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Lucas Polynomial

The $w$ Polynomials obtained by setting $p(x)=x$ and $q(x)=1$ in the Lucas Polynomial Sequence. The first few are

$\displaystyle F_1(x)$ $\textstyle =$ $\displaystyle x$  
$\displaystyle F_2(x)$ $\textstyle =$ $\displaystyle x^2+2$  
$\displaystyle F_3(x)$ $\textstyle =$ $\displaystyle 3x^3+3x$  
$\displaystyle F_4(x)$ $\textstyle =$ $\displaystyle x^4+4x^2+2$  
$\displaystyle F_5(x)$ $\textstyle =$ $\displaystyle x^5+5x^3+5x.$  

The corresponding $W$ Polynomials are called Fibonacci Polynomials. The Lucas polynomials satisfy


where the $L_{n}$s are Lucas Numbers.

See also Fibonacci Polynomial, Lucas Number, Lucas Polynomial Sequence

© 1996-9 Eric W. Weisstein