The entire function
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
and in a connection between Brownian motion and a generalized heat equation. First the asymptotic behavior of
at infinity is considered, then the asymptotic expansion of
is computed. It is also shown that
can be approximated by the Bessel function. In the final part of the paper the properties of the zeroes of
are investigated. It is added as a note that meanwhile the conjecture that all zeroes of
are simple has been verified.