The roots of Bessel functions of order one-half are special cases of roots of transcendental equations of the form
tan
z
=
A
(
z
)
/
B
(
z
)
\tan z = A(z)/B(z)
, where
A
(
z
)
,
B
(
z
)
A(z),B(z)
are polynomials and
A
(
z
)
/
B
(
z
)
A(z)/B(z)
is odd. We prove that the function
f
(
z
)
=
B
(
z
)
sin
z
−
A
(
z
)
cos
z
,
f
(
z
)
f(z) = B(z)\sin z - A(z)\cos z,f(z)
even or odd, satisfies the conditions of Hadamard’s factorization theorem, and derive recurrences for sums of the form
S
l
=
∑
k
=
1
∞
α
k
−
2
l
,
l
=
1
,
2
,
⋯
{S_l} = \sum \nolimits _{k = 1}^\infty {\alpha _k^{ - 2l},l = 1,2, \cdots }
, where the
α
k
{\alpha _k}
’s are the nonzero roots of
f
(
z
)
f(z)
. We also prove under what conditions on
A
(
z
)
A(z)
and
B
(
z
)
B(z)
is
S
l
=
π
−
2
l
−
2
ζ
(
2
l
+
2
)
{S_l} = {\pi ^{ - 2l - 2}}\zeta (2l + 2)
or
S
l
=
π
−
2
l
−
2
ζ
(
2
l
+
2
)
(
2
2
l
+
2
−
1
)
{S_l} = {\pi ^{ - 2l - 2}}\zeta (2l + 2)({2^{2l + 2}} - 1)
, where
ζ
\zeta
is the Riemann zeta function. We prove that, although Bessel functions of positive half-order,
J
l
+
1
/
2
{J_{l + 1/2}}
, have only real roots, perturbation of any one of its coefficients introduces nonreal roots for
l
>
2
l > 2
.