We study the umbral polynomials A^(k)_n(x, α) = x(x–k·α)ⁿ⁻¹, by means of which a wide range of formal power series identities, including Lagrange inversion formula, can be usefully manipulated. We apply this syntax within cumulant theory, and show how moments and its formal cumulants (classical, free and Boolean) are represented by polynomials A^(k)_n(α, γ) for suitable choices of umbrae α and γ.