  For non-negative integers $l_{1}, l_{2},\ldots, l_{n}$, we define character
sums $\varphi_{(l_{1}, l_{2},\ldots, l_{n})}$ and $\psi_{(l_{1}, l_{2},\ldots,
l_{n})}$ over a finite field which are generalizations of Jacobsthal and
modified Jacobsthal sums, respectively. We express these character sums in
terms of Greene's finite field hypergeometric series. We then express the
number of points on the hyperelliptic curves $y^2=(x^m+a)(x^m+b)(x^m+c)$ and
$y^2=x(x^m+a)(x^m+b)(x^m+c)$ over a finite field in terms of the character sums
$\varphi_{(l_{1}, l_{2}, l_{3})}$ and $\psi_{(l_{1}, l_{2}, l_{3})}$, and
finally obtain expressions in terms of the finite field hypergeometric series.
