Gromov–Witten invariant: Difference between revisions

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In [[mathematics]], a '''Lehmer number''' is a generalization of a [[Lucas sequence]].
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==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]]

Latest revision as of 06:29, 29 December 2014

Hello, my title is Andrew and my wife doesn't like it at all. Since he was eighteen he's been working as an information officer but he plans on altering it. Her family members lives in Alaska but her husband wants them to move. What I adore doing is football but I don't have the time recently.

My website; psychic phone readings (kpupf.com)

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