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Given the Dedekind eta function $\eta(\tau)$ and complex number $\tau$. I came across these family of functions,

$${f_2(\tau)= \frac{i}{\sqrt{2}}\frac{\,_2F_1\left(\tfrac14,\tfrac34,1,\,1-\alpha_2\right)}{\,_2F_1\left(\tfrac14,\tfrac34,1,\,\alpha_2\right)}=\tau}$$


$${f_3(\tau)= \frac{i}{\sqrt{3}}\frac{\,_2F_1\left(\tfrac13,\tfrac23,1,\,1-\alpha_3\right)}{\,_2F_1\left(\tfrac13,\tfrac23,1,\,\alpha_3\right)}=\tau}$$


$${f_4(\tau)= \frac{i}{\sqrt{4}}\frac{\,_2F_1\left(\tfrac12,\tfrac12,1,\,1-\alpha_4\right)}{\,_2F_1\left(\tfrac12,\tfrac12,1,\,\alpha_4\right)}=\tau}$$


where,

$$\alpha_2 =\frac{64}{64+\Big(\frac{\eta(\tau)}{\eta(2\tau)}\Big)^{24}},\quad \alpha_3 =\frac{27}{27+\Big(\frac{\eta(\tau)}{\eta(3\tau)}\Big)^{12}},\quad \alpha_4 =\frac{16}{16+\Big(\frac{\eta(\tau)}{\eta(4\tau)}\Big)^{8}},$$

So the input variable is $\tau$ and the output is also $\tau$. Presumably these are identity functions $f(x)=x$?

Q: What are other not-so-trivial examples of identity functions?

P.S. There is a $f_1(\tau)$ using $\,_2F_1\left(\tfrac16,\tfrac56,1,\,\alpha_1\right)$ but it uses the j-function, instead of the Dedekind eta function.

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  • $\begingroup$ Essentially the same eta quotients are used in this post. $\endgroup$ – Tito Piezas III Jan 4 at 12:10

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