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I am implementing a floating point function to compute the q-Pochhammer Euler function (1, 2) $$\phi(q)=(q;q)_{\infty}=\prod_{k=1}^{\infty}(1-q^k)$$ for real $-1 \le q \le 1$. The basic algorithm uses the Euler identity (Pentagonal number theorem) $$\phi(q)=\sum_{n=-\infty}^{\infty}(-1)^n q^{(3n^2-n)/2}$$

For $|q| \approx 1$ the summation suffers from rounding errors and cancellation (which cannot be avoided because $\phi \rightarrow 0$.)

For $q \rightarrow 1$ there is the asymptotic expression (2, formula 8)

$$\phi(q)=\sqrt{\frac{2\pi}{t}} \exp\left( -\frac{\pi^2}{6t} + \frac{t}{24}\right)$$ with $t=-\ln q.$

This expression is remarkably efficient with a relative error $\le 10^{-18}$ for $q>0.5$ (I guess it comes from the Jacobi Theta function representation of $\phi$ in [2], formulas 6,7.)

Question: Is there a similar asymptotic expression for $\phi(q)$ for $q\rightarrow -1?$

Unfortunately I could not find such a result, and Maple or Wolfram Alpha refuse to help me.

Update: With the result of @Professor Vector I get (using algebraic manipulations) the asymptotic expression

$$\phi(-q)=\frac{\phi(q^2)^3}{\phi(q)\phi(q^4)} \sim \sqrt{\frac{\pi}{t}} \exp\left( -\frac{\pi^2}{24t} + \frac{t}{24}\right)$$

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From the product, you can easily derive $$\phi(-q)=\frac{\phi(q^2)^3}{\phi(q)\,\phi(q^4)},$$ due to the elementary identity $(1+x)(1-x)=1-x^2$, so you can use the asymptotics for $q\rightarrow1$ to get those for $q\rightarrow-1$.
EDIT: No problem, I'll add some details. It's clear that with $q\rightarrow -q$, only the terms with odd powers of $q$ will change. Let's define $$\phi_{odd}(q)=\prod^\infty_{k=1}(1-q^{2k-1}).$$ Then, it's obvious that $$\phi_{odd}(q)=\phi(q)/\phi(q^2),$$ and from the identity I mentioned, $$\phi_{odd}(q)\,\phi_{odd}(-q)=\phi_{odd}(q^2).$$ Thus, we have $$\phi(-q)=\phi(q^2)\,\phi_{odd}(-q)=\phi(q^2)\,\phi_{odd}(q^2)/\phi_{odd}(q),$$ and substituting the above expression for $\phi_{odd}$ will give the result.
Don't you worry, I found that a bit tricky myself, and since I couldn't find a reference, either, I did make some numerical checks myself, just to be on the safe side. :)

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  • $\begingroup$ Sorry to bother, but after many hours of trying and searching I was not able to prove your formula. My best result besides the numerical verification is, that for finite products the low-order terms of $\phi(-q)\phi(q)\phi(q^4) -{\phi(q^2)^3}$ vanish. I would appreciate if you could give a reference or an extended hint. $\endgroup$ Commented Jun 5, 2017 at 14:49
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    $\begingroup$ @gammatester I've added some details, I hope it's clearer, now. $\endgroup$ Commented Jun 5, 2017 at 19:04
  • $\begingroup$ Thank you very much! With your details I can prove your result, assuming that it is allowed to reorder the infinite products. $\endgroup$ Commented Jun 5, 2017 at 21:45
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    $\begingroup$ @gammatester It is allowed. If in doubt, take logarithms: the resulting infinite series is absolutely convergent, as long as $|q|<1$. $\endgroup$ Commented Jun 6, 2017 at 4:50
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Let $F(z)$ be defined as

$$ F(z)=\phi(e^{2\pi iz})^{-1}=\prod_{n\ge1}(1-e^{2\pi inz})^{-1}.\tag1 $$

Then it is known that for any coprime integers $h,k$ such that $k\ne0$ and for any $u$ with $\Re(u)>0$,

$$ F\left(\frac hk+{iu\over k}\right)=e^{\pi is(h,k)}\exp\left[{\pi\over12k}\left(\frac1u-u\right)\right]F\left({h'\over k}+{iu^{-1}\over k}\right),\tag2 $$

where $h'$ is chosen such that $hh'\equiv-1\pmod k$ and $s(h,k)$ is called Dedekind sum, which is defined as

$$ s(h,k)=\sum_{1\le r<k}\frac rk\left({hr\over k}-\left\lfloor hr\over k\right\rfloor-\frac12\right).\tag3 $$

This transformation formula would produce asymptotic formulae for $\phi(q)$ when $q$ approaches any root of unity from the interior of the unit circle centered at origin. Moreover, (2) is used by Hardy & Ramanujan and Rademacher to derive asymptotics for the partition function $p(n)$:

$$ p(n)\sim{1\over4n\sqrt3}\exp\left(\pi\sqrt{2n\over3}\right).\tag4 $$

For a proof of (2), see Rademacher's 1932 paper Zur Theorie der Modulfunktionen. For proofs of (4), see Hardy & Ramanujan's 1918 paper Asymptotic Formulaae in Combinatory Analysis and Rademacher's 1938 paper On the Partition Function $p(n)$.

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