Entry 51

Level 24, Part 2. As pointed out in Part 1, Level 24 is unusual since two triples \((a,b,c)\) have the special property $$\frac{a_1}{b_1\, c_1} = \frac{a_2}{b_2\, c_2}$$ These also form linear dependencies but involve non-monster functions. But if we combine the two, plus some level 12 functions, then there is a relationship with only monster functions. Define \(d_k=\eta(k\tau)\) with the Dedekind eta function \(\eta(\tau)\). Then $$\quad\left(\frac{d_2\, d_{12} }{d_4\, d_ 6}\right) \left(\frac{d_3\, d_8 }{d_1\, d_{24} }\right)^2+1=\frac{d_2^5\, d_3\, d_8\, d_{12}^5 }{\big(d_1\, d_4\, d_6\, d_{24}\big)^3}\\ \qquad \left(\frac{d_6\, d_{12} }{d_2\, d_ 4}\right) \left(\frac{d_3\, d_8 }{d_1\, d_{24} }\right)^2-1=\frac{\big(d_4\, d_6\big)^4 }{d_1\, d_2^2\, d_3\, d_8\, d_{12}^2\, d_{24} }$$ Expressed as the triple of eta quotients \((a,b,c)\) such that $$a+1 =b\\ a-1=c$$ so \((m,n)=(1,-1)\). Then $$\begin{align}\frac{ab}{c} &= \left(\frac{d_2^2\,d_3\,d_8\,d_{12}^2}{d_1\,d_4^2\,d_6^2\,d_{24}}\right)^4\\ \color{blue}{\frac{bc}{a}} &= \left(\frac{d_2\, d_4\, d_6\, d_{12}}{d_1\, d_3\, d_8\, d_{24}}\right)^2 = T_{24B}\\ \frac{ac}{b} &=  \left(\frac{d_4\, d_6}{d_2\,d_{12}}\right)^6 \end{align}$$ where \((a,b,c)\) are the McKay-Thompson series of class 24I (A138688). They obey $$\frac{bc}a = a-\frac1a$$ $$\left(\frac{4a}{bc}+\frac{bc}{a}\right)+2a  = \frac{ab}c+ \frac{bc}a+ \frac{ac}{b} $$ This is not one of the 9 linear dependencies by Conway et al since one of the terms, again \(\left(\frac{ab}c\right)\), is not a moonshine function. But just like in Part 1, the \(\left(\frac{bc}a\right)\) term is the McKay-Thompson series of class 24B (A212771).