---
title: Two-Variable Jacobi Polynomials
url: https://www.emergentmind.com/topics/two-variable-jacobi-polynomials
type: topic
---

# Two-Variable Jacobi Polynomials

Two-variable Jacobi polynomials are bivariate extensions of the Jacobi family that arise in several inequivalent constructions. The most developed classical realization is the Proriol–Koornwinder system on the triangle, where the polynomials \(J_{n,k}^{(a,b,c)}(x,y)\) are orthogonal with respect to the Jacobi weight \(x^a y^b (1-x-y)^c\). Other constructions include a family built from the combinations \(x\pm \sqrt{y}\), root-system \(BC_2\) Jacobi polynomials, and Jacobi-type extensions in the \(q=-1\) and Konhauser settings. This suggests that the expression “two-variable Jacobi polynomials” is an umbrella term for several structurally related but non-equivalent bivariate systems [2507.07766] [2009.11269].

## 1. Principal constructions

The literature represented here contains several standard models of two-variable Jacobi polynomials.

| Family | Defining form | Domain or structure |
|---|---|---|
| Triangle (Proriol–Koornwinder) | \(J_{n,k}^{(a,b,c)}(x,y)=J_{n-k}^{(a,b+c+2k+1)}(x)(1-x)^k J_k^{(b,c)}\!\left(\frac{y}{1-x}\right)\) | Orthogonal on \(T=\{(x,y)\mid x\ge 0,\ y\ge 0,\ x+y\le 1\}\) |
| \(x\pm\sqrt{y}\) family | \(P_n^{(\alpha,\beta)}(x,y)\) via hypergeometric and generating-function formulas in \(x\pm \sqrt{y}\) | Real or complex variables; mixed differential recurrences |
| \(BC_2\)-type Jacobi | \(P_{n,k}^{(\alpha,\beta,\gamma)}(\xi,\eta)\) with weight \((1-\xi+\eta)^\alpha(1+\xi+\eta)^\beta(\xi^2-4\eta)^\gamma\) | Root-system \(BC_2\), dominance orthogonality |

In the triangle model, the one-variable building blocks are the shifted Jacobi polynomials \(J_n^{(a,b)}(x)=P_n^{(a,b)}(1-2x)\), and the indices satisfy \(n\ge k\ge 0\). The factor \((1-x)^k\) and the barycentric variable \(y/(1-x)\) make the construction explicitly adapted to the simplex geometry [2507.07766].

The \(x\pm \sqrt{y}\) model instead packages the two-variable dependence through the combinations \(x-\sqrt{y}\) and \(x+\sqrt{y}\). Its defining series, hypergeometric forms, and generating functions show that it is not simply a reparametrization of the triangular system. The central differential operator in that setting is \(\Xi=\partial_z+\sqrt{w}\,\partial_w\), which has no direct analogue in the basic Proriol–Koornwinder definition [2009.11269].

The \(BC_2\)-type family is organized by Weyl-group symmetry and dominance orthogonality rather than by simplex barycentric coordinates. In that setting, the natural variables are \((\xi,\eta)\) or symmetric coordinates \((x+y,xy)\), and quadratic transformations are intrinsic to the theory [1512.09294].

## 2. Orthogonality on the triangle and on the simplex

For real parameters \(a,b,c>-1\), the two-variable Jacobi polynomials on the triangle are
\[
J_{n,k}^{(a,b,c)}(x,y)=J_{n-k}^{(a,b+c+2k+1)}(x)\,(1-x)^k\, J_k^{(b,c)}\!\left(\frac{y}{1-x}\right),\qquad n\ge k\ge 0.
\]
They are orthogonal on
\[
T=\{(x,y)\in\mathbb{R}^2\mid 0\le x,\;0\le y,\;x+y\le 1\}
\]
with respect to
\[
w^{(a,b,c)}(x,y)=x^a y^b (1-x-y)^c,
\]
and satisfy
\[
\int_T J_{n,k}^{(a,b,c)}(x,y)\,J_{n',k'}^{(a,b,c)}(x,y)\,x^a y^b(1-x-y)^c\,dx\,dy
= h_{n,k}\,\delta_{n,n'}\delta_{k,k'}.
\]
The explicit squared norm is
\[
h_{n,k}=
\dfrac{(b+c+2k+2)_{\,n-k}\,(b+c+k+1)_k}{(a+b+c+2n+2)\,(n-k)!\,k!}\,
\dfrac{\Gamma(a+n-k+1)\,\Gamma(b+k+1)\,\Gamma(c+k+1)}
{\Gamma(a+b+c+n+k+2)}.
\]
For \(a,b,c>-1\), the system \(\{J_{n,k}^{(a,b,c)}\}_{n\ge k\ge 0}\) forms a complete orthogonal basis of \(L^2(T,w^{(a,b,c)})\) [2507.07766].

