This article is within the scope of WikiProject Mathematics , a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.Mathematics Wikipedia:WikiProject Mathematics Template:WikiProject Mathematics mathematics articles Low This article has been rated as Low-priority on the project's priority scale .
Moved unreferenced content that someone added. Given the recent sockpuppetry involving IP addresses in Austria and User:A. Pichler in our articles on special functions, please provide citations before adding such content. Sławomir Biały (talk ) 00:48, 20 October 2012 (UTC) Reply
Another important series expansion is given by
∑
n
=
0
∞
C
n
(
α
)
(
x
)
(
2
α
+
n
−
1
n
)
t
n
n
!
=
Γ
(
α
+
1
2
)
e
t
x
J
α
−
1
2
(
t
1
−
x
2
)
(
1
2
t
1
−
x
2
)
α
−
1
2
,
{\displaystyle \sum _{n=0}^{\infty }{\frac {C_{n}^{(\alpha )}(x)}{2\alpha +n-1 \choose n}}{\frac {t^{n}}{n!}}=\Gamma \left(\alpha +{\frac {1}{2}}\right)e^{tx}{\frac {J_{\alpha -{\frac {1}{2}}}\left(t{\sqrt {1-x^{2}}}\right)}{\left({\frac {1}{2}}t{\sqrt {1-x^{2}}}\right)^{\alpha -{\frac {1}{2}}}}},}
where
J
α
{\displaystyle J_{\alpha }}
is the Bessel function .
The Askey–Gasper inequality has the generalization
∑
j
=
0
n
C
j
α
(
x
)
(
2
α
+
j
−
1
j
)
s
j
=
2
F
1
(
α
−
1
2
,
1
2
;
α
+
1
2
;
s
2
(
1
−
x
)
(
1
+
x
)
1
−
2
s
x
+
s
2
)
1
−
2
x
s
+
s
2
≥
0
(
−
1
≤
x
≤
1
,
−
1
≤
s
≤
1
,
α
≥
0
)
.
{\displaystyle \sum _{j=0}^{n}{\frac {C_{j}^{\alpha }(x)}{2\alpha +j-1 \choose j}}s^{j}={\frac {\,_{2}F_{1}\left(\alpha -{\frac {1}{2}},{\frac {1}{2}};\alpha +{\frac {1}{2}};{\frac {s^{2}(1-x)(1+x)}{1-2sx+s^{2}}}\right)}{\sqrt {1-2xs+s^{2}}}}\geq 0\qquad (-1\leq x\leq 1,\,-1\leq s\leq 1,\,\alpha \geq 0).}
for Gegenbauer polynomials.