Multiplicative duality, q-triplet and (μ,ν,q)-relation derived from the one-to-one correspondence between the (μ,ν)-multinomial coefficient and Tsallis entropy Sq

We derive the multiplicative duality “q↔1/q” and other typical mathematical structures as the special cases of the (μ,ν,q)-relation behind Tsallis statistics by means of the (μ,ν)-multinomial coefficient. Recently the additive duality “q↔2-q” in Tsallis statistics is derived in the form of the one-to-one correspondence between the q-multinomial coefficient and Tsallis entropy. A slight generalization of this correspondence for the multiplicative duality requires the (μ,ν)-multinomial coefficient as a generalization of the q-multinomial coefficient. This combinatorial formalism provides us with the one-to-one correspondence between the (μ,ν)-multinomial coefficient and Tsallis entropy Sq, which determines a concrete relation among three parameters μ,ν and q, i.e., ν(1-μ)+1=q which is called “(μ,ν,q)-relation” in this paper. As special cases of the (μ,ν,q)-relation, the additive duality and the multiplicative duality are recovered when ν=1 and ν=q, respectively. As other special cases, when ν=2-q, a set of three parameters (μ,ν,q) is identified with the q-triplet (qsen,qrel,qstat) recently conjectured by Tsallis. Moreover, when ν=1/q, the relation 1/(1-qsen)=1/αmin-1/αmax in the multifractal singularity spectrum f(α) is recovered by means of the (μ,ν,q)-relation