View source for U n(x)=Sum (-1)^k n!/((2k+1)!(n-2k-1)!)(1-x^2)^(k+1/2)x^(n-2k-1)

==Theorem==
The following formula holds:
$$U_n(x) = \displaystyle\sum_{k=0}^{\left\lfloor \frac{n-1}{2} \right\rfloor} \dfrac{(-1)^k n!}{(2k+1)!(n-2k-1)!} (1-x^2)^{k+\frac{1}{2}}x^{n-2k-1},$$
where $U_n$ denotes [[Chebyshev U]] and $\lfloor \frac{n-1}{2} \rfloor$ denotes the [[floor]].

==Proof==

==References==
* {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=T n(x)=Sum (-1)^k n!/((2k)! (n-2k)!) (1-x^2)^k x^(n-2k)|next=findme}}: Theorem 7.2 (ii)

[[Category:Theorem]]
[[Category:Unproven]]