A normalized overlap form makes the representation-theoretic meaning explicit:
\[
\langle n,k;a,b,c\mid x,y\rangle
=\sqrt{\frac{x^{a}y^{b}(1-x-y)^c}
{N_k^{(b,c)}N_{n-k}^{(a,b+c+2k+1)}}}\,
J_{n,k}^{(a,b,c)}(x,y),
\]
so the polynomials become matrix elements between coordinate and spectral bases [2509.07949].

The same family is the two-dimensional specialization of Jacobi polynomials on the simplex. In that formulation one writes \(T^2=\{(x_1,x_2)\mid x_1\ge 0,\ x_2\ge 0,\ x_1+x_2\le 1\}\) and the weight
\[
W_\kappa(x_1,x_2)=x_1^{\kappa_1}x_2^{\kappa_2}(1-x_1-x_2)^{\kappa_3},\qquad \kappa_i>-1.
\]
An explicit orthogonal basis is
\[
P_{\nu_1,\nu_2}(x_1,x_2)
=
\frac{p_{\nu_1}^{(a_1,\kappa_1)}(2x_1-1)}{p_{\nu_1}^{(a_1,\kappa_1)}(1)}
(1-x_1)^{\nu_2}
\frac{p_{\nu_2}^{(\kappa_3,\kappa_2)}\!\big(\frac{2x_2}{1-x_1}-1\big)}
{p_{\nu_2}^{(\kappa_3,\kappa_2)}(1)},
\qquad
a_1=\kappa_2+\kappa_3+2\nu_2+1.
\]
This simplex perspective is the basis for the appearance of several-variable Hahn polynomials as connection coefficients, generated by Jacobi polynomials on the simplex [1309.1510].

Two standard one-variable reductions are immediate in the triangle model. If \(k=0\), then
\[
J_{n,0}^{(a,b,c)}(x,y)=J_n^{(a,b+c+1)}(x),
\]
and if \(n=k\), then
\[
J_{k,k}^{(a,b,c)}(x,y)=(1-x)^k J_k^{(b,c)}\!\left(\frac{y}{1-x}\right).
\]
Hence the univariate Jacobi hierarchy is built directly into the triangular system [2507.07766].

## 3. The \(x\pm\sqrt{y}\) construction and mixed differential recurrences

A different two-variable generalization begins with
\[
P_n^{(\alpha,\beta)}(x,y)
=\sum_{k=0}^{n}
\frac{(1+\alpha)_n\,(1+\alpha+\beta)_{n+k}}
{k!\,(n-k)!\,(1+\alpha)_k\,(1+\alpha+\beta)_n}
\left(x-\sqrt{y}\right)^k,
\qquad \Re(\alpha),\Re(\beta)>-1.
\]
Equivalent representations include a hypergeometric form,
\[
P_n^{(\alpha,\beta)}(x,y)
=
\frac{(1+\alpha)_n}{n!}\,
{}_2F_1\!\left(
-n,\,
1+\alpha+\beta+n;\,
1+\alpha;\,
\frac{\sqrt{y}-x}{2}
\right),
\]
and generating functions built from
\[
\mu=\sqrt{1-2xt+yt^2},
\]
for example
\[
\sum_{n=0}^\infty P_n^{(\alpha,\beta)}(x,y)t^n
=
\mu^{-1}
\left(1+\frac{yt}{\mu}\right)^{-\beta}
\left(1-\frac{yt}{\mu}\right)^{-\alpha}.
\]
An Appell-type generating function is also available through \(F_4\), the Appell hypergeometric function of two variables [2009.11269].

