Namespaces
Variants
Actions

Lucas sequence

From Encyclopedia of Mathematics
Jump to: navigation, search

A recursive sequence of integers defined by two integer parameters. Given integers , Q with D =P^2 - 4Q \neq 0, the Lucas sequences of the first kind, U_n = U_n(P,Q) and of the second kind V_n = V_n(P,Q) are the recursive sequences defined by the same linear recurrence relation \begin{equation}\label{eq:1} X_{n+1} = P.X_n - Q.X_{n-1} \end{equation} but with starting values U_0 = 0, U_1 = 1 and V_0 = 2, V_1 = P.

The characteristic equation of the recurrence relation is X^2 = P.X - Q with roots \alpha,\beta = (P \pm \sqrt{D})/2. In terms of \alpha,\beta we have U_n = \frac{\alpha^n - \beta^n}{\alpha - \beta}\ , V_n = \alpha^n + \beta^n \ , so that \alpha^n,\beta^n = \frac{U_n \pm V_n\sqrt{D}}{2} \ .

In the case P=1,Q=1 the Fibonacci numbers and Lucas numbers are the Lucas sequences of the first and second kind respectively.

The z-transform of a Lucas sequence \sum_{n=0}^\infty X_n z^{-n} satisfying \eqref{eq:1} is a rational function with denominator z^2 - Pz + Q. Indeed, \sum_{n=0}^\infty U_n z^{-n} = \frac{1}{\alpha-\beta}\left(\frac{z}{z-\alpha} - \frac{z}{z-\beta}\right) = \frac{z}{z^2 - Pz + Q} \ . \sum_{n=0}^\infty V_n z^{-n} = \frac{z}{z-\alpha} + \frac{z}{z-\beta} = \frac{z(2z-P)}{z^2 - Pz + Q} \ .

Lehmer numbers

The Lehmer numbers are a modified form of Lucas sequence of the first kind, defined in terms of \alpha, \beta algebraic numbers with \alpha\beta and (\alpha+\beta)^2 integers: W_n = \begin{cases}\frac{\alpha^n - \beta^n}{\alpha - \beta},&n\,\text{odd},\\ \frac{\alpha^n - \beta^n}{\alpha^2 - \beta^2},&n\,\text{even}.\end{cases}

References

  • Paulo Ribenboim, "My Numbers, My Friends: Popular Lectures on Number Theory". Springer-Verlag (2002) ISBN 0-387-98911-0.
  • Arthur T. Benjamin; Jennifer J. Quinn, "Proofs that Really Count". Mathematical Association of America (2003) ISBN 0883853337.
  • Wladyslaw Narkiewicz, "Rational Number Theory in the 20th Century: From PNT to FLT", Springer Monographs in Mathematics, Springer (2011) ISBN 0857295322
How to Cite This Entry:
Lucas sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lucas_sequence&oldid=54564