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  "body": "\n## Summary\n\nA Dunkl operator is a [[partial derivative|09EGC6F8PT]] plus a sum of difference terms, one for each reflection of a finite [[reflection group|G6FND7X69J]].\nCharles Dunkl introduced these operators and proved that, for a fixed group and multiplicity function, they commute with each other.\nThey give a theory of special functions attached to root systems.\nIn it, Dunkl harmonics, the Dunkl transform and Dunkl kernels generalize spherical harmonics, the Fourier transform and the exponential.\nThey are also used to study the Calogero–Moser–Sutherland quantum systems, whose eigenfunctions in type $A$ are related to the [[Jack functions|WW7H1KNXK8]].\n\n## Explanations\n\n### L1 human\n\nAn ordinary derivative measures how a function changes along a direction.\nA Dunkl operator adds to this the difference between the function and its mirror image in each mirror of a symmetry group, divided by the distance to that mirror.\nOn the real line with the single mirror $x \\mapsto -x$, it is $T f(x) = f'(x) + k \\frac{f(x) - f(-x)}{x}$.\nIt sends $x^2$ to $2x$, like the derivative, but sends $x^3$ to $(3 + 2k) x^2$.\n",
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