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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 :

$P_0(x) = 2$

$P_1(x) = x$

$P_2(x) =x^2- 2$

$P_3(x) =x^3-3x$

$P_4(x) = x^4-4x^2+2$

$\vdots$

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2 Answers

up vote 4 down vote accepted

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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Pedja? –  anon Feb 18 '12 at 8:49
    
@WillJagy,Thanks...:-),I think that these polynomials can be used for testing primality of Generalized Fermat numbers...for example $P_4(x)$ can be used for testing primality of ordinary Fermat numbers... –  pedja Feb 18 '12 at 8:51
    
I think this should be the accepted answer... :D –  J. M. Feb 18 '12 at 9:04
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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:

$$\begin{align*} 2^{-n}((x+\sqrt{x^2-4})^n+(x-\sqrt{x^2-4})^n)&=\left(\frac{x}{2}+\sqrt{\frac{x^2}{4}-1}\right)^n+\left(\frac{x}{2}-\sqrt{\frac{x^2}{4}-1}\right)^n\\ &=\left(\frac{x}{2}+\sqrt{\left(\frac{x}{2}\right)^2-1}\right)^n+\left(\frac{x}{2}-\sqrt{\left(\frac{x}{2}\right)^2-1}\right)^n\\ &=2T_n\left(\frac{x}{2}\right) \end{align*}$$

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