  In this paper we extend our recent results [Physica A340 (2004)110] on
q-nonextensive statistics with non-Tsallis entropies. In particular, we combine
an axiomatics of Renyi with the q-deformed version of Khinchin axioms to obtain
the entropy which accounts both for systems with embedded self-similarity and
q-nonextensivity. We find that this entropy can be uniquely solved in terms of
a one-parameter family of information measures. The corresponding entropy
maximizer is expressible via a special function known under the name of the
Lambert W-function. We analyze the corresponding "high" and "low-temperature"
asymptotics and make some remarks on the possible applications.
