Recall that $(a;\,q)\_\infty$ is the [$q$-Pochhammer symbol](http://mathworld.wolfram.com/q-PochhammerSymbol.html):
$$(a;\,q)\_\infty=\prod\_{n=0}^\infty(1-a \, q^n).\tag1$$
Its important special case $(q;\,q)\_\infty=\prod\_{n=1}^\infty(1-q^n)$ is sometimes called the [Euler function](http://mathworld.wolfram.com/EulerFunction.html). It appears in Euler's [pentagonal number theorem](http://mathworld.wolfram.com/PentagonalNumberTheorem.html), and its reciprocal $(q;\,q)\_\infty^{-1}$ is the generating function for the [partition numbers](http://mathworld.wolfram.com/PartitionFunctionP.html). It is also related to [Jacobi theta functions](http://mathworld.wolfram.com/JacobiThetaFunctions.html) and [Ramanujan theta functions](http://mathworld.wolfram.com/RamanujanThetaFunctions.html).


Let
$$f(x) = \frac{(-q;\,-q)\_\infty}{(q;\,q)\_\infty},\quad\text{where}\,\,q=e^{-\pi\sqrt x}.\tag2$$
In the OEIS entry [A080054](http://oeis.org/A080054) there is an empirical observation by Simon Plouffe that apparently $f(1)=\sqrt[8]2$. 



> 
> **Empirical** (Simon Plouffe, Feb. 20, 2011):
>  $$\sum\_{n=0}^{\infty}e^{-\pi n}a(n) =\sqrt[8]2.$$
> 
> 
> 


I did some numerical experiments related to this observation, and the outcomes suggest a fascinating stronger conjecture:


*Conjecture:* For every $p\in\mathbb Q,\,p>0$, the value $f(p)$
is an algebraic number.


For example, it appears that
$$f(3/5) = \sqrt[8]{2} \cdot \sqrt[4]{9 \sqrt{5}+5 \sqrt{15}-11 \sqrt{3}-19},$$
and $f(13/7)$ is an algebraic number of degree $96$ whose minimal polynomial is
$$x^{96}-647442063456 \, x^{88}+16702438371168 \, x^{80}-529345497357824
 \, x^{72}+4159684203040512 \, x^{64}-12099397290541056
 \, x^{56}+16408771708010496 \, x^{48}-10607690933600256
 \, x^{40}+2651923007078400 \, x^{32}-367001600 \, x^{24}+257949696
 \, x^{16}-100663296 \, x^8+16777216$$
and an isolating rational interval is $(37/36,\,6/5)$.


Is this conjecture new? Is it known to be true? If not, can you suggest any ideas how to (dis-)prove it?



A conjecture about algebraic values of $(-q;\,-q)_\infty/(q;\,q)_\infty$
Yes, it is always algebraic, because it is a modular function
evaluated at a CM (complex multiplication) point.


"$(q;q)\_\infty$" is $q^{-1/24} \eta(\tau)$ where $q = e^{2\pi i \tau}$, so
"$(q;q)\_\infty / (-q;-q)\_\infty$" is a root of unity times
 $\eta(\tau) \, / \, \eta(\tau+1/2)$,
which is modular for some congruence subgroup of ${\rm SL}\_2({\bf Z})$,
i.e. a rational function on some modular curve $X$.
If $q = e^{-\pi \sqrt x}$ then $\tau = (i/2)\sqrt x$
is an imaginary quadratic irrationality, and thus a CM point on $X$.
It is known that every CM point is algebraic, whence the value of
$\eta(\tau) \, / \, \eta(\tau+1/2)$ at the point is also algebraic.


The CM theory also provides further information about the
degree of such algebraic numbers, their Galois group (always solvable),
and conjugates (values of the same function at other CM points).


