Algebraic Leonard trio approach to rational functions: the Hahn case

Abstract: The finite families of Hahn polynomials and associated biorthogonal rational functions are interpreted algebraically in the framework of Leonard trios. We introduce the trio Hahn algebra and prove that it is isomorphic to the meta Hahn algebra, thereby clarifying the structural connection between Leonard trios and meta algebras. Finite dimensional realizations in terms of difference operators are constructed, and the functions of interest arise as overlaps between eigensolutions of ordinary eigenvalue problems. Their bispectral and biorthogonality properties follow naturally from the algebraic framework.

• The paper introduces the trio Hahn algebra, proving its isomorphism with the meta Hahn algebra for rational Hahn functions.
• It constructs explicit difference operator realizations that reveal the Leonard trio structure and corresponding eigenbasis interrelations.
• The findings rigorously derive recurrence relations and biorthogonality of Hahn polynomials, unifying bispectral theories.

Algebraic Leonard Trio Structures and Rational Hahn Functions

Introduction

The paper "Algebraic Leonard trio approach to rational functions: the Hahn case" (2605.19475) advances the algebraic foundation of the theory of bispectral functions by leveraging the recent concept of Leonard trios in the context of Hahn polynomials and associated biorthogonal rational functions. The work rigorously constructs the so-called trio Hahn algebra, proves its isomorphism to the known meta Hahn algebra, and relates these abstract algebraic entities to concrete realizations via difference operators. These developments not only clarify the structural role of Leonard trios but also unify and extend prior work on meta algebras, leading to deeper algebraic insights into the bispectral properties of rational functions associated with the Hahn class.

Algebraic Setting: Meta Hahn and Trio Hahn Algebras

The meta Hahn algebra, $m\mathfrak{H}$, is introduced as an associative algebra generated by elements $X$, $V$, and $Z$ with additional central elements $\eta$, $\xi$, and a Casimir $Q$. The algebra encapsulates the commutator structure associated with orthogonal polynomials of Hahn type. Notably, in finite-dimensional representations, the meta Hahn algebra is realized via difference operators acting on $\mathbb{C}_N[x]$, the vector space of polynomials of degree at most $N$.

A pivotal technical development in the paper is the passage to the "trio" viewpoint: recognizing that in contexts where $Z$ is invertible, one can define a third operator $X$0. By considering the algebra generated by $X$1, $X$2, and $X$3 (along with $X$4), the authors axiomatize the trio Hahn algebra, $X$5, via a set of nontrivial commutation and anti-commutation relations. They then prove an explicit isomorphism between $X$6 (with $X$7 adjoined) and $X$8, showing these algebraic frameworks are structurally equivalent. This bridges Leonard trios with the prevailing meta-algebraic perspective and supports their use as an organizing algebraic principle.

Difference Operator Realizations and Leonard Trio Construction

Explicit realizations are constructed for both $X$9 and $V$0 as families of difference operators in the basis of polynomials. These realizations, parameterized by constants $V$1, $V$2, and $V$3, produce concrete operator expressions for $V$4, $V$5, $V$6, and $V$7.

The Leonard trio structure is made manifest through these realizations. The following key properties hold:

• In one basis, $V$8 is diagonal while $V$9 and $Z$0 are tridiagonal.
• In a second basis, $Z$1 is diagonal while $Z$2 and $Z$3 are tridiagonal.

This tridiagonal/diagonal interplay defines the Leonard trio and facilitates a transparent algebraic derivation of the salient properties of both Hahn polynomials and associated rational functions.

Hahn Polynomials: Algebraic Properties via Leonard Trios

Within this framework, the overlap coefficients between eigenbases of $Z$4 and $Z$5 (a shifted version of $Z$6 and $Z$7) are exactly Hahn polynomials. The algebraic perspective directly yields their three-term recurrence relations, difference equations, and orthogonality relations via operator identities and basis transformations. The connection coefficients and explicit expressions for eigenvectors in terms of Pochhammer symbols allow for closed-form summation formulas and recursion relations.

Proposition 5.1 presents detailed identities expressing each polynomial basis in terms of the others, with overlap coefficients given by terminating $Z$8 hypergeometric series—the defining form of Hahn polynomials.

Bispectral Rational Hahn Functions and Biorthogonality

Extending the bispectral framework, the paper analyzes rational functions of Hahn type, realized as linear combinations (via explicit connection formulas) between polynomial eigenbases of different operators. These functions have bispectral properties, i.e., they are eigenfunctions of two distinct commuting difference operators. The overlap identities and biorthogonality relations for these rational functions are algebraically derived using the representation theory of the trio Hahn algebra. The weight functions and normalization factors for the biorthogonality relations are specified explicitly.

The algebraic derivation recovers the known analytical results for the biorthogonality of biorthogonal rational Hahn functions, but with proofs that are entirely representation-theoretic in origin, utilizing the interplay between operators in the Leonard trio.

Implications and Future Directions

The unification and equivalence of the trio Hahn algebra and meta Hahn algebra clarify the fundamental algebraic underpinnings of bispectrality for Hahn-type rational functions. This result supports the broader program of classifying and constructing all bispectral (biorthogonal) rational functions via the combinatorics and representation theory of meta algebras and Leonard trios.

Practically, this opens up systematic algebraic methods to derive recurrence, difference, and orthogonality properties in polynomial and rational families further up the ($Z$9-)Askey scheme, and for their degenerations. The explicit difference realizations provide connections to finite oscillator models and spectral theory in mathematical physics.

Theoretically, a full classification of Leonard trios and their associated algebras would provide an analogous framework to Leonard pairs and the ($\eta$0-)Askey scheme for orthogonal polynomials. The paper directly suggests generalizations to the $\eta$1-Hahn, Racah, and Wilson families. These directions are likely to yield new algebraic insights in the theory of bispectrality, deepen the understanding of the meta algebra hierarchy, and potentially lead to new families of special functions.

Conclusion

This work rigorously establishes the algebraic basis for the theory of rational Hahn functions using the framework of Leonard trios. By introducing and characterizing the trio Hahn algebra and proving its isomorphism with the meta Hahn algebra, the paper integrates and strengthens two principal algebraic approaches to bispectrality. The explicit realizations illuminate the operator-theoretic origins of the bispectral and biorthogonality properties of rational Hahn functions, and the results underscore the utility of Leonard trios as a general organizing structure for bispectral rational functions. Future research is expected to build on these algebraic foundations to produce further systematic theory for other hypergeometric and $\eta$2-hypergeometric families.