Two-Variable Jacobi Polynomials

• Two-variable Jacobi polynomials are bivariate extensions of classical Jacobi polynomials, defined on simplices with specific weight functions and domain constraints.
• They are constructed via multiple paradigms including the triangle (Proriol–Koornwinder) system, x±√y representations, and BC2-type frameworks, each offering unique orthogonality and recurrence properties.
• Their bispectrality and interplay with algebraic structures—such as rank-two Jacobi and Racah algebras—highlight deep connections between differential operators and orthogonal polynomial bases.

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 (Crampe et al., 10 Jul 2025, Makky et al., 2020).

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
$x^a y^b (1-x-y)^c$0-type Jacobi	$x^a y^b (1-x-y)^c$1 with weight $x^a y^b (1-x-y)^c$2	Root-system $x^a y^b (1-x-y)^c$3, dominance orthogonality

In the triangle model, the one-variable building blocks are the shifted Jacobi polynomials $x^a y^b (1-x-y)^c$4, and the indices satisfy $x^a y^b (1-x-y)^c$5. The factor $x^a y^b (1-x-y)^c$6 and the barycentric variable $x^a y^b (1-x-y)^c$7 make the construction explicitly adapted to the simplex geometry (Crampe et al., 10 Jul 2025).

The $x^a y^b (1-x-y)^c$8 model instead packages the two-variable dependence through the combinations $x^a y^b (1-x-y)^c$9 and $x\pm \sqrt{y}$0. 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 $x\pm \sqrt{y}$1, which has no direct analogue in the basic Proriol–Koornwinder definition (Makky et al., 2020).

The $x\pm \sqrt{y}$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 $x\pm \sqrt{y}$3 or symmetric coordinates $x\pm \sqrt{y}$4, and quadratic transformations are intrinsic to the theory (Koornwinder, 2015).

2. Orthogonality on the triangle and on the simplex

For real parameters $x\pm \sqrt{y}$5, the two-variable Jacobi polynomials on the triangle are

$x\pm \sqrt{y}$6

They are orthogonal on

$x\pm \sqrt{y}$7

with respect to

$x\pm \sqrt{y}$8

and satisfy

$x\pm \sqrt{y}$9

The explicit squared norm is

$BC_2$0

For $BC_2$1, the system $BC_2$2 forms a complete orthogonal basis of $BC_2$3 (Crampe et al., 10 Jul 2025).

A normalized overlap form makes the representation-theoretic meaning explicit: $BC_2$4 so the polynomials become matrix elements between coordinate and spectral bases (Crampé et al., 9 Sep 2025).

The same family is the two-dimensional specialization of Jacobi polynomials on the simplex. In that formulation one writes $BC_2$5 and the weight

$BC_2$6

An explicit orthogonal basis is

$BC_2$7

This simplex perspective is the basis for the appearance of several-variable Hahn polynomials as connection coefficients, generated by Jacobi polynomials on the simplex (Xu, 2013).

Two standard one-variable reductions are immediate in the triangle model. If $BC_2$8, then

$BC_2$9

and if $q=-1$0, then

$q=-1$1

Hence the univariate Jacobi hierarchy is built directly into the triangular system (Crampe et al., 10 Jul 2025).

3. The $q=-1$2 construction and mixed differential recurrences

A different two-variable generalization begins with

$q=-1$3

Equivalent representations include a hypergeometric form,

$q=-1$4

and generating functions built from

$q=-1$5

for example

$q=-1$6

An Appell-type generating function is also available through $q=-1$7, the Appell hypergeometric function of two variables (Makky et al., 2020).

For differential-operator purposes, the variables are complexified to $q=-1$8, giving

$q=-1$9

with $$J_{n,k}^{(a,b,c)}(x,y)=J_{n-k}^{(a,b+c+2k+1)}(x)(1-x)^k J_k^{(b,c)}\!\left(\frac{y}{1-x}\right)$$0, $$J_{n,k}^{(a,b,c)}(x,y)=J_{n-k}^{(a,b+c+2k+1)}(x)(1-x)^k J_k^{(b,c)}\!\left(\frac{y}{1-x}\right)$$1. The key operator is

$$J_{n,k}^{(a,b,c)}(x,y)=J_{n-k}^{(a,b+c+2k+1)}(x)(1-x)^k J_k^{(b,c)}\!\left(\frac{y}{1-x}\right)$$2

Its action is particularly simple because

$$J_{n,k}^{(a,b,c)}(x,y)=J_{n-k}^{(a,b+c+2k+1)}(x)(1-x)^k J_k^{(b,c)}\!\left(\frac{y}{1-x}\right)$$3

The main mixed recurrence is

$$J_{n,k}^{(a,b,c)}(x,y)=J_{n-k}^{(a,b+c+2k+1)}(x)(1-x)^k J_k^{(b,c)}\!\left(\frac{y}{1-x}\right)$$4

This lowers the degree by one and shifts both parameters by $$J_{n,k}^{(a,b,c)}(x,y)=J_{n-k}^{(a,b+c+2k+1)}(x)(1-x)^k J_k^{(b,c)}\!\left(\frac{y}{1-x}\right)$$5. The paper interprets $$J_{n,k}^{(a,b,c)}(x,y)=J_{n-k}^{(a,b+c+2k+1)}(x)(1-x)^k J_k^{(b,c)}\!\left(\frac{y}{1-x}\right)$$6 accordingly as a lowering operator in degree and a raising operator in parameters (Makky et al., 2020).

