---
title: 'Racah Polynomials: Hypergeometric Orthogonal Systems'
url: https://www.emergentmind.com/topics/racah-polynomials
type: topic
---

# Racah Polynomials: Hypergeometric Orthogonal Systems

Searching arXiv for recent and foundational papers on Racah polynomials to ground the encyclopedia entry.
Racah polynomials are a finite family of terminating Saalschützian hypergeometric orthogonal polynomials on a quadratic lattice. In the standard hypergeometric form they are given by
\[
R_n(X(x);\alpha,\beta,\gamma,\delta)
={}_4F_3\!\left(\begin{matrix}
-n,\ n+\alpha+\beta+1,\ -x,\ x+\gamma+\delta+1\\
\alpha+1,\ \beta+\delta+1,\ \gamma+1
\end{matrix};1\right),
\qquad n=0,1,2,\dots,N,
\]
with
\[
X(x)=x(x+\gamma+\delta+1),
\]
and one of the truncation conditions
\[
\alpha+1=-N \quad\text{or}\quad \beta+\delta+1=-N \quad\text{or}\quad \gamma+1=-N
\]
for a non-negative integer \(N\) [1412.7115]. They belong to the classical Racah level of the Askey scheme and are among the most general finite-variable hypergeometric orthogonal polynomials; many other discrete orthogonal polynomial systems arise as degenerations or limits of Racah polynomials [1412.7115].

## 1. Hypergeometric form and quadratic-lattice structure

The defining hypergeometric series is built from the generalized hypergeometric function
\[
{}_pF_q\!\left(\begin{matrix} a_1,a_2,\dots,a_p\\ b_1,b_2,\dots,b_q\end{matrix};z\right)
=\sum_{n=0}^\infty \frac{(a_1)_n(a_2)_n\cdots(a_p)_n}{n!(b_1)_n(b_2)_n\cdots(b_q)_n}z^n,
\]
where \((a)_n\) is the rising factorial [1412.7115]. In the Racah case the \({}_4F_3(1)\) is terminating and Saalschützian, and the natural variable is not \(x\) itself but the quadratic lattice \(X(x)\) [1412.7115].

A standard univariate normalization used in multivariate and divided-difference treatments is
\[
r_n(\alpha,\beta,\gamma,\delta;s)
=(\alpha+1)_n(\beta+\delta+1)_n(\gamma+1)_n
\,{}_4F_3\!\left(
\begin{matrix}
-n,\ n+\alpha+\beta+1,\ -s,\ s+\gamma+\delta+1\\
\beta+1,\ \delta+\beta+1,\ \gamma+1
\end{matrix};1\right),
\]
which is of degree \(n\) in the quadratic lattice
\[
\eta(s)=s(s+\gamma+\delta+1)
\]
[1605.09280]. In discrete quantum-mechanical notation, ordinary Racah polynomials also appear as
\[
P_n(x;\lambda)= {}_4F_3\!\left(
\begin{matrix}
-n,\ n+\tilde d,\ -x,\ x+d\\
a,\ b,\ c
\end{matrix};1\right),
\]
with
\[
\eta(x;\lambda)=x(x+d), \qquad \mathcal E_n(\lambda)=n(n+\tilde d),
\]
and \(\tilde d=a+b+c-d-1\) [1102.0812, 1805.00345].

The classical difference-equation side is equally central. In the univariate quadratic-lattice formulation, Racah polynomials satisfy
\[
\phi(\eta(s))D_\eta^2 r_n(s)+\tau(\eta(s))S_\eta D_\eta r_n(s)+\lambda_n r_n(s)=0,
\]
with divided-difference and averaging operators
\[
D_\eta f(s)=\frac{f(s+\tfrac12)-f(s-\tfrac12)}{\eta(s+\tfrac12)-\eta(s-\tfrac12)},
\qquad
S_\eta f(s)=\frac{f(s+\tfrac12)+f(s-\tfrac12)}{2}
\]
[1605.09280]. This quadratic-lattice formulation is the basis for multivariate generalizations and for the interpretation of Racah polynomials as bispectral objects.

