The Gauss-Airy functions and their properties

Abstract

 In this  paper, in connection with the generating function of three-variable Hermite polynomials, we  introduce  the Gauss-Airy function
 $${\rm {GAi}}(x;y,z)=\frac{1}{\pi}\int_{0}^\infty e^{-yt^2}\cos(xt+ \frac{zt^3}{3})dt,\quad y\geq0,\; x,z\in\mathbb{R}.$$
 Some   properties of this function  such as    behaviors of zeros,   orthogonal relations, corresponding inequalities and their integral transforms are investigated.