Let $P\in \Bbb Q_p[x,y]$, $s\in \Bbb C$ with sufficiently large real part, and consider the integral operator $ (A_{ P,s}f)(y):=\frac{ 1}{ 1-p^{ -1}}\int_{ \Bbb Z_p}|P(x,y)|^sf(x) |dx| $ on $L^2(\Bbb Z_p)$. We show that if $P$ is homogeneous then for each character $\chi$ of $\Bbb Z_p^\times$ the characteristic function $\det(1-uA_{ P,s,\chi})$ of the restriction $A_{ P,s,\chi}$ of $A_{ P,s}$ to the eigenspace $L^2(\Bbb Z_p)_\chi$ is the $q$-Wronskian of a set of solutions of a (possibly confluent) $q$-hypergeometric equation. In particular, the nonzero eigenvalues of $A_{ P,s,\chi}$ are the reciprocals of the zeros of such $q$-Wronskian.
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