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We find all factorized duality functions for a class of interacting particlesystems. The functions we recover are self-duality functions for interactingparticle systems such as zero-range processes, symmetric inclusion andexclusion processes, as well as duality and self-duality functions for theircontinuous counterparts. The approach is based on, firstly, a general relationbetween factorized duality functions and stationary product measures and,secondly, an intertwining relation provided by generating functions. For theinteracting particle systems, these self-duality and duality functions turn outto be generalizations of those previously obtained in [9] and, more recently,in [8]. Thus, we discover that only these two families of dualities cover allpossible cases. Moreover, the same method discloses all self-duality functionsfor interacting diffusion systems such as the Brownian energy process, whereboth the process and its dual are in continuous variables.

Factorized duality, stationary product measures and generating functions