Gromov–Witten invariant: Difference between revisions

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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

a+b=R
ab=Q

under the following conditions:

Then, the corresponding Lehmer numbers are:

Un(R,Q)=anbnab

for n odd, and

Un(R,Q)=anbna2b2

for n even.

Their companion numbers are:

Vn(R,Q)=an+bna+b

for n odd and

Vn(R,Q)=an+bn

for n even.

Recurrence

Lehmer numbers form a linear recurrence relation with

Un=(R2Q)Un2Q2Un4=(a2+b2)Un2a2b2Un4

with initial values U0=0,U1=1,U2=1,U3=RQ=a2+ab+b2. Similarly the companions sequence satisfies

Vn=(R2Q)Vn2Q2Vn4=(a2+b2)Vn2a2b2Vn4

with initial values V0=2,V1=1,V2=R2Q=a2+b2,V3=R3Q=a2ab+b2.

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