Equivalence of the four versions of Tsallis statistics

In spite of its undeniable success, there are still open questions regarding Tsallis non-extensive statistical formalism, whose founding stone was laid in 1988 in JSTAT. Some of them are concerned with the so-called normalization problem of just how to evaluate expectation values. The Jaynes MaxEnt approach for deriving statistical mechanics is based on the adoption of (1) a specific entropic functional form S and (2) physically appropriate constraints. The literature on non-extensive thermostatistics has considered, in its historical evolution, four possible choices for the evaluation of expectation values: (i) 1988 Tsallis-original (TO), (ii) Curado-Tsallis (CT), (iii) Tsallis-Mendes- Plastino (TMP), and (iv) the same as (iii), but using centered operators as constraints (OLM). The 1988 was promptly abandoned and replaced, mostly with versions ii) and iii). We will here (a) show that the 1988 is as good as any of the others, (b) demonstrate that the four cases can be easily derived from just one (any) of them, i.e., the probability distribution function in each of these four instances may be evaluated with a unique formula, and (c) numerically analyze some consequences that emerge from these four choices.