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.

Differential equation

Orthogonal polynomials satisfy the 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.

Recurrence relations

Orthogonal polynomials obey a recurrence relation

f_{n+1}=(a_n+xb_n)f_n - c_nf_{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_nx^n+k_n'x^{n-1}+...

and

h_n

is the normalization, defined below.

Rodrigues formula

The orthogonal polynomials can be obtained through 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 weight function defined above,

e_n

a constant depending only on n, and

g(x)

is a polynomial independent of n.

List of orthogonal polynomials

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.

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

References

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

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