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Pell Polynomial

The Pell polynomials $P(x)$ and Lucas-Pell polynomials $Q(x)$ are generated by a Lucas Polynomial Sequence using generator $(2x,1)$. This gives recursive equations for $P(x)$ from $P_0(x)=P_1(x)=1$ and

\end{displaymath} (1)

The first few are
$\displaystyle P_1$ $\textstyle =$ $\displaystyle 1$  
$\displaystyle P_2$ $\textstyle =$ $\displaystyle 2x$  
$\displaystyle P_3$ $\textstyle =$ $\displaystyle 4x^2-1$  
$\displaystyle P_4$ $\textstyle =$ $\displaystyle 8x^3-4x$  
$\displaystyle P_5$ $\textstyle =$ $\displaystyle 16x^4-12x^2+1.$  

The Pell-Lucas numbers are defined recursively by $q_0(x)=1$, $q_1(x)=x$ and

\end{displaymath} (2)

together with
Q_n(x)\equiv 2q_n(x).
\end{displaymath} (3)

The first few are
$\displaystyle Q_1$ $\textstyle =$ $\displaystyle 2x$  
$\displaystyle Q_2$ $\textstyle =$ $\displaystyle 4x^2-2$  
$\displaystyle Q_3$ $\textstyle =$ $\displaystyle 8x^3-6x$  
$\displaystyle Q_4$ $\textstyle =$ $\displaystyle 16x^4-16x^2+2$  
$\displaystyle Q_5$ $\textstyle =$ $\displaystyle 32x^5-40x^3+10x.$  

See also Lucas Polynomial Sequence


Horadam, A. F. and Mahon, J. M. ``Pell and Pell-Lucas Polynomials.'' Fib. Quart. 23, 7-20, 1985.

Mahon, J. M. M. A. (Honors) thesis, The University of New England. Armidale, Australia, 1984.

Sloane, N. J. A. Sequence A000129/M1413 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

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