\[T_{n}\left(x\right)=\tfrac{1}{2}n\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}%
\frac{(-1)^{\ell}(n-\ell-1)!}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}=2^{n-1}x^{n}{{}%
_{2}F_{1}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop 1-n};\frac{1}%
{x^{2}}\right),\]