How to find a power series representation for a divergent product?

Not sure if there is a precise sense in which this is meaningful, as the product is divergent for all $x\neq 0$, but the function $$ f(x):=\frac{1}{\Gamma\left(1-\frac{x}{\pi}\right)} $$ has simple zeroes at precisely the positive multiples of $\pi$, and satisfies $f(0)=1$. The reflection formula for $\Gamma$ shows that $$ f(x)f(-x)=\frac{\sin(x)}{x}=\prod_{n=1}^\infty \left(1-\frac{x}{n\pi}\right)\left(1+\frac{x}{n\pi}\right). $$