The Louardi Polynomial Scheme: A Three-Index Family Generalizing Classical Orthogonal Polynomials via Higher-Order Exponentials Eₙᵏ

We introduce the Louardi polynomial scheme: a family LN(n,k,α) indexed by a triple (n, k, α)—order, twist, and parameter—defined by the generating function Nk(t) / Pn(t)α, where Pn(t) = ∏l=0n−1(1 − eωlx t) is the characteristic polynomial of order n and Nk(t) is the twist-k numerator polynomial. At α = 1, the scheme recovers all generalized exponentials: LN(n,k,1) = Enk(Nx). At n = 2, the base family reduces to the classical Gegenbauer, Chebyshev, Legendre, and Hermite polynomials. The twist index k plays a structural role: for n ≥ 3, no single family closes under differentiation, but the full n-component vector LN(α) = (LN(n,0,α), …, LN(n,n−1,α))T satisfies dLN(α)/dx = A(α) LN(α) + α C LN(α+1) with C circulant, yielding an explicit vector Rodrigues formula. The base polynomials are orthogonal with respect to the weight |Vn|2α on the compact eigenvalue configuration (the circular β-ensemble with β = 2α), with closed-form normalization constants identified with the Macdonald–Mehta–Selberg integral via a theorem of Kadell.