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

Abstract

We derive the multiplicative duality &#34;q<->1/q&#34; and other typical mathematical structures as the special cases of the (mu,nu,q)-relation behind Tsallis statistics by means of the (mu,nu)-multinomial coefficient. Recently the additive duality &#34;q<->2-q&#34; 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 (mu,nu)-multinomial coefficient as a generalization of the q-multinomial coefficient. This combinatorial formalism provides us with the one-to-one correspondence between the (mu,nu)-multinomial coefficient and Tsallis entropy Sq, which determines a concrete relation among three parameters mu, nu and q, i.e., nu(1-mu)+1=q which is called &#34;(mu,nu,q)-relation&#34; in this paper. As special cases of the (mu,nu,q)-relation, the additive duality and the multiplicative duality are recovered when nu=1 and nu=q, respectively. As other special cases, when nu=2-q, a set of three parameters (mu,nu,q) is identified with the q-triplet (q_{sen},q_{rel},q_{stat}) recently conjectured by Tsallis. Moreover, when nu=1/q, the relation 1/(1-q_{sen})=1/alpha_{min}-1/alpha_{max} in the multifractal singularity spectrum f(alpha) is recovered by means of the (mu,nu,q)-relation.