  Generic axiomatic-nonextensive statistics characterized by two asymptotic
properties, to each of them a scaling function is assigned, characterized by c
and d for first and second scaling property, respectively, is formulated in a
grand-canonical ensemble with finite volume in the thermodynamic limit. The
thermodynamic properties of a relativistic ideal gas of hadron resonances are
studied. It is found that this generic statistics satisfies the requirements of
the equilibrium thermodynamics. Essential aspects of thermodynamic
self-consistency are clarified. Analytical expressions are proposed for the
statistical fits of various transverse momentum distributions measured in
most-central collisions at different collision energies and colliding systems.
Estimations for the freezeout temperature and chemical potential and both c and
d are determined. The earlier are found compatible with the parameters deduced
from Boltzmann-Gibbs (BG) statistics (extensive), while the latter refer to
generic nonextensivity. The resulting equivalence class (c,d) is associated to
stretched exponentials, where Lambert function reaches its asymptotic
stability. In some measurements, the resulting nonextensive entropy is linearly
composed of extensive entropies. Apart from power-scaling, the particle ratios
and yields are excellent quantities to highlighting whether the particle
production takes place, (non)extensively. Various particle ratios and yields
measured by the STAR experiment in central collisions at 200, 62.4 and 7.7 GeV
are fitted to this novel approach. We found that both c and d<1, i.e. referring
to neither BG- nor Tsallis-type statistics, but to (c,d)-entropy, where Lambert
functions exponentially raise. We conclude that the particle production at STAR
energies is likely a nonextensive process but not necessarily BG or Tsallis
type.