Two one-variable slices make the Jacobi analogy explicit. At $$J_{n,k}^{(a,b,c)}(x,y)=J_{n-k}^{(a,b+c+2k+1)}(x)(1-x)^k J_k^{(b,c)}\!\left(\frac{y}{1-x}\right)$$7,

$$J_{n,k}^{(a,b,c)}(x,y)=J_{n-k}^{(a,b+c+2k+1)}(x)(1-x)^k J_k^{(b,c)}\!\left(\frac{y}{1-x}\right)$$8

while at $$J_{n,k}^{(a,b,c)}(x,y)=J_{n-k}^{(a,b+c+2k+1)}(x)(1-x)^k J_k^{(b,c)}\!\left(\frac{y}{1-x}\right)$$9,

$T=\{(x,y)\mid x\ge 0,\ y\ge 0,\ x+y\le 1\}$0

These are univariate-type derivative recurrences induced by the two-variable operator (Makky et al., 2020).

The same framework contains the two-variable Legendre case. Setting $T=\{(x,y)\mid x\ge 0,\ y\ge 0,\ x+y\le 1\}$1 yields

$T=\{(x,y)\mid x\ge 0,\ y\ge 0,\ x+y\le 1\}$2

with generating function

$T=\{(x,y)\mid x\ge 0,\ y\ge 0,\ x+y\le 1\}$3

This construction therefore extends the two-variable Legendre family exactly as ordinary Jacobi polynomials extend Legendre in one variable (Makky et al., 2020).

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 $T=\{(x,y)\mid x\ge 0,\ y\ge 0,\ x+y\le 1\}$4 and of difference operators in $T=\{(x,y)\mid x\ge 0,\ y\ge 0,\ x+y\le 1\}$5. The principal differential operator is

$T=\{(x,y)\mid x\ge 0,\ y\ge 0,\ x+y\le 1\}$6

together with

$T=\{(x,y)\mid x\ge 0,\ y\ge 0,\ x+y\le 1\}$7

The joint eigenvalue equations are

$T=\{(x,y)\mid x\ge 0,\ y\ge 0,\ x+y\le 1\}$8

$T=\{(x,y)\mid x\ge 0,\ y\ge 0,\ x+y\le 1\}$9

These operators diagonalize the family in the variable representation (Crampe et al., 10 Jul 2025).

Multiplication by the coordinate functions

$x\pm\sqrt{y}$0

produces the recurrence side of the bispectral pair. The operator $x\pm\sqrt{y}$1 gives a three-term recurrence in $x\pm\sqrt{y}$2 at fixed $x\pm\sqrt{y}$3, while $x\pm\sqrt{y}$4 gives a nine-term recurrence mixing shifts in both $x\pm\sqrt{y}$5 and $x\pm\sqrt{y}$6. On the degree side, the dual operators are $x\pm\sqrt{y}$7, together with

$x\pm\sqrt{y}$8

The equality

$x\pm\sqrt{y}$9

encodes the differential–difference duality (Crampe et al., 10 Jul 2025).

These operators generate the quadratic rank-two Jacobi algebra

$P_n^{(\alpha,\beta)}(x,y)$0

with basic relations

$P_n^{(\alpha,\beta)}(x,y)$1

The algebra closes quadratically under triple commutators. A typical relation is

$P_n^{(\alpha,\beta)}(x,y)$2

The construction contains several rank-one Jacobi subalgebras, including $P_n^{(\alpha,\beta)}(x,y)$3, $P_n^{(\alpha,\beta)}(x,y)$4, $P_n^{(\alpha,\beta)}(x,y)$5, and $P_n^{(\alpha,\beta)}(x,y)$6, as well as a rank-one Racah subalgebra generated by $P_n^{(\alpha,\beta)}(x,y)$7 after fixing the eigenvalue of $P_n^{(\alpha,\beta)}(x,y)$8 (Crampe et al., 10 Jul 2025).

The same algebra acquires a representation-theoretic interpretation in terms of overlaps between bases. The coordinate basis $P_n^{(\alpha,\beta)}(x,y)$9 diagonalizes $x\pm \sqrt{y}$0 and $x\pm \sqrt{y}$1; the spectral bases $x\pm \sqrt{y}$2 and $x\pm \sqrt{y}$3 diagonalize $x\pm \sqrt{y}$4 and $x\pm \sqrt{y}$5, 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 (Crampé et al., 9 Sep 2025).

