Gromov–Witten invariant

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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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