Two-variable Jacobi polynomials on the triangle are realized as representation overlaps of the rank two Jacobi algebra J_2, whose pentagonal subalgebra structure also yields Racah expansions under variable permutations.
The rank two Jacobi algebra
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abstract
The quadratic rank two Jacobi algebra is identified from the relations obeyed by the bispectral operators of the two variable Jacobi polynomials orthogonal on the triangle. It is seen to admit as subalgebras Racah and Jacobi algebras of rank one. The dual realizations in terms of differential operators in the variable representation and in terms of difference operators in the degree representation are provided. Structure relations for the two variable Jacobi polynomials are obtained as a by product.
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Algebraic interpretation of the two-variable Jacobi polynomials on the triangle: the pentagonal way
Two-variable Jacobi polynomials on the triangle are realized as representation overlaps of the rank two Jacobi algebra J_2, whose pentagonal subalgebra structure also yields Racah expansions under variable permutations.