The entire function

$\phi \left(x\right)={\int}_{-\infty}^{\infty}exp(-{w}^{2m}+ixw)dw$,

$m\in \mathbb{N}$ is studied. It appears in many areas: in Waring’s problem, as a solution of a special form of Turrittin’s differential equation, as a generalization of the Airy function, in questions about analytic hypoellipticity of the tangential Cauchy-Riemann operator, in the representation of the Bergman and Szegő kernel of weakly pseudoconvex domains in

${\u2102}^{2}$ and in a connection between Brownian motion and a generalized heat equation. First the asymptotic behavior of

$\phi $ at infinity is considered, then the asymptotic expansion of

$\phi $ is computed. It is also shown that

$\phi $ can be approximated by the Bessel function. In the final part of the paper the properties of the zeroes of

$\phi $ are investigated. It is added as a note that meanwhile the conjecture that all zeroes of

$\phi $ are simple has been verified.