Trio Hahn Algebra Overview

• Trio Hahn algebra is a three-generator structure related to Hahn polynomials, Hahn-type biorthogonal rational functions, and discrete superintegrable systems.
• It unifies different Hahn algebra formulations via explicit difference operator realizations and embeddings in Lie algebras such as sl2.
• Finite-dimensional modules yield concrete overlap coefficients, special functions, and recurrence relations that encapsulate bispectrality.

“Trio Hahn algebra” denotes a family of closely related three-generator algebraic structures attached to Hahn polynomials, Hahn-type biorthogonal rational functions, and discrete superintegrable systems. In the literature represented here, the term appears for a quadratic algebra $R_H$ generated by three difference operators $X,Y,Z$ on the uniform grid (Tsujimoto et al., 2020), for the meta-Hahn algebra $m\mathfrak{H}$ with generators $X,Z,V$ that contains both the Hahn algebra and the rational-Hahn algebra (Vinet et al., 2020), for a simplified “meta-Hahn (Trio) algebra” used to unify finite Hahn polynomial and rational-function families (Tsujimoto et al., 2024), and for a trio Hahn algebra $t\mathfrak{h}$ proved to be isomorphic to the meta-Hahn algebra after adjoining $Z^{-1}$ (Crampé et al., 19 May 2026). A recurrent theme is bispectrality: the same algebra controls both recurrence relations and difference or generalized eigenvalue equations.

1. Terminology and principal formulations

In the papers surveyed here, the expression “Trio Hahn algebra” labels several three-generator structures attached to Hahn-type bispectral problems.

Formulation	Generators	Role
$R_H$ (Tsujimoto et al., 2020)	$X,Y,Z$	Quadratic algebra for ${}_3F_2$ biorthogonal rational functions
$m\mathfrak{H}$ (Vinet et al., 2020)	$X,Y,Z$0	Unified algebra containing Hahn and rational-Hahn algebras
Meta-Hahn (Trio) algebra (Tsujimoto et al., 2024)	$X,Y,Z$1	Unified treatment of Hahn polynomials and Hahn-type rational functions
$X,Y,Z$2 (Crampé et al., 19 May 2026)	$X,Y,Z$3	Leonard-trio formulation, isomorphic to $X,Y,Z$4
2D symmetry algebra (Iliev et al., 2017)	$X,Y,Z$5	Hidden symmetry of Hahn-polynomial superintegrable systems

This nonuniform usage is structurally significant rather than merely terminological. In each case, the algebra is three-generated, admits a central element or Hamiltonian, and organizes a finite-dimensional representation in which one basis diagonalizes one generator while another generator is tridiagonal or bidiagonal. The one-variable Hahn algebra itself appears as a two-generator subalgebra or degeneration in several of these constructions (Genest et al., 2015).

2. The difference-operator trio Hahn algebra $X,Y,Z$6

A concrete three-generator quadratic algebra was given on the $X,Y,Z$7-dimensional space

$X,Y,Z$8

with parameters $X,Y,Z$9, by the operators

$m\mathfrak{H}$0

$m\mathfrak{H}$1

$m\mathfrak{H}$2

where

$m\mathfrak{H}$3

$m\mathfrak{H}$4

$m\mathfrak{H}$5

In the monomial basis $m\mathfrak{H}$6, $m\mathfrak{H}$7 and $m\mathfrak{H}$8 are bidiagonal and $m\mathfrak{H}$9 is tridiagonal (Tsujimoto et al., 2020).

These operators satisfy the quadratic commutation relations

$X,Z,V$0

$X,Z,V$1

$X,Z,V$2

with structure constants

$X,Z,V$3

$X,Z,V$4

These equations define the Trio Hahn quadratic algebra $X,Z,V$5. The cubic central element $X,Z,V$6 is explicit, and in the difference-operator realization one finds

$X,Z,V$7

(Tsujimoto et al., 2020).

The algebra encodes the bispectrality of the Hahn-type rational functions through the generalized eigenvalue problem

$X,Z,V$8

with normalized solution

$X,Z,V$9

The paper further gives three equivalent $t\mathfrak{h}$0-dimensional irreducible representations: the monomial basis, a “Pochhammer-ratio” basis

$t\mathfrak{h}$1

and the rational-function basis $t\mathfrak{h}$2, in which all three generators become tridiagonal (Tsujimoto et al., 2020).

