Calculating statistical distributions from operator relations: the statistical distributions of various intermediate statistics

In this paper, we give a general discussion on the calculation of the statistical distribution from a given operator relation of creation, annihilation, and number operators. Our result shows that as long as the relation between the number operator and the creation and annihilation operators can be expressed as $a^{\dagger}b=\Lambda\left(N\right)$ or $N=\Lambda^{-1} \left( a^{\dagger}b\right)$, where $N$, $a^{\dagger}$, and $b$ denote the number, creation, and annihilation operators, i.e., $N$ is a function of quadratic product of the creation and annihilation operators, the corresponding statistical distribution is the Gentile distribution, a statistical distribution in which the maximum occupation number is an arbitrary integer. As examples, we discuss the statistical distributions corresponding to various operator relations. In particular, besides Bose-Einstein and Fermi-Dirac cases, we discuss the statistical distributions for various schemes of intermediate statistics, especially various $q$-deformation schemes. Our result shows that the statistical distributions corresponding to various $q$-deformation schemes are various Gentile distributions with different maximum occupation numbers which are determined by the deformation parameter $q$. This result shows that the results given in much literature on the $q$-deformation distribution are inaccurate or incomplete.