Let $\mu$ be a positive real number smaller than $1$. Define $${k\choose{r}}_{\mu}=\frac{\prod_{i=1}^{k}(1-\mu^{2i})}{\prod_{i=1}^{k-r} (1-\mu^{2i}) \prod_{i=1}^{r} (1-\mu^{2i})}$$ Then I need to show that there exists a constant $c$ (independent of $k$ and $r$) such that $${k\choose{r}}_{\mu}\leq c$$
I tried to get a bound using the following method.
I know that the numerator is less than $1$. If I can show that the denominator has a lower bound independent of $k$ and $r$ than the job will be done. But in all the attempts I have made so far, the lower bound I am able to find involves $k$. I will appreciate if anyone can show any direction to go about it. Thanks in advance.