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Jacobsthal Polynomial

The Jacobsthal polynomials are the Polynomials obtained by setting $p(x)=1$ and $q(x)=2x$ in the Lucas Polynomial Sequence. The first few Jacobsthal polynomials are

$\displaystyle J_1(x)$ $\textstyle =$ $\displaystyle 1$  
$\displaystyle J_2(x)$ $\textstyle =$ $\displaystyle 1$  
$\displaystyle J_3(x)$ $\textstyle =$ $\displaystyle 1+2x$  
$\displaystyle J_4(x)$ $\textstyle =$ $\displaystyle 1+4x$  
$\displaystyle J_5(x)$ $\textstyle =$ $\displaystyle 4x^2+6x+1,$  

and the first few Jacobsthal-Lucas polynomials are
$\displaystyle j_1(x)$ $\textstyle =$ $\displaystyle 1$  
$\displaystyle j_2(x)$ $\textstyle =$ $\displaystyle 4x+1$  
$\displaystyle j_3(x)$ $\textstyle =$ $\displaystyle 6x+1$  
$\displaystyle j_4(x)$ $\textstyle =$ $\displaystyle 8x^2+8x+1$  
$\displaystyle j_5(x)$ $\textstyle =$ $\displaystyle 20x^2+10x+1.$  

Jacobsthal and Jacobsthal-Lucas polynomials satisfy



where $J_n$ is a Jacobsthal Number and $j_n$ is a Jacobsthal-Lucas Number.

© 1996-9 Eric W. Weisstein