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Infinitesimal moments in free and c-free probability and Motzkin paths

Infinitesimal moments associated with infinitesimal freeness and infinitesimal conditional freeness are studied. For free random variables, we consider continuous deformations of moment functionals associated with Motzkin paths $w$, which provide a decomposition of their moments, and we compute their derivatives at zero. We show that the first-order derivative of each functional vanishes unless the path has exactly one local maximum. Geometrically, this means that $w$ is a pyramid path, which is consistent with the characteristic formula for alternating moments of infinitesimally free centered random variables. In this framework, infinitesimal Boolean independence is also obtained and it corresponds to flat paths. A similar approach is developed for infinitesimal conditional freeness, for which we show that the only moment functionals that have a non-zero first-order derivative are associated with concatenations of a pyramid path and a flat path. This charaterization leads to a Leibniz-type definition of infinitesimal conditional freeness at the level of moments.