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Geometric volume analogues for other orthogonal polynomials

Determine whether the Legendre volume identity χ_n(t) = n! · mes_n(E_{n,t}) for t ≥ 1, where E_{n,t} = {x ∈ R^n : Σ_{j=1}^n |x_j| + |1 − Σ_{j=1}^n x_j| ≤ t}, is a particular case of a more general pattern that extends to other classes of orthogonal polynomials, specifically Chebyshev and Jacobi polynomials. In particular, construct explicit families of convex sets and volume formulas that characterize these polynomials analogously to the Legendre case, or prove that no such geometric characterizations exist.

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Background

The paper proves that the volume of the convex set E_{n,γ} equals χn(γ)/n! (Theorem 4.1), yielding a geometric characterization of standardized Legendre polynomials via volumes of convex polyhedra: χ_n(t) = n! * mes_n(E{n,t}) for t ≥ 1.

The authors ask whether analogous geometric volume identities exist for other orthogonal polynomial families, such as Chebyshev or Jacobi polynomials, suggesting a broader framework that could unify polynomial identities with convex-geometric measures.

References

There is an interesting open problem related to the above-mentioned equality th_2_1_1. The question arises about analogues of Legendre for other classes of orthogonal polynomials, such as Chebyshev polynomials or, more generally, Jacobi polynomials: Is the equality Legendre a particular case of a more general pattern?

th_2_1_1:

(En,γ)=12nn!∑i=0n(ni)2(γ−1)n−i(γ+1)i=χn(γ)n!.(E_{n,\gamma}) = \frac{1}{2^n n!} \sum_{i=0}^{n} {n \choose i}^2 (\gamma - 1)^{n-i} (\gamma + 1)^i = \frac{\chi_n(\gamma)}{n!}.

Legendre:

χn(t)=n! mesn(En,t),\chi_n(t)= n!\,{\rm mes}_n(E_{n,t}),

— Optimal Lagrange Interpolation Projectors and Legendre Polynomials (2405.01254 - Nevskii, 2 May 2024) in Section 4 (Legendre polynomials and the measure of E_{n,γ}), following equations (4.1)–(4.2)