Polynomials obtained depending on the
Abstract
Special polynomials: Laguerre, Hermite, Legendre, Tchebycheff and Gegenbauer are obtained through wellknown linear algebra methods based on SturmLiouville 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:
where
There are different cases obtained, depending on the kind of the

Canonical form and weight function  Example 

Constant 

When

First degree 

When

Second degree: with two different real roots 

Eq. (6) is the Jacobi equation, considering

Second degree: with one double real root 

When

Second degree: with two complex roots 

Eq. (10) is the Romanovski equation; considering

The SturmLiouville Theory is covered in most advanced physics and engineering courses. In this context, an eigenvalue equation sometimes takes the more general selfadjoint form:
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 LaguerreGaussian laser beams [4]. 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 [5].
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
in which
By use of the abbreviations
Eq. (14) becomes
where
in which
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 [6], etc.
2.2 Hermite
Hermite polynomials were defined into the theory of probability by PierreSimon Laplace in 1810, and Charles Hermite extended them to include several variables and named them in 1864 [7].
Hermite polynomials are used to describe the transversal profile of HermiteGaussian laser beams [4], but mainly to analyze the quantum mechanical simple harmonic oscillator [8]. For a potential energy
The oscillating particle has mass
in which
where
where
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 AdrienMarie 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
In the separation of variables of Laplace’s equation, Helmholtz’s or the spacedependence of the classical wave equation, and the Schrödinger wave equation for central force fields,
the angular dependence, coming entirely from the Laplacian operator, is
The separated azimuthal equation is
with an orthogonal and normalized solution,
Splitting off the azimuthal dependence, the polar angle dependence (
Taking the product of Eqs. (24) and (25) to define,
These
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, steadystate heat conduction problems in spherical problems inside a homogeneous solid sphere, and so forth [11].
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,
and then cuts off the series after the
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 [12], another one can be found in aircraft aerodynamics [13], 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 Positivedefinite functions [14].
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
the Laguerre differential equation,
the Hermite differential equation,
the Legendre differential equation,
the Tchebycheff differential equation,
and the Gegenbauer differential equation,
with
3.2 Rodrigues formula
For polynomials
and with
is known as the
3.3 Generating function and contour integral
Let
The
In the following section, Laguerre [2], Hermite [17], Legendre, Tchebycheff [18] and Gegenbauer [3] 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
with
Taking the first derivative of the above polynomial (x), one obtains the polynomial
which may be written as
Taking the second derivative of the polynomial (Eq. (37)) one obtains
which may be written as
Using Eq. (40), Eq. (39) may be written as
therefore, the first derivative operator
Doing the same for Eq. (41),
the second derivative operator
4.1 Laguerre
The Laguerre differential operator is given by.
substituting Eqs. (41) and (44) into Eq. (47),
which may be written as
For simplicity, the Laguerre differential operator, as a
The eigenvalues of a matrix
Substituting eigenvalue
the elements of this eigenvector correspond to the first Laguerre polynomial,
Substituting eigenvalue
the elements of this eigenvector correspond to the second Laguerre polynomial,
Substituting eigenvalue
the elements of this eigenvector correspond to the third Laguerre polynomial,
Substituting eigenvalue
the elements of this eigenvector correspond to the fourth Laguerre polynomial,
4.2 Hermite
The Hermite differential operator is given by
substituting Eqs. (41) and (44) into Eq. (56),
which may be written as
For simplicity, the Hermite differential operator, as a
The eigenvalues of a matrix
Substituting eigenvalue
the elements of this eigenvector correspond to the first Hermite polynomial,
Substituting eigenvalue
the elements of this eigenvector correspond to the second Hermite polynomial,
Substituting eigenvalue
the elements of this eigenvector correspond to the third Hermite polynomial,
Substituting eigenvalue
the elements of this eigenvector correspond to the fourth Hermite polynomial,
4.3 Legendre
The Legendre differential operator is given by
substituting Eqs. (41) and (44) into Eq. (65),
which may be written as
For simplicity, the Legendre differential operator, as a
The eigenvalues of a matrix
Substituting eigenvalue
the elements of this eigenvector correspond to the first Legendre polynomial,
Substituting eigenvalue
the elements of this eigenvector correspond to the second Legendre polynomial,
Substituting eigenvalue
the elements of this eigenvector correspond to the third Legendre polynomial,
Substituting eigenvalue
the elements of this eigenvector correspond to the fourth Legendre polynomial,
4.4 Tchebycheff
The Tchebycheff differential operator is given by
substituting Eqs. (41) and (44) into Eq. (74),
which may be written as
For simplicity, the Tchebycheff differential operator, as a
The eigenvalues of a matrix
Substituting eigenvalue
the elements of this eigenvector correspond to the first Tchebycheff polynomial,
Substituting eigenvalue
the elements of this eigenvector correspond to the second Tchebycheff polynomial,
Substituting eigenvalue
the elements of this eigenvector correspond to the third Tchebycheff polynomial,
Substituting eigenvalue
the elements of this eigenvector correspond to the fourth Tchebycheff polynomial,
4.5 Gegenbauer
The Gegenbauer differential operator is given by
substituting (41) and (44) into (83),
which may be written as
For simplicity, the Gegenbauer differential operator, as a
The eigenvalues of a matrix
Substituting eigenvalue
the elements of this eigenvector correspond to the first Gegenbauer polynomial,
Substituting eigenvalue
the elements of this eigenvector correspond to the second Gegenbauer polynomial,
Substituting eigenvalue
the elements of this eigenvector correspond to the third Gegenbauer polynomial,
Substituting eigenvalue
the elements of this eigenvector correspond to the fourth Gegenbauer polynomial,
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
Acknowledgments
V. Aboites acknowledges support and useful conversations with Prof. Ernst Wintner from TUWien, Dr. Matei Tene from TUDelft 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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