Let
P
∈
Q
p
[
x
,
y
]
P\in \Bbb Q_p[x,y]
,
s
∈
C
s\in \Bbb C
with sufficiently large real part, and consider the integral operator
(
A
P
,
s
f
)
(
y
)
≔
1
1
−
p
−
1
∫
Z
p
|
P
(
x
,
y
)
|
s
f
(
x
)
|
d
x
|
\begin{equation*} (A_{P,s}f)(y)≔\frac {1}{1-p^{-1}}\int _{\Bbb Z_p}|P(x,y)|^sf(x) |dx| \end{equation*}
on
L
2
(
Z
p
)
L^2(\Bbb Z_p)
. We show that if
P
P
is homogeneous of degree
d
d
then for each character
χ
\chi
of
Z
p
×
\Bbb Z_p^\times
the characteristic function
det
(
1
−
u
A
P
,
s
,
χ
)
\det (1-uA_{P,s,\chi })
of the restriction
A
P
,
s
,
χ
A_{P,s,\chi }
of
A
P
,
s
A_{P,s}
to the eigenspace
L
2
(
Z
p
)
χ
L^2(\Bbb Z_p)_\chi
is the
q
q
-Wronskian of a set of solutions of a (possibly confluent)
q
q
-hypergeometric equation, where
q
=
p
−
1
−
d
s
q=p^{-1-ds}
. In particular, the nonzero eigenvalues of
A
P
,
s
,
χ
A_{P,s,\chi }
are the reciprocals of the zeros of such
q
q
-Wronskian.