Through the classical umbral calculus, we provide a unifying syntax for single and multivariate kstatistics, polykays and multivariate polykays. From a combinatorial point of view, we revisit the theory as exposed by Stuart and Ord, taking into account the Doubilet approach to symmetric functions. Moreover, by using exponential polynomials rather than set partitions, we provide a new formula for k-statistics that results in a very fast algorithm to generate such estimators. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2008, Vol. 14, No. 2, 440-468. This reprint differs from the original in pagination and typographic detail. 1350-7265

We provide an algebraic setting for cumulants and factorial moments via the classical umbral calculus. Our main tools are the compositional inverse of the unity umbra, this being related to logarithmic power series, and a new umbra here introduced, the singleton umbra. We develop formulae that express cumulants, factorial moments and central moments as umbral functions

We revisit the theory of Sheffer sequences by means of the formalism introduced in Rota and Taylor (SIAM J Math Anal 25(2):694–711, 1994) and developed in Di Nardo and Senato (Umbral nature of the Poisson random variables. Algebraic combinatorics and computer science, pp 245–256, Springer Italia, Milan, 2001, European J Combin 27(3):394–413, 2006). The advantage of this approach is twofold. First, this new syntax allows us noteworthy computational simplification and conceptual clarification in several topics involving Sheffer sequences, most of the open questions proposed in Taylor (Comput Math Appl 41:1085–1098, 2001) finds answer. Second, most of the results presented can be easily implemented in a symbolic language. To get a general idea of the effectiveness of this symbolic approach, we provide a formula linking connection constants and Riordan arrays via generalized Bell polynomials, here defined. Moreover, this link allows us to smooth out many results involving Bell Polynomials and Lagrange inversion formula

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

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