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Pell Polynomial

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PellPolynomial

The Pell polynomials P(x) are the W-polynomials generated by the Lucas polynomial sequence using the generator p(x)=2x, q(x)=1. This gives recursive equations for P(x) from P_0(x)=0, P_1(x)=1, and

P_(n+2)(x)=2xP_(n+1)(x)+P_n(x).
(1)

They are related to the Fibonacci polynomials by

P_n(x)=F_n(2x).
(2)

The first few are

P_1(x)=1
(3)
P_2(x)=2x
(4)
P_3(x)=4x^2+1
(5)
P_4(x)=8x^3+4x
(6)
P_5(x)=16x^4+12x^2+1
(7)

(OEIS A115322).

See also

Lucas Polynomial Sequence, Pell-Lucas Polynomial, Pell Number

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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 A115322 in "The On-Line Encyclopedia of Integer Sequences."

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Pell Polynomial

Cite this as:

Weisstein, Eric W. "Pell Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PellPolynomial.html

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