In this paper the Christoffel numbers
a
v
,
n
(
λ
)
G
a_{v,n}^{(\lambda )G}
for ultraspherical weight functions
w
λ
{w_\lambda }
,
w
λ
(
x
)
=
(
1
−
x
2
)
λ
−
1
/
2
{w_\lambda }(x) = {(1 - {x^2})^{\lambda - 1/2}}
, are investigated. Using only elementary functions, we state new inequalities, monotonicity properties and asymptotic approximations, which improve several known results. In particular, denoting by
θ
v
,
n
(
λ
)
\theta _{v,n}^{(\lambda )}
the trigonometric representation of the Gaussian nodes, we obtain for
λ
∈
[
0
,
1
]
\lambda \in [0,1]
the inequalities
\[
π
n
+
λ
sin
2
λ
θ
v
,
n
(
λ
)
{
1
−
λ
(
1
−
λ
)
2
(
n
+
λ
)
2
sin
2
θ
v
,
n
(
λ
)
}
≤
a
v
,
n
(
λ
)
G
≤
π
n
+
λ
sin
2
λ
θ
v
,
n
(
λ
)
\begin {array}{*{20}{c}} {\frac {\pi }{{n + \lambda }}{{\sin }^{2\lambda }}\theta _{v,n}^{(\lambda )}\left \{ {1 - \frac {{\lambda (1 - \lambda )}}{{2{{(n + \lambda )}^2}{{\sin }^2}\theta _{v,n}^{(\lambda )}}}} \right \}} \\ { \leq a_{v,n}^{(\lambda )G} \leq \frac {\pi }{{n + \lambda }}\;{{\sin }^{2\lambda }}\theta _{v,n}^{(\lambda )}} \\ \end {array}
\]
and similar results for
λ
∉
(
0
,
1
)
\lambda \notin (0,1)
. Furthermore, assuming that
θ
v
,
n
(
λ
)
\theta _{v,n}^{(\lambda )}
remains in a fixed closed interval, lying in the interior of
(
0
,
π
)
(0,\pi )
as
n
→
∞
n \to \infty
, we show that, for every fixed
λ
>
−
1
/
2
\lambda > - 1/2
,
\[
a
v
,
n
(
λ
)
G
=
π
n
+
λ
sin
2
λ
θ
v
,
n
(
λ
)
{
1
−
λ
(
1
−
λ
)
2
(
n
+
λ
)
2
sin
2
θ
v
,
n
(
λ
)
−
λ
(
1
−
λ
)
[
3
(
λ
+
1
)
(
λ
−
2
)
+
4
sin
2
θ
v
,
n
(
λ
)
]
8
(
n
+
λ
)
4
sin
4
θ
v
,
n
(
λ
)
}
+
O
(
n
−
7
)
.
a_{v,n}^{(\lambda )G} = \frac {\pi }{{n + \lambda }}\;{\sin ^{2\lambda }}\theta _{v,n}^{(\lambda )}\left \{ {1 - \frac {{\lambda (1 - \lambda )}}{{2{{(n + \lambda )}^2}{{\sin }^2}\theta _{v,n}^{(\lambda )}}} - \frac {{\lambda (1 - \lambda )\;[3(\lambda + 1)(\lambda - 2) + 4{{\sin }^2}\theta _{v,n}^{(\lambda )}]}}{{8{{(n + \lambda )}^4}{{\sin }^4}\theta _{v,n}^{(\lambda )}}}} \right \} + O({n^{ - 7}}).
\]