Gromov–Witten invariant: Difference between revisions
Jump to navigation
Jump to search
notation consistency for moduli space (mathcal M) |
en>Polytope24 template |
||
Line 1: | Line 1: | ||
In [[mathematics]], a '''Lehmer number''' is a generalization of a [[Lucas sequence]]. | |||
==Algebraic relations== | |||
If a and b are [[complex number]]s with | |||
:<math>a + b = \sqrt{R}</math> | |||
:<math>ab = Q</math> | |||
under the following conditions: | |||
* ''Q'' and ''R'' are [[relatively prime]] nonzero [[integer]]s | |||
* <math>a/b</math> is not a [[root of unity]]. | |||
Then, the corresponding Lehmer numbers are: | |||
:<math>U_n(\sqrt{R},Q) = \frac{a^n-b^n}{a-b}</math> | |||
for ''n'' odd, and | |||
:<math>U_n(\sqrt{R},Q) = \frac{a^n-b^n}{a^2-b^2}</math> | |||
for ''n'' even. | |||
Their companion numbers are: | |||
:<math>V_n(\sqrt{R},Q) = \frac{a^n+b^n}{a+b}</math> | |||
for ''n'' odd and | |||
:<math>V_n(\sqrt{R},Q) = a^n+b^n</math> | |||
for ''n'' even. | |||
== Recurrence == | |||
Lehmer numbers form a linear [[recurrence relation]] with | |||
:<math>U_n=(R-2Q)U_{n-2}-Q^2U_{n-4}=(a^2+b^2)U_{n-2}-a^2b^2U_{n-4}</math> | |||
with initial values <math>U_0=0,U_1=1,U_2=1,U_3=R-Q=a^2+ab+b^2</math>. Similarly the companions sequence satisfies | |||
:<math>V_n=(R-2Q)V_{n-2}-Q^2V_{n-4}=(a^2+b^2)V_{n-2}-a^2b^2V_{n-4}</math> | |||
with initial values <math>V_0=2,V_1=1,V_2=R-2Q=a^2+b^2,V_3=R-3Q=a^2-ab+b^2</math>. | |||
{{numtheory-stub}} | |||
[[Category:Integer sequences]] |
Revision as of 18:44, 25 October 2013
In mathematics, a Lehmer number is a generalization of a Lucas sequence.
Algebraic relations
If a and b are complex numbers with
under the following conditions:
- Q and R are relatively prime nonzero integers
- is not a root of unity.
Then, the corresponding Lehmer numbers are:
for n odd, and
for n even.
Their companion numbers are:
for n odd and
for n even.
Recurrence
Lehmer numbers form a linear recurrence relation with
with initial values . Similarly the companions sequence satisfies