# Dunkl operator

$T_i f(x) = \frac{\partial}{\partial x_i} f(x) + \sum_{v\in R_+} k_v \frac{f(x) - f(x \sigma_v)}{\left\langle x, v\right\rangle} v_i$
where $v_i$ is the i-th component of v, 1 ≤ iN, 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 $T_i (T_j f(x)) = T_j (T_i f(x))$ just as partial derivatives do. Thus Dunkl operators represent a meaningful generalization of partial derivatives.