For differential-operator purposes, the variables are complexified to \(z,w\in\mathbb{C}\), giving
\[
P_n^{(\alpha,\beta)}(z,w)
=
\sum_{k=0}^{n}
\frac{(1+\alpha)_n\,(1+\alpha+\beta)_{n+k}}
{k!\,(n-k)!\,(1+\alpha)_k\,(1+\alpha+\beta)_n}
\left(z-\sqrt{w}\right)^k,
\]
with \(|z|<1\), \(|w|<1\). The key operator is
\[
\Xi=\frac{\partial}{\partial z}+\sqrt{w}\,\frac{\partial}{\partial w}.
\]
Its action is particularly simple because
\[
\Xi(z-\sqrt{w})^k=\frac{k}{2}(z-\sqrt{w})^{k-1}.
\]
The main mixed recurrence is
\[
\frac{\partial}{\partial z}P_n^{(\alpha,\beta)}(z,w)
+\sqrt{w}\,\frac{\partial}{\partial w}P_n^{(\alpha,\beta)}(z,w)
=
\frac{1+\alpha+\beta+n}{4}\,
P_{n-1}^{(1+\alpha,1+\beta)}(z,w),
\qquad n\ge 1.
\]
This lowers the degree by one and shifts both parameters by \(+1\). The paper interprets \(\Xi\) accordingly as a lowering operator in degree and a raising operator in parameters [2009.11269].

Two one-variable slices make the Jacobi analogy explicit. At \(w=1\),
\[
\frac{\partial}{\partial z}P_n^{(\alpha,\beta)}(z,1)
=
\frac{1+\alpha+\beta+n}{4}\,
P_{n-1}^{(1+\alpha,1+\beta)}(z,1),
\]
while at \(z=1\),
\[
\sqrt{w}\,\frac{\partial}{\partial w}P_n^{(\alpha,\beta)}(1,w)
=
-\frac{1+\alpha+\beta+n}{4}\,
P_{n-1}^{(1+\alpha,1+\beta)}(1,w).
\]
These are univariate-type derivative recurrences induced by the two-variable operator [2009.11269].

The same framework contains the two-variable Legendre case. Setting \(\alpha=\beta=0\) yields
\[
P_n(x,y)=P_n^{(0,0)}(x,y)
=
\sum_{k=0}^n
\frac{(n+k)!}{(k!)^2 (n-k)!}
\left(\frac{x-\sqrt{y}}{2}\right)^k,
\]
with generating function
\[
\sum_{n=0}^{\infty} P_n(x,y)t^n
=
\left(1-2xt+yt^2\right)^{-1/2}.
\]
This construction therefore extends the two-variable Legendre family exactly as ordinary Jacobi polynomials extend Legendre in one variable [2009.11269].

## 4. Bispectrality and the rank-two Jacobi algebra

For the triangle family, the central analytic feature is bispectrality: the same polynomials are eigenfunctions of differential operators in \((x,y)\) and of difference operators in \((n,k)\). The principal differential operator is
\[
\begin{aligned}
L^{(a,b,c)}&=
x(1-x)\partial_{xx}
+y(1-y)\partial_{yy}
-2xy\partial_{xy}\\
&\quad +(a+1-(a+b+c+3)x)\partial_x
+(b+1-(a+b+c+3)y)\partial_y,
\end{aligned}
\]
together with
\[
L_1^{(b,c)}
=
((b+1)(1-x)-(b+c+2)y)\partial_y
+y(1-x-y)\partial_{yy}.
\]
The joint eigenvalue equations are
\[
L[J_{n,k}^{(a,b,c)}]
=
-n(n+a+b+c+2)\,J_{n,k}^{(a,b,c)},
\]
\[
L_1[J_{n,k}^{(a,b,c)}]
=
-k(k+b+c+1)\,J_{n,k}^{(a,b,c)}.
\]
These operators diagonalize the family in the variable representation [2507.07766].

Multiplication by the coordinate functions
\[
X_1:=x,\qquad X_3:=1-x-y
\]
produces the recurrence side of the bispectral pair. The operator \(X_1\) gives a three-term recurrence in \(n\) at fixed \(k\), while \(X_3\) gives a nine-term recurrence mixing shifts in both \(n\) and \(k\). On the degree side, the dual operators are \(\hat X_1,\hat X_3\), together with
\[
\hat L=-n(n+a+b+c+2),\qquad
\hat L_1=-k(k+b+c+1).
\]
The equality
\[
O\,J_{n,k}(x,y)=\hat O\,J_{n,k}(x,y),
\qquad
O\in\{L,L_1,L_3,X_1,X_3\},
\]
encodes the differential–difference duality [2507.07766].

