Yet Another Characterisation of Classical Orthogonal Polynomials?

TL;DR

This paper revisits the classification of classical orthogonal polynomials using a functional-analytic approach, unifying continuous and discrete cases and clarifying the structural relationships among polynomial families.

Contribution

It extends Maroni's duality-based classification to include all known families and clarifies the algebraic and topological structures underlying classical orthogonal polynomials.

Findings

Recover all known families as special cases

Unify continuous and discrete polynomials within a dual-topological framework

Show algebraic equivalences are often treated as distinct in traditional classifications

Abstract

The NIST Handbook of Mathematical Functions (2010) and the NIST Digital Library of Mathematical Functions (2025) classify classical orthogonal polynomials through Bochner's 1929 algebraic-differential characterisation and its discretisation. Yet this classification rests on a narrow reading of Bochner's work and on a restricted notion of orthogonality that becomes inadequate once polynomials are characterised by their algebraic properties. As a result, algebraically equivalent families are treated as distinct, parameter domains are restricted, and families already implicit in Bochner's scheme are excluded. In the mid-1980s, Maroni challenged this view by extending the notion of classical orthogonal polynomials through duality theory on locally convex spaces, thereby reaching the algebraic limits latent in Bochner's framework. Yet when the notion was later enlarged to include further families, Maroni's criteria and rationale were largely set aside. To clarify this history, we revisit a less familiar line of development and use it to obtain a classification of classical orthogonal polynomials on linear lattices within Maroni's functional-analytic setting, beyond the positive-definite case. This classification recovers all known families as special cases, preserves orthogonality and the defining algebraic properties, places supposedly new families in their proper structural context, and shows that algebraically identical polynomials are often treated as distinct. Moreover, through a limit process in the weak topology of the continuous dual, we recover families implicit in Bochner's work and unify the continuous and discrete cases within a dual-topological framework. Thus, neither Bochner's classical characterisation nor its discrete analogue is modified to produce ad hoc families; both are recovered at the level of their intrinsic algebraic structure.