Chemistry - Orthogonal polynomials

In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative "weight function" w precisely if

$\int_{x_1}^{x_2} f(x)g(x)w(x)\,dx=0.$

In other words, if polynomials are treated as vectors and the inner product of two polynomials f(x) and g(x) is defined as

$\langle f,g \rangle=\int_{x_1}^{x_2} f(x)g(x)\,w(x)\,dx$

then the orthogonal polynomials are simply orthogonal vectors in this inner product space.

A polynomial sequence p_n(x) for n = 0, 1, 2, ... , where p_n(x) has degree n, is said to be a sequence of orthogonal polynomials with respect to a "weight function" w when any two of them are orthogonal with respect to that weight function, i.e.,

$\langle p_n, p_m \rangle=\int_{x_1}^{x_2} p_n(x) p_m(x)\,w(x)\,dx=0\ \mbox{whenever}\ n\neq m.$

The sequence of orthogonal polynomials can be successively constructed by carrying out the Gram-Schmidt process with the sequence of powers $x^k, \; k \ge 0$ , where the positive-definite inner product $\langle p,q \rangle$ on the space of polynomials is given by the integral above.

Contents

1 General properties of orthogonal polynomials

2 The classical orthogonal polynomials

3 See also

4 References

General properties of orthogonal polynomials

Three-term recurrence relations

For each non-negative weight function the corresponding orthogonal polynomials obey a recurrence relation

f n + 1 = (a n + x b n) f n - c n f n - 1

where the constants $a n$ , $b n$ and $c n$ are given by

$b_n=\frac{k_{n+1}}{k_n}$

$a_n=b_n \left(\frac{k_{n+1}'}{k_{n+1}} - \frac{k_n'}{k_n} \right)$

$c_n=\frac{k_{n+1}k_{n-1}h_n} {k_n^2 h_{n-1}}$

and $k n$ and $k n'$ are the leading terms in the expansion of the polynomial:

f n (x) = k n x n + k n' x n - 1 + ...

and $h n$ is the normalization, defined below.

The classical orthogonal polynomials

The collective name "classical orthogonal polynomials" refers to a class of orthogonal polynomials which are distinguished by several characteristic properties. They occur in many applications including mathematical physics, interpolation theory , the theory of random matrices and many others and have been therefore studied in mathematics since a long time. Some of their characteristic properties will be outlined in the following subsections:

Differential equation

The classical orthogonal polynomials satisfy a second-order differential equation

g 2 (x) f n''(x) + g 1 (x) f n'(x) + d n f n (x) = 0

where $g 1 (x)$ and $g 2 (x)$ are independent of n and $d n$ is a constant that depends only on n. The coefficient function $g 2 (x)$ is a polynomial of degree $\le 2$ , the coefficient $g 1 (x)$ is a polynomial of degree $\le 1$ .

Existence of a Rodrigues formula

Every classical orthogonal polynomial can be obtained via a so-called Rodrigues formula:

$f_n(x)=\frac{1}{e_n w(x)}\, \frac{d^n}{dx^n} w(x)[g(x)]^n$

where $w (x)$ is the defining weight function of the series of orthogonal polynomials (defined in the list below) and $e n$ is a constant depending only on n, and $g (x)$ is a polynomial independent of n.

The orthogonality relationship is

$\int_{x_1}^{x_2}p_n(x)p_m(x)w(x)\,dx=\delta_{mn}h_n$

where δ_mn is the Kronecker delta and h_{n_{is defined in the table below for each weight function.}}

Table Of Classical Orthogonal Polynomials
Name	x₁	x₂	w(x)	h_n
Chebyshev polynomials (first kind)	$- 1$	$1$	$(1 - x 2) - 1 / 2$	$\left\{ \begin{matrix} \pi &:~n=0 \\ \pi/2 &:~n\ne 0 \end{matrix}\right.$
Chebyshev polynomials (second kind)	$- 1$	$1$	$(1 - x 2) 1 / 2$	$π / 2$
Legendre polynomials	$- 1$	$1$	$1$	$\frac{2}{2n+1}$
Laguerre polynomials	$0$	$\infty$	$e - x$	$1$
Hermite polynomials	$-\infty$	$\infty$	$e^{-x^2}$	$n!\,\sqrt{2\pi}$

References

Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 . (See chapter 22).

Categories: Polynomials | Special functions