The rank two Jacobi algebra <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {J}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">J</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> is used to provide an interpretation of the two-variable Jacobi polynomials <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$J_{n,k}^{(a,b,c)}(x,y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>J</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> on the triangle, as overlaps between two representation bases. The subalgebra structure of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {J}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">J</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> depicted via a pentagonal graph is exploited to find the explicit expression of the two-variable functions in terms of univariate Jacobi polynomials. It is also seen to provide an explanation for the fact that the expansion on the basis <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$J_{n,k}^{(a,b,c)}(x,y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>J</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of the polynomials obtained from the latter by permuting the variables <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$x,y, z=1-x-y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>=</mo> <mn>1</mn> <mo>-</mo> <mi>x</mi> <mo>-</mo> <mi>y</mi> </mrow> </math></EquationSource> </InlineEquation> and the parameters (a, b, c) is given in terms of Racah polynomials. The underlying order-three symmetry is discussed.

Algebraic interpretation of the two-variable Jacobi polynomials on the triangle: the pentagonal way

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2026年3月4日

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Nicolas Crampé,
Quentin Labriet,
Lucia Morey,
Satoshi Tsujimoto,
Luc Vinet,
Alexei Zhedanov

摘要

The rank two Jacobi algebra $\mathcal {J}_2$ $J_{2}$ is used to provide an interpretation of the two-variable Jacobi polynomials $J_{n,k}^{(a,b,c)}(x,y)$ $J_{n, k}^{(a, b, c)} (x, y)$ on the triangle, as overlaps between two representation bases. The subalgebra structure of $\mathcal {J}_2$ $J_{2}$ depicted via a pentagonal graph is exploited to find the explicit expression of the two-variable functions in terms of univariate Jacobi polynomials. It is also seen to provide an explanation for the fact that the expansion on the basis $J_{n,k}^{(a,b,c)}(x,y)$ $J_{n, k}^{(a, b, c)} (x, y)$ of the polynomials obtained from the latter by permuting the variables $x,y, z=1-x-y$ $x, y, z = 1 - x - y$ and the parameters (a, b, c) is given in terms of Racah polynomials. The underlying order-three symmetry is discussed.