∫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
They can be expressed via repeated differentiation of a "basis" function: 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 ∫abpn(x)pm(x)w(x)dx=hnδnmintegral from a to b of p sub
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 Orthogonal Polynomials
is the Kronecker delta. These polynomials are foundational in mathematical physics, numerical analysis, and approximation theory. 1. Identify the core families