial, P3x=52x332x.

### 4.4 Tchebycheff

The Tchebycheff differential operator is given by

1x2d2dx2xddx;E74

substituting Eqs. (41) and (44) into Eq. (74),

1x22a2+6a3x++nn1anxn2xa1+2a2x+3a3x2++nanxn1=2a2+6a3a1x+12a44a2x2+20a59a3x3+n^2an,E75

which may be written as

002000010600004012000090000000n2a0a1a2a3an=2a26a3a112a44a220a59a3n2an.E76

For simplicity, the Tchebycheff differential operator, as a 4x4matrix, is represented by

1x2d2dx2xddx0020010600400009.E77

The eigenvalues of a matrix Mare the values that satisfy the equation DetMλI=0. However, since the matrix (Eq. (77)) is a triangular matrix, the eigenvalues λiof this matrix are the elements of the diagonal, namely: λ1=0, λ2=1, λ3=4, λ4=9. The corresponding eigenvectors are the solutions of the equation MλiI·v=0, where the eigenvector v=a0a1a2a3T;

0λi02001λi06004λi00009λia0a1a2a3=0000.E78

Substituting eigenvalue λ1=0in Eq. (78), we obtain eigenvector v1:

v1=1000;E79

the elements of this eigenvector correspond to the first Tchebycheff polynomial, T0x=1.

Substituting eigenvalue λ2=1in Eq. (78), we obtain eigenvector v2:

v2=0100;E80

the elements of this eigenvector correspond to the second Tchebycheff polynomial, T1x=x.

Substituting eigenvalue λ3=4in Eq. (78), we obtain eigenvector v3:

v3=1020;E81

the elements of this eigenvector correspond to the third Tchebycheff polynomial, T2x=2x21.

Substituting eigenvalue λ4=9in Eq. (78), we obtain eigenvector v4:

v4=0304.E82

the elements of this eigenvector correspond to the fourth Tchebycheff polynomial, T3x=4x33x.

### 4.5 Gegenbauer

The Gegenbauer differential operator is given by

1x2d2dx22λ+1xddx;E83

substituting (41) and (44) into (83),

1x22a2+6a3x++nn1anxn22λ+1xa1+2a2x+3a3x2++nanxn1=2a2+6a32λ+1a1x+12a44λ+1a2x2+20a532λ+3a3x3+n2+2λnan,E84

which may be written as

00200002λ+10600004λ+1012000032λ+30000000n22λna0a1a2a3an=2a26a32λ+1a112a44λ+1a220a532λ+3a3n2+2λnan.E85

For simplicity, the Gegenbauer differential operator, as a 4x4matrix, is represented by

1x2d2dx22λ+1xddx002002λ+106004λ+1000032λ+3.E86

The eigenvalues of a matrix Mare the values that satisfy the equation DetMλI=0. However, since the matrix (Eq. (86)) is a triangular matrix, the eigenvalues λiof this matrix are the elements of the diagonal, namely: λ1=0, λ2=2λ+1, λ3=4λ+1, λ4=32λ+3. The corresponding eigenvectors are the solutions of the equation MλiI·v=0, where the eigenvector v=a0a1a2a3T;

0λi02002λ+1λi06004λ+1λi000032λ+3λi.a0a1a2a3=0000.E87

Substituting eigenvalue λ1=0in Eq. (87), we obtain eigenvector v1:

v1=1000;E88

the elements of this eigenvector correspond to the first Gegenbauer polynomial, C0λx=1.

Substituting eigenvalueλ2=2λ+1in Eq. (87), we obtain eigenvector v2:

v2=02λ00;E89

the elements of this eigenvector correspond to the second Gegenbauer polynomial, C1λx=2λx.

Substituting eigenvalue λ3=4λ+1in Eq. (87), we obtain eigenvector v3:

v3=λ02λ1+λ0;E90

the elements of this eigenvector correspond to the third Gegenbauer polynomial, C2λx=λ+2λ1+λx2.

Substituting eigenvalue λ4=32λ+3in Eq. (87), we obtain eigenvector v4:

v4=02λ1+λ043λ1+λ2+λ;E91

the elements of this eigenvector correspond to the fourth Gegenbauer polynomial, C3λx=2λ1+λx+43λ1+λ2+λx3.

## 5. Conclusions

Laguerre, Hermite, Legendre, Tchebycheff and Gegenbauer polynomials are obtained in a simple and straightforward way using basic linear algebra concepts, such as the eigenvalue and the eigenvector of a matrix. Once the matrix of the corresponding differential operator is obtained, the eigenvalues of this matrix are found, and the elements of its eigenvectors correspond to the coefficients of each kind of polynomials. Using a larger matrix, higher order polynomials may be found; however, the general case for an nxnmatrix was not obtained since it seems that in this general case, standard methods would be easier to use. The main advantage of this method lies in its easiness, since it relies on simple linear algebra concepts. This method contrasts in simplicity with standard methods based on solving the differential equation using power series, using the generating function, using the Rodrigues formula, or using a contour integral.

## Acknowledgments

V. Aboites acknowledges support and useful conversations with Prof. Ernst Wintner from TU-Wien, Dr. Matei Tene from TU-Delft and Dr. Klaus Huber from Berlin. The authors acknowledge the professional English proof reading service provided by Mr. Mario Ruiz Berganza.

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Vicente Aboites and Miguel Ramírez (January 24th 2019). Simple Approach to Special Polynomials: Laguerre, Hermite, Legendre, Tchebycheff, and Gegenbauer, Applied Mathematics, Bruno Carpentieri, IntechOpen, DOI: 10.5772/intechopen.83029. Available from:

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