## 2. Algebraic and representation-theoretic interpretations

The algebraic structure behind Racah polynomials is the Racah algebra, a quadratic algebra generated by two algebraically independent operators \(A,B\) with commutator \(C=[A,B]\) and defining relations
\[
[A,B]=C,
\]
\[
[B,C]=B^2+\{A,B\}+d\,B+e_1,
\]
\[
[C,A]=A^2+\{A,B\}+d\,A+e_2,
\]
together with a Casimir operator
\[
Q= \{A^2,B\}+\{A,B^2\}+A^2+B^2+C^2+(d+1)\{A,B\}+(2e_1+d)A+(2e_2+d)B
\]
[1309.3540]. In finite-dimensional irreducible modules, one may choose a basis in which \(A\) is diagonal and \(B\) is tridiagonal, and vice versa; thus \((A,B)\) form a Leonard pair, and the transition coefficients between the eigenbasis of \(A\) and that of \(B\) are exactly Racah polynomials [1309.3540].

This Leonard-pair viewpoint is sharpened by the \(\mathfrak{su}(1,1)\) Racah problem. For three mutually commuting copies of \(\mathfrak{su}(1,1)\), with intermediate Casimirs \(\mathcal Q^{(12)},\mathcal Q^{(23)},\mathcal Q^{(31)}\), setting
\[
A=-\frac{\mathcal Q^{(12)}}{2},\qquad B=-\frac{\mathcal Q^{(23)}}{2}
\]
realizes the Racah algebra, and the overlap coefficients between eigenbases of \(\mathcal Q^{(12)}\) and \(\mathcal Q^{(23)}\) are the Racah coefficients, or \(6j\)-symbols, hence Racah polynomials [1309.3540]. The corresponding one-variable reduction yields explicit hypergeometric Sturm–Liouville operators \(\mathcal S_{12},\mathcal S_{23},\mathcal S_{31}\) whose polynomial eigenfunctions are hypergeometric and whose overlap coefficients are the polynomial objects of Racah type [1309.3540].

The same recoupling picture extends to \(\mathfrak{sl}(2)\)-type representation theory in a different guise. For Rankin–Cohen brackets, the coefficients in the recoupling identity
\[
[[f_1,f_2]_k,f_3]_{n-k}
=
\sum_{p=0}^n
X^{\lambda_1,\lambda_2;k}_{\lambda_3;n,p}\,
[f_1,[f_2,f_3]_p]_{n-p}
\]
are expressed in terms of the Racah values
\[
R_{p,k} =
{}_4F_3\!\left(
\begin{matrix}
-p,\ p+\lambda_2+\lambda_3-1,\ -k,\ k+\lambda_1+\lambda_2-1\\
\lambda_2,\ \lambda_1+\lambda_2+\lambda_3+n-1,\ -n
\end{matrix};1\right),
\]
so the transition matrix between two coupling bases of a triple tensor product is again given by Racah polynomials [2304.10803]. This places the classical hypergeometric formulas and the recoupling-coefficient interpretation within a common algebraic framework.

A more recent extension is the meta Racah algebra, generated by \(X,V,Z\) with
\[
[Z,X]=Z^2+X,\qquad
[X,V]=\{V,Z\}+2\zeta X+2\zeta^2 Z+\xi I,\qquad
[V,Z]=V+2X+2\zeta Z+\eta I,
\]
whose finite-dimensional representations yield both classical Racah polynomials and finite families of Racah-type biorthogonal rational functions as overlap coefficients [2603.28839]. In this setting the standard Racah family arises from overlaps between two ordinary eigenbases, while the rational extensions arise from overlaps involving a generalized eigenvalue problem [2603.28839].

## 3. Recoupling problems, superintegrability, and multivariate Racah polynomials

For four copies of \(\mathfrak{su}(1,1)\), the recoupling scheme produces bivariate Racah polynomials. In the basis \(|\nu_{12},\nu_{123},\nu\rangle\) associated with the chain \(\{Q^{(12)},Q^{(123)}\}\), and the basis \(|\nu_{23},\nu_{234},\nu\rangle\) associated with \(\{Q^{(23)},Q^{(234)}\}\), the overlap coefficients are precisely bivariate Racah polynomials of Tratnik type [1504.03705]. The same problem is equivalent to the generic quantum superintegrable system on \(S^3\), whose Hamiltonian is the total Casimir of four \(\mathfrak{su}(1,1)\) copies, and the overlap coefficients are the expansion coefficients between separated eigenbases in different spherical coordinate systems [1504.03705].

