Formally, let G be a Coxeter group with reduced root system R and kv a multiplicity function on R (so ku = kv whenever the reflections σu and σv corresponding to the roots u and v are conjugate in G). Then, the Dunkl operator is defined by:
where is the i-th component of v, 1 ≤ i ≤ N, x in RN, and f a smooth function on RN.
Dunkl operators were introduced by Charles Dunkl (1989). One of Dunkl's major results was that Dunkl operators "commute," that is, they satisfy just as partial derivatives do. Thus Dunkl operators represent a meaningful generalization of partial derivatives.