These operators generate the quadratic rank-two Jacobi algebra
\[
\{L,\ L_1,\ L_3,\ X_1,\ X_3\},
\]
with basic relations
\[
[L,L_1]=[L,L_3]=[X_1,X_3]=0.
\]
The algebra closes quadratically under triple commutators. A typical relation is
\[
[N_1,X_1]=-2X_1^2+2X_1,
\qquad
N_1=[L,X_1].
\]
The construction contains several rank-one Jacobi subalgebras, including \(\langle L,X_1\rangle\), \(\langle L,X_3\rangle\), \(\langle L_1,X_3\rangle\), and \(\langle L_3,X_1\rangle\), as well as a rank-one Racah subalgebra generated by \(\langle L_1,L_3\rangle\) after fixing the eigenvalue of \(L\) [2507.07766].

The same algebra acquires a representation-theoretic interpretation in terms of overlaps between bases. The coordinate basis \(|x,y\rangle\) diagonalizes \(X_1\) and \(X_3\); the spectral bases \(|n,k;a,b,c\rangle\) and \(|n,k;a,b,c\rangle^\pi\) diagonalize \((L,L_1)\) and \((L,L_3)\), respectively. The two-variable Jacobi polynomials are precisely the overlaps between these bases. In this formulation, the factorized Proriol–Koornwinder formula is recovered by moving along edges of a pentagonal graph of Jacobi subalgebras [2509.07949].

## 5. Symmetry, connection coefficients, and \(BC_2\) realizations

The triangle family carries a full \(S_3\)-symmetry through permutations of
\[
(x,y,z)=(x,y,1-x-y)
\]
and of the parameters \((a,b,c)\). The basic reflections described in the literature are \(\pi\), \(\sigma\), and \(\tau\), corresponding to the exchanges \(x\leftrightarrow z\), \(x\leftrightarrow y\), and \(y\leftrightarrow z\). A particularly simple formula is
\[
J_{n,k}^{(a,c,b)}(x,1-x-y)=(-1)^k J_{n,k}^{(a,b,c)}(x,y),
\]
which follows from the univariate symmetry \(J_n^{(a,b)}(1-x)=(-1)^nJ_n^{(b,a)}(x)\) [2509.07949].

At fixed total degree \(n\), the change of basis between the standard family
\[
J_{n,k}^{(a,b,c)}(x,y)
\]
and the permuted family
\[
J_{n,\ell}^{(c,b,a)}(1-x-y,y)
\]
is governed by Racah polynomials. In the algebraic formulation, the pair of bases diagonalizing \(L_1\) and \(L_3\) at fixed \(n\) forms a Leonard pair for the Racah algebra, and the overlaps are orthonormalized Racah polynomials. This explains why permutations of variables and parameters are encoded by Racah connection coefficients rather than by Jacobi coefficients [2509.07949].

A different symmetry-rich realization is provided by the \(BC_2\)-type Jacobi polynomials. On
\[
\Omega=\{(\xi,\eta)\in\mathbb{R}^2\mid 1-\xi+\eta>0,\ 1+\xi+\eta>0,\ \xi^2-4\eta>0\},
\]
the weight is
\[
w_{\alpha,\beta,\gamma}(\xi,\eta)
=
(1-\xi+\eta)^\alpha(1+\xi+\eta)^\beta(\xi^2-4\eta)^\gamma,
\]
and the corresponding \(P_{n,k}^{(\alpha,\beta,\gamma)}(\xi,\eta)\) are the \(BC_2\)-type Jacobi polynomials. In symmetric coordinates on
\[
A=\{(x,y)\mid -1<y<x<1\},
\]
the weight becomes
\[
W_{\alpha,\beta,\gamma}(x,y)
=
(1-x)^\alpha(1-y)^\alpha(1+x)^\beta(1+y)^\beta(x-y)^{2\gamma}.
\]
These polynomials are symmetric under \(x\leftrightarrow y\) and arise as eigenfunctions of commuting differential operators associated with the \(BC_2\) root system [1512.09294].

