Entry 43

Define \(d_k=\eta(k\tau)\) with the Dedekind eta function \(\eta(\tau)\). Then for Level 10 $$\left(\frac{d_2^2\,d_5}{d_1\,d_{10}^2}\right)^2-1=\left(\frac{d_2\,d_5^5}{d_1\,d_{10}^5}\right)\\ \left(\frac{d_2^2\,d_5}{d_1\,d_{10}^2}\right)^2-5=\left(\frac{d_1^3\,d_5}{d_2\,d_{10}^3}\right)$$ Expressed as the triple of eta quotients \((a,b,c)\) such that $$a-1 =b\\ a-5=c$$ so \((m,n)=(-1,-5)\). Then $$ \frac{ab}{c} = \left(\frac{d_2\, d_5}{d_1\,  d_{10}}\right)^{6}\\ \frac{bc}{a} = \left(\frac{d_1\, d_5}{d_2\,  d_{10}}\right)^{4}\\ \frac{ac}{b} = \left(\frac{d_1\, d_2}{d_5\,  d_{10}}\right)^{2}$$ where \((a,b,c)\) are the McKay-Thompson series of class 10E (A138516). They obey $$\left(\frac{16a}{bc}+\frac{bc}a\right)+2a  = \frac{ab}c+ \frac{bc}a+ \frac{ac}{b}$$ which 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 \((6, 10, 12, 18, 30)\).