# properties of orthogonal polynomials

A countable system of orthogonal polynomials

 $\displaystyle p_{0}(x),\,p_{1}(x),\,p_{2}(x),\,\ldots$ (1)

on an interval  $[a,\,b]$,  where a inner product of two functions

 $(f,\,g)\;:=\;\int_{a}^{b}\!f(x)g(x)W(x)\,dx$

is defined with respect to a weighting function $W(x)$, satisfies the orthogonality condition (http://planetmath.org/OrthogonalVectors)

 $(p_{m},\,p_{n})\;=\;0\quad\mbox{always when}\quad m\neq n.$

One also requires that

 $\deg\left(p_{n}(x)\right)\;=\;n\quad\mbox{for all }n.$

Such a system (1) may be used as basis for the vector space of functions defined on  $[a,\,b]$, i.e. certain such functions $f$ may be expanded as a series (http://planetmath.org/FunctionSeries)

 $f(x)\;=\;c_{0}p_{0}(x)+c_{1}p_{1}(x)+c_{2}p_{2}(x)+\ldots$

where the coefficients $c_{n}$ have the expression

 $c_{n}\;=\;\int_{a}^{b}\!f(x)p_{n}(x)W(x)\,dx.$

Other properties

• The basis property of the system (1) comprises that any polynomial $P(x)$ of degree $n$ can be uniquely expressed as a finite linear combination$\!$

 $P(x)\;=\;c_{0}p_{0}(x)+c_{1}p_{1}(x)+\ldots+c_{n}p_{n}(x).$
• Every member $p_{n}(x)$ of (1) is orthogonal to any polynomial $P(x)$ of degree less than $n$.

• There is a recurrence relation

 $p_{n+1}(x)\;=\;(a_{n}x\!+\!b_{n})p_{n}(x)+c_{n}p_{n-1}(x)$

enabling to determine a .

• The zeros of $p_{n}(x)$ are all real and belong to the open interval$(a,\,b)$;  between two of those zeros there are always zeros of $p_{n+1}(x)$.

• The Sturm–Liouville differential equation

 $\displaystyle Q(x)p^{\prime\prime}+L(x)p^{\prime}+\lambda p\;=\;0,$ (2)

where $Q(x)$ is a polynomial of at most degree 2 and $L(x)$ a linear polynomial, gives under certain conditions, as http://planetmath.org/node/8719solutions $p$ a system of orthogonal polynomials $p_{0},\,p_{1},\,\ldots$  corresponding suitable values (eigenvalues) $\lambda_{0},\,\lambda_{1},\,\ldots$  of the parametre $\lambda$.  Those satisfy the Rodrigues formula

 $p_{n}(x)\;=\;\frac{k_{n}}{W(x)}\frac{d^{n}}{dx^{n}}\left(W(x)[Q(x)]^{n}\right),$

where $k_{n}$ is a constant and

 $W(x)\;:=\;\frac{1}{Q(x)}e^{\int\frac{L(x)}{Q(x)}dx}.$

The classical Chebyshev (http://planetmath.org/ChebyshevPolynomial), Hermite (http://planetmath.org/HermitePolynomials), Laguerre (http://planetmath.org/LaguerrePolynomial), and Legendre polynomials all satisfy an equation (2).