Matrix elements of some irreducible unitary representation of compact and noncompact groups can be represented in terms of special functions. Any unitary matrix can be associated with a bistochastic matrix. Sums of elements of such matrix are equal both in columns and rows to one. It is possible to use the elements of the latter matrices as probabilities. Symmetry properties allow to connect the representation aspects of groups and algebras with properties of special functions [1]. Shanon entropy determines the probability distribution of the random variable (rv) which appears as a result of an experiment with a finite number of outcomes. For two rv’s the joint probability distribution can be obtained. This distribution is connected with N = N1
N2 outcomes. N1 and N2 are observations of the first and the second rv’s, respectively. By Sklar’s theorem the joint distribution function and the dependence between two rv’s determine two marginal distribution functions [2]. For these distributions the Shannon entropies that satisfy the subadditivity condition can be calculated. The entropic inequalities for bipartite systems are used in [3] in the framework of the tomographic probability representation of quantum mechanics to characterize two degrees of quantum correlations in the systems. For systems without subsystems these inequalities are introduced in [4]. We can apply the subadditivity condition in all cases, where the set of nonnegative numbers or functions with unity sums is arisen. For Lie subgroups like SU(2) and SU(1; 1) unitary irreducible representations in terms of Jacobi, Legendre and Gauss’ hypergeometric polynomials are known. We propose new inequalities for the latter polynomials. Such inequalities for the system with the spin j = 3=2 are introduced in [5]. Considering the matrix elements of the unitary irreducible representations of the groups SU(2) and SU(1; 1) and applying the subadditivity condition for the joint probability distributions constructed from these matrix elements we obtain new inequalities for the Jacobi and the Gauss’ hypergeometric polynomials. The results are illustrated by examples of the systems with the spins j = 3=2 and j = 2. Using other mappings or entropies, e.g., Tsallis entropy, other inequalities for the special functions can be written.