Define $\color{blue}{q=e^{2\pi i \tau}}$ and Dedekind eta function $\eta(\tau)$. Note: I found these relations empirically, but their consistent forms suggest they can be rigorously proven.



> 
> **I.** $p=2$
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We have,


$$\left(\frac{8}{\alpha^8+8}\right)^2+\left(\frac{\beta^8}{\beta^8+32}\right)^2=1\tag1$$
$$\alpha=\frac{\eta(\tau/2)}{\eta(2\tau)},\quad\beta=\frac{\eta(\tau)}{\eta(4\tau)}$$
If $\tau=\sqrt{-n}$, then,
$$\frac{\_2F\_1\Big(\tfrac14,\tfrac34;1;\,1-\big(\tfrac{8}{\alpha^2+8}\big)^2\Big)}{\_2F\_1\Big(\tfrac14,\tfrac34;1;\,\big(\tfrac{8}{\alpha^8+8}\big)^2\Big)}=\color{red}{\sqrt{2n}}$$



> 
> **II.** $p=3$
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We have,


$$\left(\frac{3}{\gamma^3+3}\right)^3+\left(\frac{\delta^3}{\delta^3+9}\right)^3=1\tag2$$
$$\gamma=\frac{\eta(\tau/3)}{\eta(3\tau)},\quad\delta=\frac{\eta(\tau)}{\eta(9\tau)}$$
If $\tau=\sqrt{-n}$, then,
$$\frac{\_2F\_1\Big(\tfrac13,\tfrac23;1;\,1-\big(\tfrac{3}{\gamma^3+3}\big)^3\Big)}{\_2F\_1\Big(\tfrac13,\tfrac23;1;\,\big(\tfrac{3}{\gamma^3+3}\big)^3\Big)}=\color{red}{\sqrt{3n}}$$


Note: The identity,
$$\left(\frac{x^3y + y}{x y^3 + x}\right)^3+\left(\frac{x^3 - y^3}{x y^3 + x}\right)^3=1,\quad \text{if}\; x^3+y^3=1$$ 
should guarantee *infinitely* many eta parametrizations to $(2)$. Note also that,
$$\gamma^3+3=4C^2(q)+\frac1{C(q)}\\
\delta^3+3=4C^2(q^3)+\frac1{C(q^3)}$$
with ***cubic continued fraction***,
$$C(q)=\frac{\eta(\tau)\,\eta^3(6\tau)}{\eta(2\tau)\,\eta^3(3\tau)}=\cfrac{q^{1/3}}{1+\cfrac{q+q^2}{1+\cfrac{q^2+q^4}{1+\cfrac{q^3+q^6}{1+\ddots}}}}$$


$\color{blue}{Update:}$
I just realized that the cubic *eta* identity $(2)$ is equivalent to the Borweins' cubic *theta* identity,
$$\frac{c^3(q)}{a^3(q)}+\frac{b^3(q)}{a^3(q)} = 1$$
where,
$$a(q) = 1+6\sum\_{n=0}^\infty\left(\frac{q^{3n+1}}{1-q^{3n+1}}-\frac{q^{3n+2}}{1-q^{3n+2}}\right)$$
$$b(q) = \tfrac{1}{2}\big(3a(q^3)-a(q)\big)$$
$$c(q) = \tfrac{1}{2}\big(a(q^{1/3})-a(q)\big)$$
from Ramanujan's Notebooks Vol. V, p.93.



> 
> **III.** $p=4,8$
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$$\left(\frac{\vartheta\_2(0,q)}{\vartheta\_3(0,q)}\right)^4+\left(\frac{\vartheta\_4(0,q)}{\vartheta\_3(0,q)}\right)^4 = 1\tag{3a}$$
with [Jacobi theta function](http://mathworld.wolfram.com/JacobiThetaFunctions.html) $\vartheta\_n(0,q)$. If $q=e^{2\pi i \tau}$, then equivalently,


$$\left(\frac{\sqrt2\,\eta(\tau)\,\eta^2(4\tau)}{\eta^3(2\tau)}\right)^8+\left(\frac{\eta^2(\tau)\,\eta(4\tau)}{\eta^3(2\tau)}\right)^8 = 1\tag{3b}$$


hence the addends of $(3a)$ and $(3b)$ are equal. Expressed as, $$U^8(\tau)+V^8(\tau) =1$$


