Level 16

I. Moonshine functions: Define,$$a(\tau)= \left(\frac{d_8^3}{d_4\,  d_{16}^2}\right)^{2}, \quad b(\tau) = \left(\frac{d_1^2\,d_8}{d_2\,d_{16}^2}\right), \quad c(\tau) = \left(\frac{d_2^5\,d_8}{d_1^2\,d_4^2\,d_{16}^2}\right)$$ then, $$\left(\frac{d_8^3}{d_4\,  d_{16}^2}\right)^{2}-2= \left(\frac{d_1^2\,d_8}{d_2\,d_{16}^2}\right)\quad\\ \left(\frac{d_8^3}{d_4\,  d_{16}^2}\right)^{2}+2 = \left(\frac{d_2^5\,d_8}{d_1^2\,d_4^2\,d_{16}^2}\right)$$so \(a-2 = b,\, a+2 = c\). This \(a,b,c\) also belong to the class where \(a(\tau) = -a\big(\tfrac12+\tau\big)\) and \(b(\tau) = -c\big(\tfrac12+\tau\big)\). And the three level 16 moonshine functions,$$\begin{align}
j_{16A}(\tau) &= \left(\frac{d_4\, d_8}{d_2\,  d_{16}}\right)^{4}=\sqrt{j_{8A}(2\tau)}\\
\color{red}{j_{16B}}(\tau) &= \left(\frac{d_8^3}{d_4\,  d_{16}^2}\right)^{2}\; = \; a\\
j_{16C}(\tau) &= \left(\frac{d_2^3\,d_8^3}{d_1^2\,  d_4^2\, d_{16}^2}\right)^{2} = \frac{ac}{b}
\end{align}$$
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,
$$\left(\frac{bc}{a}+16\frac{a}{bc}\right)+2a = \frac{bc}{a}+\frac{ac}{b}+\frac{ab}{c}\qquad$$ In general, $$\left(\frac{bc}{a}+(b-c)^2\frac{a}{bc}\right)+2a = \frac{bc}{a}+\frac{ac}{b}+\frac{ab}{c}\qquad$$ where \(b-c=4\). Or in terms of named functions of level 8 and 16, $$\quad\quad j_{8B}\;+\;2\color{red}{j_{16B}} = j_{8D}+j_{16C}+j_{16C'}$$where, similar to level 8, $$j_{16C'}(\tau) =-j_{16C}\big(\tfrac12+\tau\big)= \;\left(\frac{d_1^2\,d_8^3}{d_2^3\, d_{16}^2}\right)^{2}$$ The last 2 of the 6 trinomial identities involves the pair \(j_{16C}\) and \(j_{16C'}\), hence is just the same function using different arguments, $$\begin{align}j_{16C}(\tau)-2 &=\left(\frac{d_2^3\,d_8^3}{d_1^2\,  d_4^2\, d_{16}^2}\right)^{2}-2 =\frac{d_4^{10}}{d_1^2\,d_2^3\,d_8^3\,d_{16}^2}\\ j_{16C}\big(\tfrac12+\tau\big)-2 &= \; -\left(\frac{d_1^2\,d_8^3}{d_2^3\, d_{16}^2}\right)^{2}-2 \;= \; -\frac{d_1^2\,d_4^{12}}{d_2^9\,d_8^3\,d_{16}^2}\end{align}$$
III. Inter-level quadratic relations: We also have,
$$\begin{align}
j_{4C} &= j_{8E}+\frac{16}{j_{8E}}\\
j_{4D} &= j_{8E}-\frac{16}{j_{8E}}
\end{align}$$ $$\begin{align}j_{8D} &= j_{16B}-\frac{4}{j_{16B}}\\
j_{8E} &= j_{16B}+\frac{4}{j_{16B}}\quad\end{align}$$ Adding each pair leaves functions with the same argument,
$$\begin{align}
 j_{4C}+j_{4D} &= 2\color{red}{j_{8E}}\\
 j_{8D}+j_{8E} &= 2\color{red}{j_{16B}}
\end{align}$$ which are 2 of the 9 linear relations found by Conway, Norton, and Atkins.