Entry 47

Level 18, Part 1. Define \(d_k=\eta(k\tau)\) with the Dedekind eta function \(\eta(\tau)\). Then 
$$\left(\frac{d_2^2\,d_9}{d_1\,d_{18}^2}\right)-1 = \left(\frac{d_6\,d_9^3}{d_3\,d_{18}^3}\right)\quad \\ \left(\frac{d_2^2\,d_9}{d_1\,d_{18}^2}\right)-3 =\left(\frac{d_1^2\,d_6\,d_9}{d_2\,d_3\,d_{18}^2}\right)$$ Expressed as the triple of eta quotients \((a,b,c)\) such that $$a-1 =b\\ a-3=c$$ then \((m,n) = (-1,-3)\) and

$$\begin{align}\qquad\frac{ab}{c} &= \left(\frac{d_2\, d_9}{d_1\,  d_{18}}\right)^{3}\\ \frac{bc}{a} & =  \left(\frac{d_1^3\,d_6^2\, d_9^3}{d_2^3\,d_3^2 d_{18}^3}\right) = \left(\frac{d_1\,d_6^4\, d_9}{d_2^2\,d_3^2 d_{18}^2}\right)^2 -1\\ \frac{ac}{b} &= \left(\frac{d_1\, d_2}{d_9\,  d_{18}}\right) \\ d\, &= \,\left(\frac{d_3^2\, d_6^2}{d_1\, d_2\, d_9\, d_{18}}\right)^{2}\end{align}$$ 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 $$\left(\frac{4a}{bc}+\frac{bc}a\right)+5=d$$ $$\left(\frac{4a}{bc}+\frac{bc}a\right)+2a  = \frac{ab}c+ \frac{bc}a+ \frac{ac}{b}$$ 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 \(172-9=163\) dimensions. Similar identities involving only moonshine functions exist for levels \((6, 10, 12, 18, 30)\), but level \(18\) is special since the first term \(\left(\frac{4a}{bc}+\frac{bc}a\right)\) plus an integer is an eta quotient.