Pell Polynomial

The Pell polynomials and Lucas-Pell polynomials are generated by a Lucas Polynomial Sequence using generator . This gives recursive equations for from and

$\begin{displaymath} P_{n+2}(x)=2xP_{n+1}(x)+P_n(x). \end{displaymath}$

(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 , and

$\begin{displaymath} q_{n+2}(x)=2xq_{n+1}(x)+q_n(x), \end{displaymath}$

(2)

together with

$\begin{displaymath} Q_n(x)\equiv 2q_n(x). \end{displaymath}$

(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

References

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.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.