Macdonald–Mehta–Selberg integral for arbitrary root systems

Prove a closed-form evaluation of the Macdonald–Mehta–Selberg integral for every root system, including non-crystallographic root systems, thereby supplying the general proof that is currently unavailable.

Background

The Macdonald–Mehta–Selberg integral is defined by c_k = ∫_{R^N} e^{-|x|^2/2} w_k(x) dx, where w_k is the weight associated with a root system and a multiplicity function. The paper notes that closed-form evaluations were conjectured and proved for crystallographic root systems, with extensions to arbitrary crystallographic reflection groups and computer-assisted proofs for some non-crystallographic systems.

A general proof covering arbitrary root systems is explicitly identified as unavailable. Establishing such a proof would complete the evaluation theory for this fundamental normalization constant in Dunkl analysis.