All classical orthogonal polynomials share distinct mathematical properties that separate them from general orthogonal sets:
They can be expressed via repeated differentiation of a "basis" function: The Classical Orthogonal Polynomials
Any sequence of orthogonal polynomials satisfies a relation: respectively. Beyond the continuous case
They are eigenfunctions of a differential operator of the form are polynomials of degree at most 2 and 1, respectively. 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:
is the Kronecker delta. These polynomials are foundational in mathematical physics, numerical analysis, and approximation theory. 1. Identify the core families