Dual Hahn Polynomials in Discrete Orthogonal Systems

• Dual Hahn polynomials are finite, discrete orthogonal polynomials defined on a quadratic lattice using terminating {}_3F2 series.
• They exhibit bispectral duality through Leonard-pair structures, satisfying both three-term recurrences and second-order difference equations.
• They underpin applications in algebraic oscillator models, spin chain transport, and systematic deformations within the Askey scheme.

Dual Hahn polynomials are a family of finite, discrete orthogonal polynomials in the Askey scheme. In the standard formulation they are polynomials of degree $n=0,1,\dots,N$ in the quadratic lattice variable $\lambda(x)=x(x+\gamma+\delta+1)$, admit terminating ${}_3F_2(1)$ representations, are orthogonal on the finite set $x=0,1,\dots,N$, and realize a bispectral duality with Hahn polynomials through a three-term recurrence in the degree and a second-order difference equation in the lattice index (Oste et al., 2015).

1. Classical definition, lattice, and orthogonality

A standard definition is

$R_n(\lambda(x);\gamma,\delta,N) = {}_3F_2\!\left( \begin{matrix} -x,\;x+\gamma+\delta+1,\;-n\[2pt] \gamma+1,\;-N \end{matrix};1 \right), \qquad \lambda(x)=x(x+\gamma+\delta+1),$

with $n=0,1,\dots,N$ and $x=0,1,\dots,N$. Because the numerator parameter $-n$ terminates the series, $R_n$ is a polynomial of degree $n$ in $\lambda(x)=x(x+\gamma+\delta+1)$0. Typical parameter ranges ensuring positivity of the weight are $\lambda(x)=x(x+\gamma+\delta+1)$1 or $\lambda(x)=x(x+\gamma+\delta+1)$2 (Oste et al., 2015).

The orthogonality relation is

$\lambda(x)=x(x+\gamma+\delta+1)$3

with

$\lambda(x)=x(x+\gamma+\delta+1)$4

and

$\lambda(x)=x(x+\gamma+\delta+1)$5

The orthonormal functions are

$\lambda(x)=x(x+\gamma+\delta+1)$6

A common alternative notation uses parameters $\lambda(x)=x(x+\gamma+\delta+1)$7 and

$\lambda(x)=x(x+\gamma+\delta+1)$8

together with polynomials $\lambda(x)=x(x+\gamma+\delta+1)$9. This suggests that much of the literature differs mainly by normalization and parameter relabeling rather than by substance (Duran, 2021).

2. Recurrence, duality, and Leonard-pair structure

Dual Hahn polynomials satisfy the three-term recurrence

${}_3F_2(1)$0

where

${}_3F_2(1)$1

and

${}_3F_2(1)$2

This is the standard orthogonal-polynomial recurrence in the degree index ${}_3F_2(1)$3 (Oste et al., 2015).

The reason for the qualifier “dual” is Leonard duality. In one formulation, the Hahn polynomials ${}_3F_2(1)$4 satisfy a second-order difference equation

${}_3F_2(1)$5

and the duality relation

${}_3F_2(1)$6

Up to known prefactors, the degree and the discrete variable are interchanged. In the same notation, the dual Hahn family is therefore bispectral: a Jacobi-type three-term recurrence acts in ${}_3F_2(1)$7, while a second-order difference operator acts in the lattice index ${}_3F_2(1)$8 (Duran, 2021).

This duality has a linear-algebraic realization through Leonard pairs. For the family ${}_3F_2(1)$9 in the dual Hahn regime, one fixes

$x=0,1,\dots,N$0

equips $x=0,1,\dots,N$1 with

$x=0,1,\dots,N$2

and obtains a Leonard pair $x=0,1,\dots,N$3 such that $x=0,1,\dots,N$4 encodes the three-term recurrence

$x=0,1,\dots,N$5

with

$x=0,1,\dots,N$6

while $x=0,1,\dots,N$7 encodes the difference equation in the grid index $x=0,1,\dots,N$8. This is the finite-dimensional algebraic skeleton of the dual Hahn family (Huang, 4 Aug 2025).

3. Position in the Askey scheme and structural transformations

At the finite-discrete level of the Askey scheme, Racah polynomials are the top four-parameter family, Hahn and dual Hahn are limit cases of Racah polynomials, and Krawtchouk polynomials arise as further limits (Oste et al., 2015). In a complementary formulation, dual Hahn polynomials sit in the “middle layer” of the Askey scheme, just below Racah polynomials (Huang, 4 Aug 2025).