An explicit bivariate Racah family used in the divided-difference treatment is
\[
R_{n,m}(s,t;\beta_0,\beta_1,\beta_2,\beta_3,N)
=
r_n(\beta_1-\beta_0-1,\beta_2-\beta_1-1,-t-1,\beta_1+t; s)
\]
\[
\times\,
r_m(2n+\beta_2-\beta_0-1,\beta_3-\beta_2-1,n-N-1,n+\beta_2+N; t-n),
\]
with quadratic lattice variables
\[
x(s)=s(s+\beta_1),\qquad y(t)=t(t+\beta_2)
\]
[1605.09280]. These polynomials satisfy a fourth-order linear partial divided-difference equation with polynomial coefficients in \(x(s)\) and \(y(t)\),
\[
f_1D_x^2D_y^2R_{n,m}
+f_2S_xD_xD_y^2R_{n,m}
+f_3S_yD_yD_x^2R_{n,m}
+f_4S_xD_xS_yD_yR_{n,m}
\]
\[
+f_5D_x^2R_{n,m}
+f_6D_y^2R_{n,m}
+f_7S_xD_xR_{n,m}
+f_8S_yD_yR_{n,m}
+(m+n)(\beta_3-\beta_0+m+n-1)R_{n,m}=0,
\]
which is the bivariate analogue of the univariate Racah divided-difference equation [1605.09280]. The same paper derives explicit matrix coefficients for the three-term recurrence relations satisfied by any bivariate orthogonal polynomial solution of that equation, presents the monic bivariate Racah family, and solves the connection problem between two different Tratnik-type families [1605.09280].

Higher-rank algebraic generalizations are realized by the higher rank Racah algebra \(R(n)\), the subalgebra of \(\bigotimes_{i=1}^n U(\mathfrak{su}(1,1))\) generated by all intermediate Casimirs \(C_A\), \(A\subset[n]\) [1808.10520]. In rank one this recovers the classical picture where \(R(3)\) encodes the bispectrality of univariate Racah polynomials; in higher rank, connection coefficients between bases of Dunkl-harmonics diagonalizing different Abelian subalgebras are multivariate Racah polynomials, and the Geronimo–Iliev commuting Racah operators appear as the images of suitable Abelian subalgebras [1808.10520]. The same paper identifies every generator of \(R(n)\) as a discrete operator acting on multivariate Racah polynomials after suitable changes of basis [1808.10520].

## 4. Special subclasses and exact identities

A particularly rigid finite subclass is obtained by the parameter choice
\[
\alpha=\beta=0,\qquad \gamma=T,\qquad \delta=-T,
\]
for a positive integer \(T\). Then the quadratic lattice reduces to
\[
X(s)=s(s+1),
\]
and the corresponding Racah polynomials are
\[
R_n(s,T)
=
R_n(X(s);0,0,T,-T)
=
{}_4F_3\!\left(
\begin{matrix}
-n,\ n+1,\ -s,\ s+1\\
1,\ 1-T,\ 1+T
\end{matrix};1\right),
\qquad 0\le n,s\le T-1
\]
[1412.7115]. These are exactly the polynomials appearing in the Kresch–Tamvakis conjecture
\[
|R_n(s,T)|\le 1
\]
for all \(0\le n,s\le T-1\), \(T\ge 1\) [1412.7115].

For this family, an inversion formula for the discrete transform
\[
\tilde x_n=\sum_{k=0}^n(-1)^k\binom{n}{k}\binom{n+k}{k}x_k
\]
yields the exact relation
\[
\sum_{n=0}^m
(-1)^n
\frac{(2n+1)\binom{m}{n}}{(m+n+1)\binom{m+n}{n}}
\,R_n(s,T)
=
\frac{(1-T)_m(1+T)_m}{(-s)_m(s+1)_m},
\]
equivalently
\[
\frac{(-s)_m(s+1)_m}{(1-T)_m(1+T)_m}
=
\sum_{n=0}^m
\frac{(2n+1)(-m)_n}{(m+1)_{n+1}}
\,R_n(s,T),
\]
for \(0\le m\le T-1\) [1412.7115]. When \(m>s\), the factor \((-s)_m\) vanishes and one obtains the linear relations
\[
\sum_{n=0}^m
(-1)^n
\frac{(2n+1)\binom{m}{n}}{(m+n+1)\binom{m+n}{n}}
\,R_n(s,T)=0,
\qquad s+1\le m\le T-1,
\]
together with the equivalent Pochhammer form [1412.7115].

