Operational properties of a generalized Hermite transformation (Q1089867)

The author studies the generalized Hermite transformation \[ F(z)=\int^{\infty}_{0}f(t)H_ z(t)e^{-t^ 2}dt,\quad z\in {\mathbb{C}} \] on some original space, where \(H_ z\) is the Hermite function \(H_ z(t)=2^ zG(-z/2,1/2,t^ 2),\) arg(t)\(\geq 0\), \(z\in {\mathbb{C}}\), G is the confluent hypergeometric function of the second kind. Some convolution theorems and an inversion formula are obtained.