Product of numbers and gaussian function

Question

Trying to approximate a gaussian function $g(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left(-\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2}\right)}$ with another function I found the product $$P(a,b,c)=\prod_{k=0}^N\frac{ak+b}{c(k+1)^2}$$ with $a\in\mathbb{R},b\in\mathbb{R},c\in\mathbb{R}$ approximate quite well the $g(N)$, obviously for $N\ge0$. Building a suitable optimization function it's possible to find $a,b,c$ so that the approximation is good enough, knowing $\sigma$ and $\mu$. My question is: is there a deep reason for wich the $P(a,b,c)$ looks like the $g(N)$ or it is only a coincidence? Thanks.