  In this paper we introduce appropriate associated function to the sequence
$M_p=p^{\t p^{\s}}$, $p\in \N$, $\t>0$, $\s>1$, and derive its sharp asymptotic
estimates in terms of the Lambert $W$ function. These estimates are used to
prove a Paley-Wiener type theorem for compactly supported functions from
extended Gevrey classes. As an application, we discuss properties of the
corresponding wave front sets.
