  Multi-particle cumulants of azimuthal angle correlations have been compelling
tools to probe the properties of the Quark-Gluon Plasma (QGP) created in the
ultra-relativistic heavy-ion collisions and the search for the QGP in small
collision systems at RHIC and the LHC. However, only very few of them are
available and have been studied in theoretical calculations and experimental
measurements. In this paper, we present a generic recursive algorithm for
multi-particle cumulants, which enables the calculation of arbitrary order
multi-particle cumulants. Among them, the new 10-, 12-, 14-, and 16-particle
cumulants of a single harmonic, named $c_{n}\{10\}$, $c_{n}\{12\}$,
$c_{n}\{14\}$, and $c_{n}\{16\}$, and the corresponding $v_n$ coefficients,
will be discussed for the first time. These new multi-particle cumulants can be
readily used along with updates to the generic framework of multi-particle
correlations to a very high order. Finally, we propose a particular series of
mixed harmonic multi-particle cumulants, which measures the general
correlations between any moments of different flow coefficients. The
predictions of these new observables are shown based on an initial state model
MC-Glauber, a toy Monte Carlo model, and the HIJING transport model for future
comparisons between experimental data and theoretical model calculations. The
study of these new multi-particle cumulants in heavy-ion collisions will
significantly improve the understanding of the joint probability density
function which involves both different harmonics of flow and also the symmetry
planes. This will pave the way for more stringent constraints on the initial
state and help to extract more precisely how the created hot and dense matter
evolves. Meanwhile, the efforts applied to small systems could be very helpful
in the understanding of the origin of the observed collectivity at RHIC and the
LHC.
