A. Boukas

We introduce the renormalized powers of $q$-deformed white noise, for any $q$ in the open interval $(-1,1)$, and we extend to them the no--go theorem recently proved by Accardi--Boukas--Franz in the Boson case. The surprising fact is that the lower bound ( ef{basicineq}), which defines the obstruction to the positivity of the sesquilinear form, uniquely determined by the renormalized commutation relations, is independent of $q$ in the half-open interval $(-1,1]$, thus including the Boson case. The exceptional value $q=-1$, corresponding to the Fermion case, is dealt with in the last section of the paper where we prove that the argument used to prove the no--go theorem for $q \ne 0$ does not extend to this case.