What is the name for the polynomials of the form : $ P_n(x)=2^{-n} \cdot ((x+\sqrt {x^2-4})^n+ (x-\sqrt {x^2-4})^n)$? Polynomials of the form : 
$ T_n(x) =2^{-1} \cdot ((x+\sqrt {x^2-1})^n+ (x-\sqrt {x^2-1})^n)$
are known as Chebyshev polynomials of the first kind .
Consider the polynomials of the form : 
$P_n(x)=2^{-n} \cdot ((x+\sqrt {x^2-4})^n+ (x-\sqrt {x^2-4})^n)$
Have these polynomials some special name ?
First few polynomials of this form are :
$P_0(x) = 2$
$P_1(x) = x$
$P_2(x) =x^2- 2$
$P_3(x) =x^3-3x$
$P_4(x) = x^4-4x^2+2$
$\vdots$

I don't think they do have names (although they bear a superficial resemblance to the Lucas polynomials), but note that your polynomials are mere rescalings of the Chebyshev polynomial of the first kind:
$$\begin{align*}
2^{-n}((x+\sqrt{x^2-4})^n+(x-\sqrt{x^2-4})^n)&=\left(\frac{x}{2}+\sqrt{\frac{x^2}{4}-1}\right)^n+\left(\frac{x}{2}-\sqrt{\frac{x^2}{4}-1}\right)^n\\
&=\left(\frac{x}{2}+\sqrt{\left(\frac{x}{2}\right)^2-1}\right)^n+\left(\frac{x}{2}-\sqrt{\left(\frac{x}{2}\right)^2-1}\right)^n\\
&=2T_n\left(\frac{x}{2}\right)
\end{align*}$$
