  We characterise higher order Riesz transforms on the Heisenberg group and
also show that they satisfy dimension-free bounds under some assumptions on the
multipliers. Using transfer- ence theorems, we deduce boundedness theorems for
Riesz trans- forms on the reduced Heisenberg group and hence also for the Riesz
transforms associated to multiple Hermite and Laguerre ex- pansions.
