Divergent products.
In the equation $\frac{\sqrt{2 \pi}}{\Gamma(n)} = \prod_{c=0}^{\infty} (c+n)$, the right-hand side may be interpreted as a zeta-regularized product. Then by definition, we have $$\prod_{c=0}^{\infty} (c+n)=\exp(-Z'(0))$$ where $Z(s)$ is the function defined by $$Z(s)=\sum_{c=0}^\infty(c+n)^{-s}$$ for $s$ with large enough real part, and by analytic continuation as necessary. The rest of the computation is given in Example 3 on p.220 of J. R. Quine, S. H. Heydari and R. Y. Song (1993), Zeta regularized products.