The Classical Orthogonal Polynomials -

All classical orthogonal polynomials share distinct mathematical properties that separate them from general orthogonal sets:

∫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

Any sequence of orthogonal polynomials satisfies a relation: The Classical Orthogonal Polynomials

pn(x)=1enw(x)dndxn[w(x)σn(x)]p sub n open paren x close paren equals the fraction with numerator 1 and denominator e sub n w open paren x close paren end-fraction the fraction with numerator d to the n-th power and denominator d x to the n-th power end-fraction open bracket w open paren x close paren sigma to the n-th power open paren x close paren close bracket 3. Apply to modern contexts

that satisfy an orthogonality condition with respect to a specific weight function over an interval . This condition is defined by the inner product: They can be expressed via repeated differentiation of

This allows for efficient iterative calculation of high-degree terms.

They can be expressed via repeated differentiation of a "basis" function: The Classical Orthogonal Polynomials

The are a special class of polynomial sequences