The Leonard-pair viewpoint also yields nongeneric identifications between families. For dual Hahn polynomials \(\{u_i(x)\}_{i=0}^d\), if
\[
r\ne 0,\qquad r+s=0,
\]
then \(L,\left(L^*+\frac{r-d}{2}\right)^2\) is a Leonard pair, and the same polynomials become Racah polynomials with respect to the same discrete inner product [2508.02032]. In that regime the same \(u_i\) admit the \({}_4F_3\) representation
\[
u_i(\bar\theta_j)
=
{}_4F_3\!\left(
\genfrac..{0pt}{}{-i,\ i-d+r,\ -j,\ j-d-\frac12}{-d,\ \frac{r-d}{2},\ \frac{r-d+1}{2}}
\Bigg|\,1\right),
\]
with
\[
\bar\theta_j=(d-2j)(d-2j+1),
\]
so the identification is not a limit transition but a second Leonard-pair structure on the same orthogonal family [2508.02032].

## 5. Exceptional, dual, and elliptic extensions

Exceptional Racah and \(q\)-Racah polynomials arise in the framework of discrete quantum mechanics with real shifts and shape invariance. For parameters
\[
\lambda=(a,b,c,d),\qquad \delta=(1,1,1,1),
\]
with
\[
\kappa=1 \quad\text{for Racah},\qquad \kappa=q^{-1}\quad\text{for \(q\)-Racah},
\]
the ordinary Racah and \(q\)-Racah systems are deformed by a degree-\(\ell\) polynomial \(\xi_\ell\) built from twisted parameters [1102.0812]. The exceptional family \(\check P_{\ell,n}\) is defined by
\[
\check P_{\ell,n}(x;\lambda)
=
\frac{1}{\hat f_{\ell,n}(\lambda)\,\varphi(x;\lambda+\ell\delta+\delta)}
\Bigl(
v_1^B(x;\lambda+\ell\delta)\check\xi_\ell(x;\lambda)\check P_n(x+1;\lambda+\ell\delta+\delta)
\]
\[
-
v_1^D(x;\lambda+\ell\delta)\check\xi_\ell(x+1;\lambda)\check P_n(x;\lambda+\ell\delta+\delta)
\Bigr),
\]
has degree \(\ell+n\), and begins at degree \(\ell\) rather than \(0\) [1102.0812]. These families remain complete because they are the eigenbasis of a finite-dimensional Hermitian Hamiltonian, but they do not satisfy the ordinary three-term recurrence relation [1102.0812].

A different dualization procedure applies to the multi-indexed \((q)\)-Racah polynomials. If
\[
\check Q_{\mathcal D,x}(n;\lambda)
=
\frac{\check P_{\mathcal D,n}(x;\lambda)}{\check P_{\mathcal D,0}(x;\lambda)},
\]
then the dual family satisfies an ordinary three-term recurrence
\[
\mathcal E_n(\lambda)\check Q_{\mathcal D,x}(n;\lambda)
=
A^{\rm dual}_{\mathcal D,x}(\lambda)\check Q_{\mathcal D,x+1}(n;\lambda)
+
B^{\rm dual}_{\mathcal D,x}(\lambda)\check Q_{\mathcal D,x}(n;\lambda)
+
C^{\rm dual}_{\mathcal D,x}(\lambda)\check Q_{\mathcal D,x-1}(n;\lambda),
\]
but also higher-order difference equations coming from the \((1+2L)\)-term recurrences of the original multi-indexed family [1805.00345]. This makes the dual family an ordinary orthogonal family of Krall type and yields new exactly solvable discrete quantum systems with real shifts; these systems satisfy closure relations but are not shape invariant [1805.00345].

