We prove that if
ν
>
1
/
2
\nu > 1/2
, then
2
ν
−
1
Γ
(
ν
)
/
[
x
ν
/
2
e
x
K
ν
(
x
)
]
{2^{\nu - 1}}\Gamma (\nu )/[{x^{\nu /2}}{e^{\sqrt x }}{K_\nu }(\sqrt x )]
is the Laplace transform of a selfdecomposable probability distribution while
2
ν
Γ
(
ν
+
1
)
x
−
ν
/
2
e
−
x
I
ν
(
x
)
{2^\nu }\Gamma \left ( {\nu + 1} \right ){x^{ - \nu /2}}{e^{ - \sqrt x }}{I_\nu }\left ( {\sqrt x } \right )
is the Laplace transform of an infinitely divisible distribution. The former result is used to show that an estimate of
M
{\text {M}}
. Wong [13] is sharp. We also prove that the roots of the equations
\[
b
3
l
ν
−
1
(
a
z
)
/
I
ν
(
a
z
)
=
a
3
I
ν
−
1
(
b
z
)
/
I
ν
(
b
z
)
,
{b^3}{l_{\nu - 1}}\left ( {a\sqrt z } \right )/{I_\nu }\left ( {a\sqrt z } \right ) = {a^3}{I_{\nu - 1}}\left ( {b\sqrt z } \right )/{I_\nu }\left ( {b\sqrt z } \right ),
\]
and
\[
b
3
K
ν
+
1
(
a
z
)
/
K
ν
(
a
z
)
=
a
3
K
ν
+
1
(
b
z
)
/
K
ν
(
b
z
)
,
ν
>
0
,
z
≠
0
,
{b^3}{K_{\nu + 1}}\left ( {a\sqrt z } \right )/{K_\nu }\left ( {a\sqrt z } \right ) = {a^3}{K_{\nu + 1}}\left ( {b\sqrt z } \right )/{K_\nu }\left ( {b\sqrt z } \right ),\nu > 0,z \ne 0,
\]
lie in a certain sector contained in the open left half plane. This proves and extends a conjecture of
H
{\text {H}}
. Hattori arising from his work in partial differential equations.