5. Symmetry, connection coefficients, and $x\pm \sqrt{y}$6 realizations

The triangle family carries a full $x\pm \sqrt{y}$7-symmetry through permutations of

$x\pm \sqrt{y}$8

and of the parameters $x\pm \sqrt{y}$9. The basic reflections described in the literature are $x^a y^b (1-x-y)^c$00, $x^a y^b (1-x-y)^c$01, and $x^a y^b (1-x-y)^c$02, corresponding to the exchanges $x^a y^b (1-x-y)^c$03, $x^a y^b (1-x-y)^c$04, and $x^a y^b (1-x-y)^c$05. A particularly simple formula is

$x^a y^b (1-x-y)^c$06

which follows from the univariate symmetry $x^a y^b (1-x-y)^c$07 (Crampé et al., 9 Sep 2025).

At fixed total degree $x^a y^b (1-x-y)^c$08, the change of basis between the standard family

$x^a y^b (1-x-y)^c$09

and the permuted family

$x^a y^b (1-x-y)^c$10

is governed by Racah polynomials. In the algebraic formulation, the pair of bases diagonalizing $x^a y^b (1-x-y)^c$11 and $x^a y^b (1-x-y)^c$12 at fixed $x^a y^b (1-x-y)^c$13 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 (Crampé et al., 9 Sep 2025).

A different symmetry-rich realization is provided by the $x^a y^b (1-x-y)^c$14-type Jacobi polynomials. On

$x^a y^b (1-x-y)^c$15

the weight is

$x^a y^b (1-x-y)^c$16

and the corresponding $x^a y^b (1-x-y)^c$17 are the $x^a y^b (1-x-y)^c$18-type Jacobi polynomials. In symmetric coordinates on

$x^a y^b (1-x-y)^c$19

the weight becomes

$x^a y^b (1-x-y)^c$20

These polynomials are symmetric under $x^a y^b (1-x-y)^c$21 and arise as eigenfunctions of commuting differential operators associated with the $x^a y^b (1-x-y)^c$22 root system (Koornwinder, 2015).

Their most distinctive feature is a family of quadratic transformations. In angular variables $x^a y^b (1-x-y)^c$23, $x^a y^b (1-x-y)^c$24, the transformed coordinates are $x^a y^b (1-x-y)^c$25 and $x^a y^b (1-x-y)^c$26. The formulas

$x^a y^b (1-x-y)^c$27

and

$x^a y^b (1-x-y)^c$28

are the two-variable analogues of classical one-variable Jacobi quadratic transformations. The paper interprets them through the internal relation between the $x^a y^b (1-x-y)^c$29 and $x^a y^b (1-x-y)^c$30 root systems inside $x^a y^b (1-x-y)^c$31 (Koornwinder, 2015).

6. Related Jacobi-type generalizations

Beyond the classical triangle, simplex, and $x^a y^b (1-x-y)^c$32 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 $x^a y^b (1-x-y)^c$33 Jacobi polynomials are defined by

$x^a y^b (1-x-y)^c$34

with a parity-dependent coupling factor $x^a y^b (1-x-y)^c$35. They are orthogonal on a disjoint union of four triangles when $x^a y^b (1-x-y)^c$36, with weight

$x^a y^b (1-x-y)^c$37

They are joint eigenfunctions of two commuting first-order differential–reflection operators $x^a y^b (1-x-y)^c$38 and $x^a y^b (1-x-y)^c$39, with eigenvalues depending separately on $x^a y^b (1-x-y)^c$40 and $x^a y^b (1-x-y)^c$41. The special case $x^a y^b (1-x-y)^c$42 yields the two-variable Little $x^a y^b (1-x-y)^c$43 Jacobi polynomials (Genest et al., 2014).

A different biorthogonal extension is given by the bivariate Jacobi Konhauser polynomials

$x^a y^b (1-x-y)^c$44

with $x^a y^b (1-x-y)^c$45 and $x^a y^b (1-x-y)^c$46. They are biorthogonal on $x^a y^b (1-x-y)^c$47 with respect to

$x^a y^b (1-x-y)^c$48

and admit operational formulas, generating functions, integral representations, and stability under Riemann–Liouville fractional integrals and derivatives through parameter shifts $x^a y^b (1-x-y)^c$49, $x^a y^b (1-x-y)^c$50 (Özarslan et al., 2024).

These variants do not replace the classical two-variable Jacobi systems on the triangle or on $x^a y^b (1-x-y)^c$51. Rather, they show that Jacobi-type behavior in two variables can be organized through orthogonality on simplices, mixed differential operators in variables $x^a y^b (1-x-y)^c$52, root-system symmetry, reflection operators at $x^a y^b (1-x-y)^c$53, 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.