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Thursday, October 31, 2019

Entry 47

Level 18, Part 1. Define dk=η(kτ) with the Dedekind eta function η(τ). Then 
(d22d9d1d218)1=(d6d39d3d318)(d22d9d1d218)3=(d21d6d9d2d3d218) Expressed as the triple of eta quotients (a,b,c) such that a1=ba3=c then (m,n)=(1,3) and

abc=(d2d9d1d18)3bca=(d31d26d39d32d23d318)=(d1d46d9d22d23d218)21acb=(d1d2d9d18)d=(d23d26d1d2d9d18)2 where (a,b,c) are the McKay-Thompson series of class 18D (A143840, A193261) and d is the McKay-Thompson series of class 18B (A215407). They obey (4abc+bca)+5=d (4abc+bca)+2a=abc+bca+acb and the latter is one of the 9 dependencies found by Conway, Norton, and Atkin such that the moonshine functions span a linear space of 1729=163 dimensions. Similar identities involving only moonshine functions exist for levels (6,10,12,18,30), but level 18 is special since the first term (4abc+bca) plus an integer is an eta quotient.

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