The Classical Orthogonal Polynomials -

Beyond the continuous case, the theory has been "developed" into broader frameworks available in academic texts like The Classical Orthogonal Polynomials by B.G.S. Doman:

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 The Classical Orthogonal Polynomials

They can be expressed via repeated differentiation of a "basis" function: Beyond the continuous case, the theory has been

∫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 Define universal characterizations They can be expressed via

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: