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
is the Kronecker delta. These polynomials are foundational in mathematical physics, numerical analysis, and approximation theory. 1. Identify the core families 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 Orthogonal Polynomials
Any sequence of orthogonal polynomials satisfies a relation: The Classical Orthogonal Polynomials
They can be expressed via repeated differentiation of a "basis" function: