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Thursday, October 31, 2019

Level 16

I. Moonshine functions: Define,a(τ)=(d38d4d216)2,b(τ)=(d21d8d2d216),c(τ)=(d52d8d21d24d216)
 then, (d38d4d216)22=(d21d8d2d216)(d38d4d216)2+2=(d52d8d21d24d216)
so a2=b,a+2=c. This a,b,c also belong to the class where a(τ)=a(12+τ) and b(τ)=c(12+τ). And the three level 16 moonshine functions,j16A(τ)=(d4d8d2d16)4=j8A(2τ)j16B(τ)=(d38d4d216)2=aj16C(τ)=(d32d38d21d24d216)2=acb

II. Relations: Using S1, this a,b,c explains 4 of the 6 trinomial identities of level 16 (prefixed with t16) by Somos. They also obey,
(bca+16abc)+2a=bca+acb+abc
In general, (bca+(bc)2abc)+2a=bca+acb+abc
 where bc=4. Or in terms of named functions of level 8 and 16, j8B+2j16B=j8D+j16C+j16C
where, similar to level 8, j16C(τ)=j16C(12+τ)=(d21d38d32d216)2
The last 2 of the 6 trinomial identities involves the pair j16C and j16C, hence is just the same function using different arguments, j16C(τ)2=(d32d38d21d24d216)22=d104d21d32d38d216j16C(12+τ)2=(d21d38d32d216)22=d21d124d92d38d216

III. Inter-level quadratic relations: We also have,
j4C=j8E+16j8Ej4D=j8E16j8E
j8D=j16B4j16Bj8E=j16B+4j16B
Adding each pair leaves functions with the same argument,
j4C+j4D=2j8Ej8D+j8E=2j16B
which are 2 of the 9 linear relations found by Conway, Norton, and Atkins.

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