Characteristic functions of polynomial-normal distributions
The polynomial-normal distribtions with densities \[ f(x)=p_{2n}(x)\exp{(-x^2/2c^2)} \] are considered, where \(p_{2n}\) is a nonnegative polynomial of the degree \(2n\geq 0\), \(c>0\). The characteristic function of the density \(f\) is given by the formula \[ \varphi(\xi)=\sum_{r=0}^{2n}(i\xi c)^r H_r \exp{(-\xi^2 c^2/2)}, \] where \(H_r\) is the Hermite polynomial of degree \(r\). The authors consider functions of the form \[ \varphi(\xi)=q_{2n}(i\xi)\exp{(-\xi^2 c^2/2)}= \sum_{r=0}^{2n}(i\xi)^r a_{r,2n} \exp{(-\xi^2 c^2/2)}, \] and give nessesary and sufficient conditions in terms of \(a_{r,2n}\) for \(\varphi(\xi)\) to be a characteristic function of a polynomial-normal distribution. See also \textit{A. Plucińska} [Demonstr. Math. 32, No. 1, 195-206 (1999; Zbl 0932.60014)].
scientific article; zbMATH DE number 1529329