3. The meta-Hahn algebra and unification of Hahn and rational-Hahn structures

A broader three-generator algebraic framework was introduced as the meta-Hahn algebra $t\mathfrak{h}$3. In its general form, one fixes real parameters $t\mathfrak{h}$4 and a central charge $t\mathfrak{h}$5, and defines the unital associative algebra over $t\mathfrak{h}$6 generated by $t\mathfrak{h}$7 with

$t\mathfrak{h}$8

$t\mathfrak{h}$9

$Z^{-1}$0

The algebra admits the central element

$Z^{-1}$1

(Vinet et al., 2020).

A key internal component is the two-generator subalgebra $Z^{-1}$2, which obeys

$Z^{-1}$3

This subalgebra is described as the real form of the “deformed Jordan plane,” and as a PBW-deformation of the Jordan plane. In finite-dimensional representations, it underlies the Hahn recurrence structure (Vinet et al., 2020).

The same paper shows two embeddings. First, if

$Z^{-1}$4

then $Z^{-1}$5 satisfy the quadratic Hahn algebra relations, so the usual Hahn algebra injects into $Z^{-1}$6 via

$Z^{-1}$7

Second, if

$Z^{-1}$8

then $Z^{-1}$9 satisfy the defining cubic commutation relations of the rational-Hahn algebra, so

$R_H$0

realizes the rational-Hahn algebra inside $R_H$1 (Vinet et al., 2020).

A simplified form of the same unifying idea was used later under the label “Meta-Hahn (Trio) Algebra,” with defining relations

$R_H$2

and Casimir

$R_H$3

For

$R_H$4

one obtains the standard Hahn algebra relations, so $R_H$5 contains the usual two-generator Hahn algebra as the subalgebra generated by $R_H$6 (Tsujimoto et al., 2024).

4. Finite-dimensional modules, overlap coefficients, and special functions

Finite-dimensional representation theory is the point at which the trio Hahn constructions become a framework for concrete special functions. In one formulation, $R_H$7 acts on an $R_H$8-dimensional real vector space $R_H$9 with nondegenerate bilinear form, and four natural bases are constructed: a GEVP basis $X,Y,Z$0, an adjoint GEVP basis $X,Y,Z$1, the eigenbasis $X,Y,Z$2 of $X,Y,Z$3, and the pencil eigenbasis $X,Y,Z$4 of $X,Y,Z$5. Their overlaps produce four families of special functions. In particular,

$X,Y,Z$6

gives the bispectral rational Hahn functions, while

$X,Y,Z$7

and

$X,Y,Z$8

are proportional to the Hahn orthogonal polynomials. The paper states biorthogonality for $X,Y,Z$9 and orthogonality for the Hahn polynomials through these overlaps (Vinet et al., 2020).

In the ${}_3F_2$0-dimensional “two-diagonal” representation of the simplified meta-Hahn algebra, one takes a basis ${}_3F_2$1, fixes ${}_3F_2$2, imposes

${}_3F_2$3

and defines

${}_3F_2$4

${}_3F_2$5

${}_3F_2$6

The associated spectral data are

${}_3F_2$7

All of the corresponding eigenvectors can be expanded explicitly on ${}_3F_2$8 in closed form by terminating ${}_3F_2$9-series (Tsujimoto et al., 2024).

Two kinds of overlap coefficients appear. The EVP-EVP overlaps

$m\mathfrak{H}$0

are proportional to the classical Hahn polynomial

$m\mathfrak{H}$1

with

$m\mathfrak{H}$2

Their orthogonality follows from the biorthonormality of the eigenbases: $m\mathfrak{H}$3

The GEVP-EVP overlaps

$m\mathfrak{H}$4

coincide, up to simple pre-factors in Pochhammer symbols, with the biorthogonal partners

$m\mathfrak{H}$5

$m\mathfrak{H}$6

with

$m\mathfrak{H}$7

The biorthogonality is stated in terms of explicit weights $m\mathfrak{H}$8 and normalizations $m\mathfrak{H}$9 (Tsujimoto et al., 2024).

