Ramanujan Once A Day
A collection of mathematical curiosities
Wednesday, November 6, 2019
Levels 126 & 252
I. Non-moonshine functions. There are no moonshine functions with uppercase index (in Atlas notation) for level >119. However, surprisingly we can still find trinomial identities for level 144 (some a consequence of level 72) and as high as level 252 (a consequence of level 126). For the latter, define the two pairs of eta quotients, a(τ)=(d27d29d3d14d18d21),b(τ)=(d21d263d2d3d21d126)c(τ)=(d214d218d6d7d9d42),d(τ)=(d22d2126d1d6d42d63)Given one moonshine function of level 21, namely j21B(τ)=(d1d3d7d21) and define, e(τ)=j21B(τ)j21B(3τ)=(d1d63d7d9) then ratios of the pairs are simply, ab=e(2τ)e2(τ)cd=e(τ)e2(2τ) They obey,a(τ)−2=b(τ)c(τ)−1=d(τ) a(12+τ)−2=b(12+τ)c(12+τ)−1=d(12+τ) which is the pair of trinomial identities each for level 126 and level 252, respectively, and the latter seems to be the highest known.
Levels 42 & 84
I. Moonshine functions. The four functions for level 42,
j42A(τ)=(√j42B+1√j42B)2j42B(τ)=(d1d6d14d21d2d3d7d42)2j42C(τ)=3√j14B(3τ)=(d3d21d6d42)j42D(τ)=(d2d6d7d21d1d3d14d42) and the three functions for level 84,
j84A(τ)=√j42A(2τ)j84B(τ)=√j42B(2τ)j84C(τ)=3√j28B(3τ)=(d26d242d3d12d21d84)
II. Non-moonshine functions. Define the two pairs of eta quotients, a(τ)=(d21d214d2d3d7d42),b(τ)=(d22d27d1d6d14d21)c(τ)=(d23d242d1d6d14d21),d(τ)=(d26d221d2d3d7d42) Interestingly, these satisfy a(τ)b(τ)c(τ)d(τ)=1 And in terms of the moonshine functions,
ab=j42Bj42Ddc=j42Bj42D They obey, a(τ)+3=b(τ)c(τ)+1=d(τ) a(12+τ)+3=b(12+τ)c(12+τ)+1=d(12+τ) which is the pair of trinomial identities each for level 42 and level 84, respectively.
j42A(τ)=(√j42B+1√j42B)2j42B(τ)=(d1d6d14d21d2d3d7d42)2j42C(τ)=3√j14B(3τ)=(d3d21d6d42)j42D(τ)=(d2d6d7d21d1d3d14d42) and the three functions for level 84,
j84A(τ)=√j42A(2τ)j84B(τ)=√j42B(2τ)j84C(τ)=3√j28B(3τ)=(d26d242d3d12d21d84)
II. Non-moonshine functions. Define the two pairs of eta quotients, a(τ)=(d21d214d2d3d7d42),b(τ)=(d22d27d1d6d14d21)c(τ)=(d23d242d1d6d14d21),d(τ)=(d26d221d2d3d7d42) Interestingly, these satisfy a(τ)b(τ)c(τ)d(τ)=1 And in terms of the moonshine functions,
ab=j42Bj42Ddc=j42Bj42D They obey, a(τ)+3=b(τ)c(τ)+1=d(τ) a(12+τ)+3=b(12+τ)c(12+τ)+1=d(12+τ) which is the pair of trinomial identities each for level 42 and level 84, respectively.
Levels 14 & 28
For certain even levels divisible by 7, the situation is now different. We can still find an eta quotient a such that a+m=b is also an eta quotient, but only for one rational m.
I. Moonshine functions: Define a(τ)=(d1d7d4d28),b(τ)=(d32d314d1d24d7d228) then a+2=b or, (d1d7d4d28)+2=(d32d314d1d24d7d228) for the unique m=2. This is the single trinomial identity of level 28. First, we have the 3 moonshine functions of level 14,
j14A(τ)=(√j14C+1√j14C)2j14B(τ)=(d1d7d2d14)3=b+4b−4j14C(τ)=(d2d7d1d14)4 and the 4 moonshine functions of level 28,
j28A(τ)=√j14A(2τ)=j28D(τ)+1j28D(τ)j28B(τ)=(d22d214d1d4d7d28)3=a+4a+4j28C(τ)=d1d7d4d28=aj28D(τ)=√j14C(2τ)=(d4d14d2d28)2 Note that j14B(12+τ)=−j28B(τ).
I. Moonshine functions: Define a(τ)=(d1d7d4d28),b(τ)=(d32d314d1d24d7d228) then a+2=b or, (d1d7d4d28)+2=(d32d314d1d24d7d228) for the unique m=2. This is the single trinomial identity of level 28. First, we have the 3 moonshine functions of level 14,
j14A(τ)=(√j14C+1√j14C)2j14B(τ)=(d1d7d2d14)3=b+4b−4j14C(τ)=(d2d7d1d14)4 and the 4 moonshine functions of level 28,
j28A(τ)=√j14A(2τ)=j28D(τ)+1j28D(τ)j28B(τ)=(d22d214d1d4d7d28)3=a+4a+4j28C(τ)=d1d7d4d28=aj28D(τ)=√j14C(2τ)=(d4d14d2d28)2 Note that j14B(12+τ)=−j28B(τ).
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