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We provide an unifying polynomial expression giving moments in terms ofcumulants, and viceversa, holding in the classical, boolean and free setting.This is done by using a symbolic treatment of Abel polynomials. As aby-product, we show that in the free cumulant theory the volume polynomial ofPitman and Stanley plays the role of the complete Bell exponential polynomialin the classical theory. Moreover via generalized Abel polynomials we constructa new class of cumulants, including the classical, boolean and free ones, andthe convolutions linearized by them. Finally, via an umbral Fourier transform,we state a explicit connection between boolean and free convolution.

Cumulants and convolutions via Abel polynomials