  In this paper, we give several characterizations of Herglotz-Nevanlinna
functions in terms of a specific type of positive semi-definite functions
called Poisson-type functions. This allows us to propose a multidimensional
analogue of the classical Nevanlinna kernel and a definition of generalized
Nevanlinna functions in several variables. Furthermore, a characterization of
the symmetric extension of a Herglotz-Nevanlinna function is also given. The
subclass of Loewner functions is also discussed, as well as an interpretation
of the main result in terms of holomorphic functions on the unit polydisk with
non-negative real part.
