All classical orthogonal polynomials share distinct mathematical properties that separate them from general orthogonal sets:
The are a special class of polynomial sequences The Classical Orthogonal Polynomials
Any sequence of orthogonal polynomials satisfies a relation: Laguerre Polynomials ( ): Defined on with weight
They are eigenfunctions of a differential operator of the form are polynomials of degree at most 2 and 1, respectively. The Classical Orthogonal Polynomials
that satisfy an orthogonality condition with respect to a specific weight function over an interval . This condition is defined by the inner product:
The "classical" label traditionally refers to three primary families (and their special cases) that satisfy a second-order linear differential equation: Defined on with weight Special Cases: Legendre polynomials ( ) and Chebyshev polynomials . Laguerre Polynomials ( ): Defined on with weight Hermite Polynomials ( ): Defined on with weight 2. Define universal characterizations
∫abpn(x)pm(x)w(x)dx=hnδnmintegral from a to b of p sub n open paren x close paren p sub m open paren x close paren w open paren x close paren space d x equals h sub n delta sub n m end-sub is a normalization constant and δnmdelta sub n m end-sub