An elliptic extension is obtained from a finite discrete reduction of the difference Heun equation. The monic elliptic Racah polynomials are defined by the principal minors
\[
p_k(E)=\det(\text{\(k\times k\) principal minor of }EI_{M+1}-H),
\]
where \(H\) is a finite tridiagonal matrix arising from the truncated elliptic Heun operator [2106.07394]. They satisfy the three-term recurrence
\[
p_{k+1}(E)=(E-b_k)p_k(E)-a_k\tilde a_{M+1-k}p_{k-1}(E),
\]
and explicit orthogonality relations on the finite spectral set \(\{E_0,\dots,E_M\}\) [2106.07394]. In the trigonometric limit \(p\to 0\), they reduce to the standard \(q\)-Racah polynomials of Askey and Wilson, so they sit naturally as an elliptic extension of the top finite discrete branch of the Askey scheme [2106.07394].

## 6. \(q\)-Racah, braid, Bethe, and Griffiths-type extensions

The \(q\)-Racah family retains the Racah recoupling interpretation in quantum-group form. In the Racah problem for \(U_q(\mathfrak{sl}_2)\), the two intermediate Casimirs \(C_{12}\) and \(C_{23}\) acting on a fixed multiplicity space \(V_\ell\) form a Leonard pair, and the recoupling matrix has entries
\[
P_{ij}
=
\beta_j\,R_i\bigl(\mu(j); q^{2a}, q^{2a}, q^{2(3a+N+1)}, q^{-2(a+N+1)}\mid q^2\bigr),
\]
equivalently a \({}_4\phi_3\) expression [2106.02416]. The same paper shows that these intermediate Casimirs are related by braid-group conjugation, and the recoupling matrix is proportional to the braid word
\[
S_1S_2S_1=m_qP,
\]
so the matrix entries of braid group representations are explicitly given in terms of \(q\)-Racah polynomials [2106.02416].

A different algebraic realization arises from scalar products of Bethe states. For a Leonard pair of \(q\)-Racah type with eigenvalue sequences
\[
\theta_M=bq^{2M}+cq^{-2M},\qquad \theta_N^*=b^*q^{2N}+c^*q^{-2N},
\]
the \(q\)-Racah polynomials
\[
R_M(\theta_N^*)
=
{}_4\phi_3\!\left(
\begin{matrix}
q^{-2M},\ bc\,q^{2M},\ q^{-2N},\ b^*c^*q^{2N}\\
bq^{2s+1}\zeta,\ cq^{2s+1}\zeta^{-1},\ q^{-4s}
\end{matrix};q^2,q^2\right)
\]
are expressed as exact ratios of scalar products of Bethe states, for example
\[
R_M(\theta_N^*)
=
\frac{\langle N^*|M\rangle}{\langle N^*|0\rangle}\,
\frac{\langle 0|0\rangle}{\langle 0|M\rangle}
\]
and its homogeneous and inhomogeneous Bethe-state variants [2211.14727]. The standard recurrence, difference, and orthogonality relations then become reinterpretations of Leonard-pair and Bethe-state structures [2211.14727].

Higher-dimensional \(q\)-Racah-type extensions appear in the bivariate Griffiths polynomials of \(q\)-Racah type,
\[
G_{i,j}(x,y;c_1,c_2,c_3,c_4;N)
=
\sum_{a=0}^{M}
w_a\!\left(\frac{c_1c_2c_3}{c_4}\right)
p_i(a;c_1,c_2,c_3;N-j)\,
p_j(y;c_3,c_0,c_4;N-a)\,
p_a(x;c_4,c_2,c_1;N-y),
\]
where each \(p_n\) is a normalized one-variable \(q\)-Racah polynomial and
\[
c_0c_1c_2c_3c_4\,q^{2N+3}=1
\]
[2407.17016]. These polynomials are built from three one-variable \(q\)-Racah blocks, are governed by the rank-2 Askey–Wilson algebra \(aw(4)\), and are bispectral and biorthogonal rather than orthogonal [2407.17016]. In the limit \(q\to1\), they reduce to the earlier Griffiths polynomials of Racah type [2407.17016].

Taken together, these developments show that Racah polynomials occupy a structurally privileged position: they are finite quadratic-lattice hypergeometric orthogonal polynomials; overlap coefficients for recoupling schemes; matrix elements of Leonard pairs and triples; eigenfunctions of difference and divided-difference operators; and starting points for multivariate, exceptional, dual, elliptic, quantum-group, and biorthogonal extensions [1309.3540, 1808.10520, 2106.07394, 2603.28839].

Source: https://www.emergentmind.com/topics/racah-polynomials