Their most distinctive feature is a family of quadratic transformations. In angular variables \(x=\cos\theta_1\), \(y=\cos\theta_2\), the transformed coordinates are \(\cos(\theta_1-\theta_2)\) and \(\cos(\theta_1+\theta_2)\). The formulas
\[
\frac{Q_{n,k}(\cos(\theta_1-\theta_2),\cos(\theta_1+\theta_2))}{Q_{n,k}(1,1)}
=
\frac{P_{n+k,n-k}(\cos\theta_1,\cos\theta_2)}{P_{n+k,n-k}(1,1)}
\]
and
\[
\frac{(\cos\theta_1+\cos\theta_2)\,R_{n,k}(\cos(\theta_1-\theta_2),\cos(\theta_1+\theta_2))}{2R_{n,k}(1,1)}
=
\frac{P_{n+k+1,n-k}(\cos\theta_1,\cos\theta_2)}{P_{n+k+1,n-k}(1,1)}
\]
are the two-variable analogues of classical one-variable Jacobi quadratic transformations. The paper interprets them through the internal relation between the \(B_2\) and \(C_2\) root systems inside \(BC_2\) [1512.09294].

## 6. Related Jacobi-type generalizations

Beyond the classical triangle, simplex, and \(BC_2\) systems, the literature contains two-variable Jacobi-type families that preserve part of the Jacobi structure while changing the underlying analytic framework.

The two-variable Big \(-1\) Jacobi polynomials are defined by
\[
\mathcal{J}_{n,k}(x,y)
=
J_{n-k}\!\left(y;\alpha,\,2k+\beta+\gamma+1,\,(-1)^k\delta\right)\,
\rho_k(y)\,
J_k\!\left(\frac{x}{y};\gamma,\beta,\frac{\delta}{y}\right),
\]
with a parity-dependent coupling factor \(\rho_k(y)\). They are orthogonal on a disjoint union of four triangles when \(|\delta|<1\), with weight
\[
W(x,y)=\theta(xy)\, |y|^{\beta+\gamma}\,(1+y)\left(1+\frac{x}{y}\right)\left(\frac{x-\delta}{y}\right)
(1-y^2)^{\frac{\alpha-1}{2}}
\left(1-\frac{x^2}{y^2}\right)^{\frac{\gamma-1}{2}}
\left(\frac{x^2-\delta^2}{y^2}\right)^{\frac{\beta-1}{2}}.
\]
They are joint eigenfunctions of two commuting first-order differential–reflection operators \(L_1\) and \(L_2\), with eigenvalues depending separately on \(n\) and \(k\). The special case \(\delta=0\) yields the two-variable Little \(-1\) Jacobi polynomials [1411.7299].

A different biorthogonal extension is given by the bivariate Jacobi Konhauser polynomials
\[
\kappa P_n^{(\alpha,\beta)}(x,y)
=
\frac{\Gamma(1+\alpha+n)}{n!}
\sum_{s=0}^n\sum_{r=0}^{n-s}
\frac{(-n)_{s+r}\,(1+\alpha+\beta+n)_s}
{s!\,r!\,\Gamma(1+\alpha+s)\,\Gamma(\beta+1+\kappa r)}
\left(\frac{1-x}{2}\right)^s y^{\kappa r},
\]
with \(\alpha,\beta>-1\) and \(\kappa\in\{1,2,\dots\}\). They are biorthogonal on \([-1,1]\times (0,\infty)\) with respect to
\[
(1-x)^\alpha(1+x)^\beta e^{-y}y^\beta,
\]
and admit operational formulas, generating functions, integral representations, and stability under Riemann–Liouville fractional integrals and derivatives through parameter shifts \(\alpha\mapsto \alpha\pm \mu\), \(\beta\mapsto \beta\pm \zeta\) [2405.15043].

These variants do not replace the classical two-variable Jacobi systems on the triangle or on \(BC_2\). Rather, they show that Jacobi-type behavior in two variables can be organized through orthogonality on simplices, mixed differential operators in variables \(x\pm \sqrt{y}\), root-system symmetry, reflection operators at \(q=-1\), or biorthogonality coupled to Konhauser theory. A plausible implication is that two-variable Jacobi theory is best understood as a family of compatible paradigms—orthogonal, bispectral, algebraic, and operational—rather than as a single canonical polynomial sequence.

Source: https://www.emergentmind.com/topics/two-variable-jacobi-polynomials