The Classical Orthogonal Polynomials -

They are eigenfunctions of a differential operator of the form are polynomials of degree at most 2 and 1, respectively.

The are a special class of polynomial sequences

that satisfy an orthogonality condition with respect to a specific weight function over an interval . This condition is defined by the inner product: The Classical Orthogonal Polynomials

They can be expressed via repeated differentiation of a "basis" function:

This allows for efficient iterative calculation of high-degree terms. They are eigenfunctions of a differential operator of

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

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 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: