A "missing" family of classical orthogonal polynomials (1011.1669v3)

Abstract: We study a family of "classical" orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl-type. These polynomials can be obtained from the little $q$-Jacobi polynomials in the limit $q=-1$. We also show that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for $q=-1$.

• The paper unveils a novel family of orthogonal polynomials that satisfy a Dunkl-type eigenvalue equation as q approaches –1.
• It details the transition of little q-Jacobi polynomials to this regime, establishing connections to the Askey-Wilson algebra.
• The study broadens classical polynomial theory by introducing non-symmetric, Dunkl-classical properties, paving the way for future research.

Overview of A “missing” family of classical orthogonal polynomials

This paper by Luc Vinet and Alexei Zhedanov explores an emergent class of orthogonal polynomials that bridge connections within the Askey scheme by considering the limit $q \rightarrow -1$. The research is anchored on the premise that classical orthogonal polynomials fulfill both the recurrence relations and the eigenvalue problems with differential or difference operators. This paper unveils a relatively unexplored path in understanding the Askey-Wilson algebra, specifically when $q$ takes the value $-1$.

Discussion on Classical Orthogonal Polynomials

Classical orthogonal polynomials traditionally satisfy a three-term recurrence relation underpinned by a differential operator eigenvalue problem. The authors illustrate that this newly considered family aligns with such criteria, featuring an eigenvalue equation governed by a Dunkl-type operator—a significant deviation from purely classical assumptions where the operator is mainly second-order differential.

Limit Transition from Little q-Jacobi Polynomials

A central facet of this paper includes delineating the transition of little $q$-Jacobi polynomials as $q \rightarrow -1$. The authors meticulously calculate this limit and manage to sustain the operator L even in this relatively unexplored regime. The result establishes these polynomials as classical and exposes their connection to Dunkl operators, which transform polynomials between families while maintaining structural polynomial integrity.

Theorical Implications and Dunkl Operators

Vinet and Zhedanov demonstrate that, unlike conventional Hermite and Gegenbauer polynomials, the new little $-1$-Jacobi polynomials satisfy a Dunkl-classical property. This pivotal revelation distinguishes these polynomials due to their non-symmetric nature and potential as the first identified case of Dunkl-classical orthogonal polynomials outside the symmetric domain. The exploration of generalized Gegenbauer and Jacobi polynomials through Dunkl operators paves an avenue to broaden analytical capabilities in polynomial theory.

Connection to Askey-Wilson Algebra

Expressions correlated with the Askey-Wilson algebra are employed to realize the mathematical structure properties of these polynomials. Particularly, the setting $q = -1$ is dissected to show compliance with AW(3) algebraic relations. This algebraic representation underpins the analytical utility of the explored polynomials, supporting the embedding of Askey-Wilson polynomial impressions.

Future Implications and Open Questions

Future developments in the landscape of orthogonal polynomials may benefit from systematically exploring similar limit transitions of other polynomial families. This paper invites further inquiry into possible asymmetries and Dunkl operator associations, potentially revealing new polynomial classes fitting unique algebraic or analytic frameworks. Moreover, broadening the comprehension of the Dunkl-type operator within a quantum or differential context could enrich the theoretical underpinnings and practical applications ranging from quantum mechanics to computational mathematics.

Conclusion

The presented research uncovers substantial theoretical implications by expanding the canonical understanding of classical orthogonal polynomials. Vinet and Zhedanov's paper invites the academic community to reflect on traditional mathematical frameworks under novel assumptions $(q \rightarrow -1)$, offering new insights into the algebraic structures governing orthogonality and polynomial transformations. This paper holds the potential to incite further advancements and refinements across mathematical and applied disciplines.