q-Pochhammer symbol asymptotics

Question

I wanrt compute a limite involving the q-Pochhammer symbol. So I'm looking for a like Stirling formula for the q-Pochhammer symbol defined by $$\prod_{i=0}^n(a+bq^n)$$ as in in this post https://mathoverflow.net/questions/58231/what-is-the-stirling-formula-for-xx1x2-xn-1. Thank you.

Answer 1

You are looking for q-analogue of the Stirling approximation, and here's the paper by Moak:

https://projecteuclid.org/download/pdf_1/euclid.rmjm/1250127592

in which he derives:

$$ \ln \Gamma_q(x) \sim \left(x - {1 \over 2}\right)\log\left[x\right]_q + {\text{Li}_2({1 - q^x}) \over {\ln q}} + C_{\hat{q}} + {1 \over 2}H(q - 1) \log q + \sum_{k=1}^{\infty}{{B_{2k}}\over{(2k)!}}\left({{\log \hat{q}} \over {\hat{q}^x - 1}}\right)^{2k - 1} {\hat{q}}^xp_{2k-3}\left(\hat{q}^x\right)\text{,}\quad x\to\infty $$

Where,

$H(\cdot)$ denotes the Heaviside step function,

$$ \hat{q} = \begin{cases} q & 0 \leq q \lt 1 \\q^{-1} & q \ge 1 \end{cases}, $$

$[x]_q = (1 - q^x)/(1-q)$, $\text{Li}_2(x)$ is dilogarithm function:

$$ \text{Li}_2(z) = - \int_{0}^{z}{\ln{(1 - t)} \over {t} }dt; \quad z\in [-\infty, 0), z\in\mathbb{C}\ $$

$p_k$ is a polynomial of degree $k$ satisfying:

$$ p_k(z) = (z-z^2)p'_{k-1}(z) + (kz +1)p_{k-1}(z), \qquad p_0=p_{-1}=1, k=1,2,\dots $$

and

$$ C_q = {{1}\over{2}}\ln(2\pi)+ {{1}\over{2}}\ln\left({q-1}\over{\ln q}\right) - {1 \over 24}\ln q + ln\left( \sum_{m = -\infty}^{\infty}{\left(r^{m(6m+1)} - r^{(2m+1)(3m+1)}\right)}\right) $$

The above is from:

https://www.researchgate.net/publication/268046368_Three_classes_of_the_Stirling_formula_for_the_q-factorial_function