properties of orthogonal polynomials
A countable system of orthogonal polynomials
${p}_{0}(x),{p}_{1}(x),{p}_{2}(x),\mathrm{\dots}$  (1) 
on an interval $[a,b]$, where a inner product of two functions^{}
$$(f,g):={\int}_{a}^{b}f(x)g(x)W(x)\mathit{d}x$$ 
is defined with respect to a weighting function $W(x)$, satisfies the orthogonality condition (http://planetmath.org/OrthogonalVectors)
$$({p}_{m},{p}_{n})=\mathrm{\hspace{0.33em}0}\mathit{\hspace{1em}}\text{always when}\mathit{\hspace{1em}}m\ne n.$$ 
One also requires that
$$\mathrm{deg}\left({p}_{n}(x)\right)=n\mathit{\hspace{1em}}\text{for all}n.$$ 
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)
$$f(x)={c}_{0}{p}_{0}(x)+{c}_{1}{p}_{1}(x)+{c}_{2}{p}_{2}(x)+\mathrm{\dots}$$ 
where the coefficients ${c}_{n}$ have the expression
$${c}_{n}={\int}_{a}^{b}f(x){p}_{n}(x)W(x)\mathit{d}x.$$ 
Other properties

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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)+\mathrm{\dots}+{c}_{n}{p}_{n}(x).$$ 
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Every member ${p}_{n}(x)$ of (1) is orthogonal^{} to any polynomial $P(x)$ of degree less than $n$.

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There is a recurrence relation
$${p}_{n+1}(x)=({a}_{n}x+{b}_{n}){p}_{n}(x)+{c}_{n}{p}_{n1}(x)$$ enabling to determine a .

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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 ${p}_{n+1}(x)$.

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The Sturm–Liouville differential equation^{}
$Q(x){p}^{\prime \prime}+L(x){p}^{\prime}+\lambda p=\mathrm{\hspace{0.33em}0},$ (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 ${p}_{0},{p}_{1},\mathrm{\dots}$ corresponding suitable values (eigenvalues^{}) ${\lambda}_{0},{\lambda}_{1},\mathrm{\dots}$ of the parametre $\lambda $. Those satisfy the Rodrigues formula^{}
$${p}_{n}(x)=\frac{{k}_{n}}{W(x)}\frac{{d}^{n}}{d{x}^{n}}\left(W(x){[Q(x)]}^{n}\right),$$ where ${k}_{n}$ is a constant and
$$W(x):=\frac{1}{Q(x)}{e}^{{\scriptscriptstyle \int {\scriptscriptstyle \frac{L(x)}{Q(x)}}\mathit{d}x}}.$$ 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).
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Title  properties of orthogonal polynomials 
Canonical name  PropertiesOfOrthogonalPolynomials 
Date of creation  20130322 19:05:34 
Last modified on  20130322 19:05:34 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  12 
Author  pahio (2872) 
Entry type  Topic 
Classification  msc 42C05 
Classification  msc 33D45 
Related topic  HilbertSpace 
Related topic  TopicsOnPolynomials 
Related topic  IndexOfSpecialFunctions 
Related topic  OrthogonalityOfLaguerrePolynomials 
Related topic  OrthogonalityOfChebyshevPolynomials 
Defines  Rodrigues formula 