  We study the spectrum of the operator $D^*D$, where the operator $D$,
introduced in \cite{KMR}, is a forward derivative on the $p$-adic tree, a
weighted rooted tree associated to $\mathbb Z_p$ via Michon's correspondence.
We show that the spectrum is closely related to the roots of a certain
$q-$hypergeometric function and discuss the analytic continuation of the zeta
function associated with $D^*D$.
