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The non-commutative Malliavin calculus on the Heisenberg–Weyl algebra is extended to the affine algebra. A differential calculus and a non-commutative integration by parts are established. As an application we obtain sufficient conditions for the smoothness of Wigner-type laws of non-commutative random variables with gamma or continuous binomial marginals. [Copyright &y& Elsevier]

The non-commutative Malliavin calculus on the Heisenberg–Weyl algebra (see (i) C. R. Acad. Sci. Paris, Se´r. I 328 (11) (1999) 1061–1066, (ii) Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (1) (2001) 11–38) is extended to the affine algebra. A differential calculus is established, which generalizes the corresponding commutative integration by parts formulas. As an application we obtain sufficient conditions for the smoothness of Wigner type laws of non-commutative random variables with gamma and continuous binomial marginals. To cite this article: U. Franz et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003). [Copyright &y& Elsevier]