A systematic deformation theory is given by the classification of dual Hahn “doubles”: pairs of coupled recurrence relations intertwining two dual Hahn families with shifted parameters. There are exactly three cases:

Double	Partner family	Christoffel parameter
Dual Hahn I	$x=0,1,\dots,N$9	$R_n(\lambda(x);\gamma,\delta,N) = {}_3F_2\!\left( \begin{matrix} -x,\;x+\gamma+\delta+1,\;-n\[2pt] \gamma+1,\;-N \end{matrix};1 \right), \qquad \lambda(x)=x(x+\gamma+\delta+1),$0
Dual Hahn II	$R_n(\lambda(x);\gamma,\delta,N) = {}_3F_2\!\left( \begin{matrix} -x,\;x+\gamma+\delta+1,\;-n\[2pt] \gamma+1,\;-N \end{matrix};1 \right), \qquad \lambda(x)=x(x+\gamma+\delta+1),$1	$R_n(\lambda(x);\gamma,\delta,N) = {}_3F_2\!\left( \begin{matrix} -x,\;x+\gamma+\delta+1,\;-n\[2pt] \gamma+1,\;-N \end{matrix};1 \right), \qquad \lambda(x)=x(x+\gamma+\delta+1),$2
Dual Hahn III	$R_n(\lambda(x);\gamma,\delta,N) = {}_3F_2\!\left( \begin{matrix} -x,\;x+\gamma+\delta+1,\;-n\[2pt] \gamma+1,\;-N \end{matrix};1 \right), \qquad \lambda(x)=x(x+\gamma+\delta+1),$3	$R_n(\lambda(x);\gamma,\delta,N) = {}_3F_2\!\left( \begin{matrix} -x,\;x+\gamma+\delta+1,\;-n\[2pt] \gamma+1,\;-N \end{matrix};1 \right), \qquad \lambda(x)=x(x+\gamma+\delta+1),$4

These are the only ways to combine two dual Hahn families so that they satisfy a coupled pair of relations of the required form. The same classification identifies the Christoffel transform parameters for which the kernel partner remains inside the dual Hahn class, namely $R_n(\lambda(x);\gamma,\delta,N) = {}_3F_2\!\left( \begin{matrix} -x,\;x+\gamma+\delta+1,\;-n\[2pt] \gamma+1,\;-N \end{matrix};1 \right), \qquad \lambda(x)=x(x+\gamma+\delta+1),$5. Each double yields a new symmetric orthogonal system and a two-diagonal Jacobi matrix with explicit eigenvalues; in the three dual Hahn cases those eigenvalues are

$R_n(\lambda(x);\gamma,\delta,N) = {}_3F_2\!\left( \begin{matrix} -x,\;x+\gamma+\delta+1,\;-n\[2pt] \gamma+1,\;-N \end{matrix};1 \right), \qquad \lambda(x)=x(x+\gamma+\delta+1),$6

respectively (Oste et al., 2015).

A distinct structural phenomenon occurs in a special Leonard-pair regime. If $R_n(\lambda(x);\gamma,\delta,N) = {}_3F_2\!\left( \begin{matrix} -x,\;x+\gamma+\delta+1,\;-n\[2pt] \gamma+1,\;-N \end{matrix};1 \right), \qquad \lambda(x)=x(x+\gamma+\delta+1),$7 and $R_n(\lambda(x);\gamma,\delta,N) = {}_3F_2\!\left( \begin{matrix} -x,\;x+\gamma+\delta+1,\;-n\[2pt] \gamma+1,\;-N \end{matrix};1 \right), \qquad \lambda(x)=x(x+\gamma+\delta+1),$8, then

$R_n(\lambda(x);\gamma,\delta,N) = {}_3F_2\!\left( \begin{matrix} -x,\;x+\gamma+\delta+1,\;-n\[2pt] \gamma+1,\;-N \end{matrix};1 \right), \qquad \lambda(x)=x(x+\gamma+\delta+1),$9

is again a Leonard pair. In that case the same polynomial family $n=0,1,\dots,N$0 is simultaneously dual Hahn and Racah, with the same discrete inner product, the same support up to reordering, and the same weights up to reordering. This is a non-limiting dual Hahn–Racah coincidence, not a generic identification (Huang, 4 Aug 2025).

4. Higher-order bispectral deformations and multi-indexing

Beyond the classical second-order difference equation, new bispectral dual Hahn families can be constructed from measure transformations. One starts from the dual Hahn measure $n=0,1,\dots,N$1, performs a Geronimus transform with respect to

$n=0,1,\dots,N$2

obtaining a measure $n=0,1,\dots,N$3 depending on a set of continuous parameters

$n=0,1,\dots,N$4

and then optionally applies a Christoffel transform

$n=0,1,\dots,N$5

The orthogonal polynomials for these measures are given by Casoratian determinant formulas, and the basic family $n=0,1,\dots,N$6 exists when the auxiliary determinants $n=0,1,\dots,N$7 do not vanish. These polynomials are eigenfunctions of a higher-order difference operator; in the basic Geronimus case the order is $n=0,1,\dots,N$8. The construction yields the first examples, from the classical discrete families, that depend on an arbitrary number of continuous parameters (Duran, 2021).

