We consider the dependence between the violation of the quadrature uncertainty relation, the sign and the existence of the photon distribution function. Using entropic inequalities for Shannon entropies new inequalities for some classical polynomials are obtained. To this end, photon dis￾tribution functions for one-, two- and multi-mode squeezed states in terms of Hermite, Laguerre, Legendre polynomials and Gauss’ hypergeometric functions are used