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:

$$
\sigma(\mathfrak{x})F''(\mathfrak{x}) + t(\mathfrak{x})F'(\mathfrak{x}) + \lambda F(\mathfrak{x}) = \mathbf{0},\tag{1}
$$

where F xð Þ is a real function of a real variable F : U ! R, where U ⊂ R is an open subset of the real line, and λ∈ R a corresponding eigenvalue, and the functions s xð Þ and t xð Þ are real polynomials of at most second order and first order, respectively.

There are different cases obtained, depending on the kind of the s xð Þ function in Eq. (1). When s xð Þ is a constant, Eq. (1) takesthe form F 00ð Þ � <sup>2</sup>αxF<sup>0</sup> x x<sup>ð</sup> Þ þ <sup>λ</sup>F x<sup>ð</sup> Þ ¼ 0, and if α ¼ 1 one obtains the Hermite polynomials. When s xð Þ is a polynomial of the first degree, Eq. (1) takes the form xF00ð Þþ <sup>x</sup> ð�α<sup>x</sup> <sup>þ</sup> <sup>β</sup> <sup>þ</sup> <sup>1</sup>Þ<sup>F</sup> <sup>x</sup> <sup>ð</sup> Þ ¼ 0, and <sup>0</sup> ð Þ þ λF x when α ¼ 1 and β ¼ 0, one obtains the Laguerre polynomials. There are three different cases when s xð Þ is a polynomial of the second degree. When the second degree polynomial has two different real roots, Eq. (1) takes the form <sup>ð</sup><sup>1</sup> � <sup>x</sup><sup>2</sup>Þ<sup>F</sup> 00ð<sup>x</sup>Þþ β � α � ðα þ β þ 2Þx�F<sup>0</sup> ½ ðxÞ þ λF xð Þ ¼ 0; this is the Jacobi equation, and for different values of α and β, one obtains particular cases of polynomials: Gegenbauer

polynomials if α ¼ β, Tchebycheff I and II if α ¼ β ¼ �1=2, and Legendre polynomials if α ¼ β ¼ 0. When the second degree polynomial has one double <sup>0</sup> root, Eq. (1) takes the form x2F 00ð Þþ <sup>x</sup> ½ð<sup>α</sup> <sup>þ</sup> <sup>2</sup>Þ<sup>x</sup> <sup>þ</sup> <sup>β</sup>�F x<sup>ð</sup> Þ þ <sup>λ</sup>F x<sup>ð</sup> Þ ¼ 0, and when α ¼ �1 and β ¼ 0, one obtains the Bessel polynomials. Finally, when the second degree polynomial has two complex roots, Eq. (1) takesthe form <sup>0</sup> ð1 þ xÞ 2 F 00ð<sup>x</sup>Þ þ <sup>ð</sup>2β<sup>x</sup> <sup>þ</sup> <sup>α</sup>ÞF x<sup>ð</sup> Þ þ <sup>λ</sup>F x<sup>ð</sup> Þ ¼ 0, which is the Romanovski equation [1]. These results are summarized in Table 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: Lu xð Þ þ λw xð Þu xð Þ ¼ 0, where L is a differential operator; h i du xð Þ <sup>L</sup>u x<sup>ð</sup> Þ ¼ <sup>d</sup> ð Þ <sup>þ</sup> q xð Þu xð Þ, <sup>λ</sup> an eigenvalue, and w xð Þ is known as <sup>a</sup> weight dx p x dx or density function. The analysis of this equation and its solutions is called the Sturm-Liouville theory. Specific forms of p xð Þ, q xð Þ, λ and w xð Þ are 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].


#### Table 1.

Polynomials obtained depending on the s xð Þ function of Eq. (1).

Simple Approach to Special Polynomials: Laguerre, Hermite, Legendre,Tchebycheff… DOI: http://dx.doi.org/10.5772/intechopen.83029

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 [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

$$-\frac{\hbar^2}{2m}\nabla^2\varphi - \frac{Ze^2}{r}\varphi = E\varphi,\tag{12}$$

in which Z ¼ 1 for hydrogen, 2 for single ionized helium, and so on. Separating variables, we find that the angular dependence of <sup>ψ</sup> is <sup>Y</sup><sup>M</sup>ðθ; <sup>φ</sup>Þ. The radial part, <sup>L</sup> R rð Þ, satisfies the equation

$$-\frac{\hbar^2}{2m}\frac{1}{\mathbf{r}^2}\frac{\mathbf{d}}{d\mathbf{r}}\left(\mathbf{r}^2\frac{d\mathbf{R}}{d\mathbf{r}}\right) - \frac{\mathbf{Z}\mathbf{e}^2}{\mathbf{r}}\mathbf{R} + \frac{\mathbf{L}(\mathbf{L}+\mathbf{1})}{\mathbf{r}^2}\mathbf{R} = \mathbf{E}\mathbf{R}.\tag{13}$$

By use of the abbreviations

$$\rho = ar,\quad\text{with }\mathbf{a}^2 = -\frac{8\mathbf{m}\mathbf{E}}{\hbar^2},\text{ E} \le \mathbf{0},\ \lambda = \frac{2\mathbf{m}\mathbf{Z}\mathbf{e}^2}{a\hbar^2},\tag{14}$$

