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

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

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

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