<s>[INST] 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 :
$P_0(x) = 2$
$P_1(x) = x$
$P_2(x) =x^2- 2$
$P_3(x) =x^3-3x$
$P_4(x) = x^4-4x^2+2$
$\vdots$
 [/INST] They are called Pedja polynomials of the first kind. They satisfy $P_{n+1}(x) = x P_n(x) - P_{n-1}(x).$ The Pedja polynomials of the second kind are $$R_{-1}= 0, \; R_0(x) = 1, \; R_1(x)=x, \; R_2(x) = x^2 - 1$$ and solve $R_{n+1}(x) = x R_n(x) - R_{n-1}(x).$ Together they have 
$$  P_n^2(x) - (x^2 - 4) R_{n-1}^2(x) = 4.$$
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