  We present a new generic framework which enables exact and fast evaluation of
all multi-particle azimuthal correlations. The framework can be readily used
along with a correction framework for systematic biases in anisotropic flow
analyses due to various detector inefficiencies. A new recursive algorithm has
been developed for higher order correlators for the cases where their direct
implementation is not feasible. We propose and discuss new azimuthal
observables for anisotropic flow analyses which can be measured for the first
time with our new framework. Effects of finite detector granularity on
multi-particle correlations are quantified and discussed in detail. We point
out the existence of a systematic bias in traditional differential flow
analyses which stems solely from the applied selection criteria on particles
used in the analyses, and is also present in the ideal case when only flow
correlations are present. Finally, we extend the applicability of our generic
framework to the case of differential multi-particle correlations.
