The probability properties of bilinear expansions in Hermite polynomiels

The present paper contains an investigation of the probability meaning of formulas (1) and (4).
According to the basic theorem 2 for sum (1) to be non negative it is nesessary and sufficient that the sequence of coefficients $\{c_k\}$ be the sequence of moments of some probability distribution in finite interval $[-1,1]$.
Some examples of densities comprised in the normal class in the sence of Fré'chet [7] are given here.