This level is also special since we can find two a(τ) namely, j18D and j18C. Using S1 of the Intro, these two explain 2×4=8 of the 14 trinomial identities of level 18 found by Somos.
I. First triple. Define,
a(τ)=(d22d9d1d218),b(τ)=(d6d39d3d318),c(τ)=(d21d6d9d2d3d218) then,
(d22d9d1d218)−1=(d6d39d3d318)(d22d9d1d218)−3=(d21d6d9d2d3d218) so a−1=b,a−3=c, and we get the moonshine functions,
j18A(τ)=(d1d2d9d18)=acbj18B(τ)=(d23d26d1d2d9d18)2j18C(τ)=(d1d46d9d22d23d218)2=bca+1j18D(τ)=(d22d9d1d218)=aj18E(τ)=(d2d9d1d18)3=abc Naturally, these also satisfy S2 and S3. We also have
j18B=j18A+32j18A+3=j18E+1j18E−1 and the linear relation between these 5 moonshine functions,
j18B+2j18D=j18A+j18C+j18E+4
II. Second triple. Define,
a(τ)=(d1d46d9d22d23d218)2,b(τ)=(d31d26d39d32d23d318),c(τ)=(d63d1d2d26d9d18) then, (d1d46d9d22d23d218)2−1=(d31d26d39d32d23d318)(d1d46d9d22d23d218)2+3=(d63d1d2d26d9d18)so a−1=b,a+3=c, and we get,
j6d=j6F+16j6Fj18B(τ)=(d23d26d1d2d9d18)2=acbj18C(τ)=(d1d46d9d22d23d218)2=aβ(τ)=(d1d26d9d2d23d18)6=abcj6F(τ)=(d3d6)8=bca where j6d and β are non-monster functions with expansions (A007263) and (A128512), respectively. Naturally, these also satisfy S1,2,3. And by adding these additional relations, we get a second linear equation for level 18 of similar form to the first,
j6d+2j18C=j18B+j6F+β−2
No comments:
Post a Comment