Generalize the Legendre volume identity to other orthogonal polynomials

Determine whether there exists a general framework yielding volume representations for other classical orthogonal polynomials (e.g., Chebyshev or Jacobi polynomials) analogous to the Legendre identity \chi_n(t)=n!\,\mathrm{mes}_n(E_{n,t}), i.e., establish whether the Legendre equality is a particular case of a broader pattern.

Background

The paper proves that the standardized Legendre polynomial \chi_n(t) equals n! times the volume of a convex polyhedron E_{n,t} defined by a linear inequality in absolute values. This gives a geometric characterization of Legendre polynomials via volumes.

The author asks whether similar volume identities exist for other families of orthogonal polynomials (Chebyshev, Jacobi), which would broaden the geometric toolkit connecting polynomial approximation and convex geometry.

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}),

— Geometric Estimates in Linear Interpolation on a Cube and a Ball (2402.11611 - Nevskii, 18 Feb 2024) in Section 5