If $\tau=\sqrt{-n}$, then,


$$\frac{\_2F\_1\big(\tfrac12,\tfrac12;1;\,1-U^8(\tau)\big)}{\_2F\_1\big(\tfrac12,\tfrac12;1;\,U^8(\tau)\big)}=\color{red}{\sqrt{4n}}$$


This has a beautiful ***octic continued fraction*** studied by Ramanujan, 
$$U(\tau)=\frac{\sqrt2\,\eta(\tau)\,\eta^2(4\tau)}{\eta^3(2\tau)}= \cfrac{\sqrt{2}\,q^{1/8}}{1+\cfrac{q}{1+q+\cfrac{q^2}{1+q^2+\cfrac{q^3}{1+q^3+\ddots}}}}$$
and [can solve the general quintic](https://math.stackexchange.com/questions/540964/how-to-solve-fifth-degree-equations-by-elliptic-functions). 




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> 
> **IV. Question**
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**Q:** Are there analogous eta quotients to parameterize the Fermat quintic $x^5+y^5=1$? And can we also use the ***Rogers-Ramanujan cfrac*** to do so?



Can we use the Rogers-Ramanujan cfrac to parameterize the Fermat quintic $x^5+y^5=1$?
(*Courtesy of a comment by Nemo who suggested Huber's paper.*) 



> 
> **Part I.** $x^5+y^5 = 1$
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> 


In "[*A Theory of Theta Functions to the Quintic base*](https://arxiv.org/pdf/1304.0684.pdf)",
Tim Huber defines four theta functions which can be ultimately expressed in terms of the [Rogers-Ramanujan identities](https://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_continued_fraction). Define $q=e^{2\pi i z}$ and,
$$P(z):=q^{11/60}H(q)=q^{11/60}\prod\_{n=1}^\infty \frac1{(1-q^{5n-2})(1-q^{5n-3})}$$
$$Q(z):=q^{-1/60}G(q)=q^{-1/60}\prod\_{n=1}^\infty \frac1{(1-q^{5n-1})(1-q^{5n-4})}$$
and the Rogers-Ramanujan continued fraction,
$$R(z)=q^{1/5}\frac{H(q)}{G(q)}=\cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\ddots}}}}$$


then Huber's four functions in simplified form are,


$$\begin{aligned}
a(\tau) &=\eta^{2/5}(\tau)\;P(\tau)\\[2mm]
b(\tau) &=\eta^{2/5}(\tau)\;Q(\tau)\\[2mm]
c(\tau) &= 5^{1/4}\phi^{1/2}\,\eta^{2/5}(5\tau)\;P\big(\tfrac{-1}{5\tau}\big)\\[2mm]
d(\tau) &= \frac{5^{1/4}}{\phi^{1/2}}\,\eta^{2/5}(5\tau)\;Q\big(\tfrac{-1}{5\tau}\big)\end{aligned}$$
with ***golden ratio*** $\phi$. These obey,
$$\Big(\frac{a\,\phi}{b}\Big)^5+\Big(\frac{c}{b}\Big)^5 = 1\tag1$$
$$-\Big(\frac{a}{b\,\phi}\Big)^5+\Big(\frac{d}{b}\Big)^5 = 1\tag2$$
It then follows that the ratio of $a,b$ as well as $c,d$ are Rogers-Ramanujan cfracs,
$$\frac{a(\tau)}{b(\tau)} = R(\tau)\tag3$$
$$\frac{c(\tau)}{d(\tau)} = \phi\, R\big(\tfrac{-1}{5\tau}\big)\tag4$$
so $R(\tau)$ ***in a way*** can parameterize the Fermat quintic. Some manipulation will also show that,
$$\frac{c(\tau)}{d(\tau)} = \phi\,\frac{1-\phi\, R(5\tau)}{\phi+R(5\tau)}\tag5$$



> 
> **Part II.** $x^5+y^5+z^5 = 1$
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> 
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It turns out that just like,
$$x^3+y^3=1$$
can be solved by the cubic continued fraction $C(q)$ and $C(q^3)$, its quintic analogue
$$x^5+y^5+z^5 = 1$$
has a beautiful solution using the Rogers-Ramanujan cfrac $R(q)$ and $R(q^5)$. This is given by,


$$\alpha^5\phi^5+\beta^5\phi^5+\alpha^5\beta^5 = 1\tag6$$


where,
$$\alpha = R(q),\quad\beta = \frac{1-\phi R(q^5)}{\phi+R(q^5)}$$
and which can be derived by combining $(1),(2)$.


