On the $q$-generalised multinomial/divergence correspondence

The asymptotic correspondence between the probability mass function of the $q$-deformed multinomial distribution and the $q$-generalised Kullback-Leibler divergence, also known as Tsallis relative entropy, is established. The probability mass function is generalised using the $q$-deformed algebra developed within the framework of nonextensive statistics, leading to the emergence of a family of divergence measures in the asymptotic limit as the system size increases. The coefficients in the asymptotic expansion yield Tsallis relative entropy as the leading-order term when $q$ is interpreted as an entropic parameter. Furthermore, higher-order expansion coefficients naturally introduce new divergence measures, extending Tsallis relative entropy through a one-parameter generalisation. Some fundamental properties of these extended divergences are also explored.