Eq. (14) becomes

$$\frac{1}{\rho^2} \frac{d}{d\rho} \left(\rho^2 \frac{d\chi(\rho)}{d\rho}\right) + \left(\frac{\lambda}{\rho} - \frac{1}{4} - \frac{L(L+1)}{\rho^2}\right) \chi(\rho) = 0,\tag{15}$$

where χ ρð Þ ¼ Rðρ=αÞ. Eq. (15) is satisfied by

$$\rho\chi(\rho) = e^{-\frac{\rho}{2}}\rho^{L+1}L\_{\lambda-L-1}^{2L+1}(\rho),\tag{16}$$

in which k is replaced by 2L þ 1 and n by λ � L � 1, in order to consider the associated Laguerre polynomials Lkð Þ<sup>ρ</sup> . <sup>n</sup>

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 Pierre-Simon 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 Hermite-Gaussian laser beams [4], but mainly to analyze the quantum mechanical simple harmonic oscillator [8]. For <sup>a</sup> potential energy <sup>V</sup> <sup>¼</sup> <sup>1</sup> Kz<sup>2</sup> <sup>¼</sup> <sup>2</sup> <sup>1</sup> mω2z<sup>2</sup> (force <sup>2</sup> F ¼ ∇V ¼ �Kz), the Schrödinger wave equation is

$$-\frac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{z}) + \frac{1}{2}\mathbf{K}\mathbf{z}^2\Psi(\mathbf{z}) = E\Psi(\mathbf{z}).\tag{17}$$

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

$$\mathbf{x} = a\mathbf{z} \text{ with } \alpha^4 = \frac{\mathbf{m}\mathbf{K}}{\hbar^2} = \frac{\mathbf{m}^2 \alpha^2}{\hbar^2}, \lambda = \frac{2\mathbf{E}}{\hbar} \left(\frac{\mathbf{m}}{\mathbf{K}}\right)^{1/2} = \frac{2\mathbf{E}}{\hbar \alpha^2},\tag{18}$$

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

$$\frac{d^2\psi(\mathbf{x})}{d\mathbf{x}^2} + \left(\lambda - \mathbf{x}^2\right)\psi(\mathbf{x}) = \mathbf{0},\tag{19}$$

where ψð Þ¼ x Ψð Þ¼ z Ψðx=αÞ. With λ ¼ 2n þ 1, Eq. (19) is satisfied by

$$\Psi \varphi\_n(\mathbf{x}) = 2^{-\frac{\pi}{2}} \pi^{-\frac{1}{4}} (n!)^{ -\frac{1}{2}} e^{-\frac{x^2}{2}} H\_n(\mathbf{x}),\tag{20}$$

where Hnð Þ x corresponds 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 s equation is s equation are related to these polynomials. They arise, for instance, when Laplace solved in spherical coordinates. Since continuous solutions of Laplace harmonic functions, these solutions are called spherical harmonics [9].

' ' 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,

$$
\nabla^2 \varphi + \mathbf{k}^2 f(r) \varphi = \mathbf{0},\tag{21}
$$

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

$$\frac{\Phi(\phi)}{\sin(\theta)}\frac{d}{d\theta}\left(\sin\theta\frac{d\Theta}{d\theta}\right) + \frac{\Theta(\theta)}{\sin^2\theta}\frac{d^2\Phi(\phi)}{d\phi^2} + n(n+1)\Theta(\theta)\Phi(\phi) = 0. \tag{22}$$

The separated azimuthal equation is

Simple Approach to Special Polynomials: Laguerre, Hermite, Legendre,Tchebycheff… DOI: http://dx.doi.org/10.5772/intechopen.83029

$$\frac{1}{\Phi(\phi)} \frac{d^2 \Phi(\phi)}{d\phi^2} = -m^2,\tag{23}$$

with an orthogonal and normalized solution,

$$\Phi\_m = \frac{1}{\sqrt{2\pi}} e^{im\phi}.\tag{24}$$

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, <sup>Θ</sup>ðθÞ ¼ <sup>P</sup> <sup>ð</sup>cosθ<sup>Þ</sup> <sup>m</sup> . Normalizing the associated Legendre function, <sup>n</sup> one obtains the orthonormal function

$$\circledast\_n^m(\cos\theta) = \sqrt{\frac{2n+1}{2} \frac{(n-m)!}{(n+m)!}} P\_n^m(\cos\theta). \tag{25}$$

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

$$Y\_n^m(\theta,\phi) \equiv \left(-1\right)^m \sqrt{\frac{2n+1}{4\pi} \frac{(n-m)!}{(n+m)!}} P\_n^m(\cos\theta) e^{im\phi}.\tag{26}$$

These <sup>Y</sup><sup>m</sup>ðθ; <sup>ϕ</sup><sup>Þ</sup> are the spherical harmonics [10]. <sup>n</sup>

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 [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,

$$f(\mathbf{x}) \sim \sum\_{i=0}^{\infty} c\_i T\_i(\mathbf{x}),\tag{27}$$

and then cuts off the series after the TN term, 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 [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 Positive-definite 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.