5. Leonard trios, concrete realizations, and Lie-theoretic embeddings

A recent reformulation introduces the trio Hahn algebra $X,Y,Z$00 with generators

$X,Y,Z$01

central parameters $X,Y,Z$02, and defining relations

$X,Y,Z$03

$X,Y,Z$04

$X,Y,Z$05

$X,Y,Z$06

The paper proves that, after adjoining $X,Y,Z$07, this algebra is isomorphic to the meta-Hahn algebra $X,Y,Z$08 under

$X,Y,Z$09

with inverse

$X,Y,Z$10

This is the explicit statement that clarifies the structural connection between Leonard trios and meta algebras (Crampé et al., 19 May 2026).

The same work gives a finite-dimensional difference-operator realization on $X,Y,Z$11, $X,Y,Z$12: $X,Y,Z$13

$X,Y,Z$14

$X,Y,Z$15

with

$X,Y,Z$16

Its eigenbases include the $X,Y,Z$17-eigenbasis

$X,Y,Z$18

the $X,Y,Z$19-eigenbasis

$X,Y,Z$20

and the generalized $X,Y,Z$21-eigenbasis

$X,Y,Z$22

The ordinary Hahn polynomials and Hahn-type rational functions arise as overlaps between these bases, and because $X,Y,Z$23 form a Leonard trio, these overlaps encode bispectrality (Crampé et al., 19 May 2026).

The meta-Hahn framework also admits differential and difference models and an embedding into $X,Y,Z$24. One differential-operator model uses

$X,Y,Z$25

$X,Y,Z$26

$X,Y,Z$27

and realizes the defining relations with $X,Y,Z$28 and $X,Y,Z$29. The paper further presents an embedding of $X,Y,Z$30 in $X,Y,Z$31, and in finite-dimensional $X,Y,Z$32-modules of spin $X,Y,Z$33 this reproduces the differential model. A Padé approximation table for the binomial function $X,Y,Z$34 is obtained as a by-product (Vinet et al., 2020).

6. Related usages: superintegrability, Heun-Hahn extensions, and $X,Y,Z$35-analogues

In the two-dimensional Hahn-polynomial system on a hexagon, the “Trio-Hahn” algebra appears as the hidden symmetry algebra generated by

$X,Y,Z$36

where

$X,Y,Z$37

The discrete Hamiltonian is

$X,Y,Z$38

and one has

$X,Y,Z$39

The operators satisfy the Kohno-Drinfeld relations

$X,Y,Z$40

and close quadratically into a three-generator algebra. In this setting, the Trio-Hahn algebra is described as the discrete hidden symmetry underlying the two-dimensional Hahn-polynomial superintegrable system (Iliev et al., 2017).

A different adjacent construction is the Heun-Hahn extension of the Hahn algebra. Starting from the Hahn generators $X,Y,Z$41, one adjoins the Heun-Hahn operator

$X,Y,Z$42

The resulting algebra is finitely closed: no new generators are produced, and one ends up with exactly three “basic” generators $X,Y,Z$43. The paper explicitly states that this three-fold closure is what it calls the “Trio Hahn algebra” (Vinet et al., 2018).

The one-variable Hahn algebra itself also arises as a degeneration of the tridiagonalization of the hypergeometric operator. In the degenerate case $X,Y,Z$44, the tridiagonalized operator becomes first order, preserves $X,Y,Z$45, and together with the Jacobi operator closes the Hahn algebra. The expansion coefficients between the corresponding eigenbases are the classical Hahn polynomials (Genest et al., 2015).

A $X,Y,Z$46-analogue appears in the Hahn specialization of the Askey-Wilson algebra $X,Y,Z$47, with generators $X,Y,Z$48 satisfying

$X,Y,Z$49

In the $X,Y,Z$50 realization, the overlap coefficients between uncoupled and coupled bases are $X,Y,Z$51-Hahn polynomials, and the algebraic relations encode their recurrence, difference, and duality properties. Here the “trio” language refers to the three-generator specialization of $X,Y,Z$52 underlying the Hahn problem for $X,Y,Z$53 (Lavrenov, 2021).

Across these formulations, the common role of the trio Hahn algebra is to package finite-dimensional bispectral data into a three-generator algebra with a central element, concrete operator realizations, and overlap coefficients given by Hahn polynomials or Hahn-type biorthogonal rational functions. The later meta-Hahn and Leonard-trio papers make this unifying role explicit by placing the polynomial and rational-function families in a single algebraic structure (Tsujimoto et al., 2024, Crampé et al., 19 May 2026).