Uniform higher-order normal ordering and positivity for Jack operators

Establish uniformly that the Sergeev–Veselov normal-ordering algorithm produces an integral form over ℤ[α] for all higher stable Jack multiplication operators, and relate this integral form to positivity in b=α−1.

Background

The paper demonstrates the cancellation of the auxiliary parameter p₀ and negative powers of α for the first nontrivial operators, Δ₂(α) and Δ₃(α), and presents the Dunkl operator at infinity as an effective low-order algorithm. It does not establish that these cancellations and integrality persist uniformly at arbitrary order.

The conclusion identifies two unresolved extensions: proving a uniform integral form over ℤ[α] and understanding how that form relates to positivity in the parameter b=α−1. These questions go beyond the conjecture’s formal coefficient assertion by asking for a systematic algorithmic proof and a connection with positivity phenomena.