A photon distribution for one-, two- and multi-mode field states can be represented
by special functions. Hermite, Laguerre, Legendre and Gauss' hypergeometric functions are used to represent photon distributions for the mixed light with a generic
Gaussian Wigner function.
These representations can be used to construct the Shannon entropies which satisfy the subadditivity condition. The entropic inequalities for bipartite systems are used in the framework of the tomographic probability representation of quantum
mechanics to characterize two degrees of quantum correlations in the systems. The
subadditivity condition can be applied when the set of nonnegative functions with
the unity sum is arisen.
We consider the polynomial representation of the photon distributions to construct
new polynomial relations and investigate the dependence between the nonobservance of the quadrature uncertainty relation and the existence of the photon
distribution function. The violation of the quadrature uncertainty relation leads to
complex values of the probability.