Gamma, Digamma, Polygamma Functions

$$z! = z\Gamma(z)=\lim_{n\rightarrow \infty}\frac{n!}{(z+1)(z+2)\cdots(z+n)}n^2$$ (2.14)

$$\ln(z!)=\lim_{n\rightarrow \infty}\left( ln(n!)+z\ln(n)-\sum_{k=1}^{n}ln(z+k)\right) \ni$$ (2.15)

$$\frac{d}{dz}\ln(z!)\equiv F(z) = \lim_{n\rightarrow \infty}\left(ln(n)-\sum_{k=1}^{n}\frac{1}{z+k}\right)$$ (2.16)