The Cauchy–Mellin integral transformation for $\Gamma(z)$ and its application

The Cauchy integral (3) for the representation of $\Gamma(z)$, when $\operatorname{Re}z<0$ is a noninteger, and the Mellin integral (4) together form the new “integral transformation of Cauchy–Mellin type” for $\Gamma(z)$, with the help of which we can find exact analytical representations in form of “nonorientable” power series for hypergeometric functions from one, two and more variables in a “pole-domain” of Euler's gamma-function.