Generalization of the possible algebraic basis of $q$-triplets

Abstract: The so called $q$-triplets were conjectured in 2004 and then found in nature in 2005. A relevant further step was achieved in 2005 when the possibility was advanced that they could reflect an entire infinite algebra based on combinations of the self-dual relations $q \to 2-q$ ({\it additive duality}) and $q \to 1/q$ ({\it multiplicative duality}). The entire algebra collapses into the single fixed point $q=1$, corresponding to the Boltzmann-Gibbs entropy and statistical mechanics. For $q \ne 1$, an infinite set of indices $q$ appears, corresponding in principle to an infinite number of physical properties of a given complex system describable in terms of the so called $q$-statistics. The basic idea that is put forward is that, for a given universality class of systems, a small number (typically one or two) of independent $q$ indices exist, the infinite others being obtained from these few ones by simply using the relations of the algebra. The $q$-triplets appear to constitute a few central elements of the algebra. During the last decade, an impressive amount of $q$-triplets have been exhibited in analytical, computational, experimental and observational results in natural, artificial and social systems. Some of them do satisfy the available algebra constructed solely with the additive and multiplicative dualities, but some others seem to violate it. In the present work we generalize those two dualities with the hope that a wider set of systems can be handled within. The basis of the generalization is given by the {\it selfdual} relation $q \to q_a(q) \equiv \frac{(a+2) -aq}{a-(a-2)q} \,\, (a \in {\cal R})$. We verify that $q_a(1)=1$, and that $q_2(q)=2-q$ and $q_0(q)=1/q$. To physically motivate this generalization, we briefly review illustrative applications of $q$-statistics, in order to exhibit possible candidates where the present generalized algebras could be useful.