Classical Orthogonal Polynomials — The

Pn+1(x)=(x−bn)Pn(x)−an2Pn−1(x)cap P sub n plus 1 end-sub open paren x close paren equals open paren x minus b sub n close paren cap P sub n open paren x close paren minus a sub n squared cap P sub n minus 1 end-sub open paren x close paren

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 The Classical Orthogonal Polynomials

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

The are a special class of polynomial sequences The Classical Orthogonal Polynomials