Finite-type multi-indexed dual Hahn polynomials arise from Darboux transformations in discrete quantum mechanics with real shifts. In the state-deleting case, one uses seed solutions of degree $n=0,1,\dots,N$9, obtaining Krein–Adler type multi-indexed orthogonal polynomials and a deformed Hamiltonian of order $x=0,1,\dots,N$0. In the state-adding case, one uses seed solutions of degree $x=0,1,\dots,N$1; after $x=0,1,\dots,N$2 steps the resulting Hamiltonian is of order $x=0,1,\dots,N$3, and the new finite type multi-indexed orthogonal polynomials still satisfy second-order difference equations. Dual Hahn is one of the twelve finite families for which this construction is carried out explicitly (Odake, 2022).

5. Related deformations: dual $x=0,1,\dots,N$4 Hahn and generalized duality

A closely related but distinct family is obtained by a $x=0,1,\dots,N$5 limit of the dual $x=0,1,\dots,N$6-Hahn polynomials: the dual $x=0,1,\dots,N$7 Hahn polynomials. They are orthogonal on a finite set of discrete points on the real axis, but in contrast to the classical orthogonal polynomials of the Askey scheme they do not exhibit the Leonard duality property. Instead, they satisfy a 4-th order difference eigenvalue equation and possess a bispectrality property. The corresponding generalized Leonard pair consists of two matrices $x=0,1,\dots,N$8 each of size $x=0,1,\dots,N$9; in the eigenbasis where $-n$0 is diagonal, $-n$1 is 3-diagonal, while in the eigenbasis where $-n$2 is diagonal, $-n$3 is 5-diagonal (Tsujimoto et al., 2011).

The associated algebraic structure is the algebra $-n$4, a two-parameter generalization of $-n$5 with an involution as additional generator. It is derived from the recurrence relation of the dual $-n$6 Hahn polynomials and realized in terms of two added $-n$7 algebras. In this realization, the Clebsch–Gordan coefficients of $-n$8 are dual $-n$9 Hahn polynomials, and one obtains an irreducible representation of $R_n$0 involving five-diagonal matrices, reflecting the five-term difference equation (Genest et al., 2012).

The same deformation appears in superintegrable models. A two-dimensional superintegrable system of singular oscillators with internal degrees of freedom has symmetry algebra equal to the dual $R_n$1 Hahn algebra, and the overlap coefficients between the uncoupled and coupled $R_n$2 bases are dual $R_n$3 Hahn polynomials. This places the $R_n$4 deformation in the representation theory of $R_n$5 and in superconformal quantum mechanics (Bernard et al., 2020).

A common misconception is to treat dual $R_n$6 Hahn as a minor normalization change of classical dual Hahn. The literature instead isolates a genuine structural break: the loss of Leonard duality and the appearance of a higher-order difference equation.

6. Multivariate forms and applications

Dual Hahn structure persists in several-variable settings. The bivariate dual Hahn polynomials of Tratnik type are defined on the triangular lattice

$R_n$7

by

$R_n$8

and satisfy the orthogonality relation

$R_n$9

under

$n$0

Their recurrence coefficients

$n$1

can be identified directly with couplings and magnetic fields in a two-dimensional spin-$n$2 XX lattice on a triangular grid, yielding an exactly solvable one-excitation Hamiltonian (Miki et al., 2020).

This solvable lattice supports perfect state transfer and fractional revival for explicit parameter choices. One family is

$n$3

for which perfect state transfer between $n$4 and $n$5 occurs at $n$6. Another is

$n$7

for which perfect state transfer occurs at $n$8; with $n$9 odd, fractional revival occurs at $\lambda(x)=x(x+\gamma+\delta+1)$00 (Miki et al., 2020).

In one-dimensional models, dual Hahn doubles yield finite oscillator systems. The position operator is represented by an explicit two-diagonal matrix, its spectrum is given by the corresponding $\lambda(x)=x(x+\gamma+\delta+1)$01, and the position wavefunctions are expressed through the symmetric orthogonal systems built from doubled dual Hahn polynomials. The same analysis also produces two-parameter extensions of the Sylvester–Kac matrix with explicit eigenvalues and eigenvectors, so dual Hahn polynomials enter both algebraic oscillator models and numerical test-matrix theory (Oste et al., 2015).

A further deformation-driven application occurs for dual $\lambda(x)=x(x+\gamma+\delta+1)$02 Hahn polynomials in XX spin chains with perfect state transfer. For odd $\lambda(x)=x(x+\gamma+\delta+1)$03 one recovers a previously known model; for even $\lambda(x)=x(x+\gamma+\delta+1)$04 one obtains a new chain whose Jacobi matrix is mirror-symmetric and whose spectrum is a Bannai–Ito grid split into two arithmetic progressions. This suggests that the role of dual Hahn polynomials in transport models extends naturally to their $\lambda(x)=x(x+\gamma+\delta+1)$05 descendants (Vinet et al., 2011).