Meijer's G-Function

$$G_{p,q}^{m,n}\left({x \vert^{a_1, \ldots, a_p}_{b_1, \ldots,... ...j=m+1}^q\Gamma(1-b_j+z)\prod_{j=n+1}^q \Gamma(q_j-z)} x^z\,dz,$$

where $\Gamma(z)$ is the Gamma Function. The Contour $\gamma_L$ and other details are discussed by Gradshteyn and Ryzhik (1980, pp. 896-903 and 1068-1071). Prudnikov et al. (1990) contains an extensive nearly 200-page listing of formulas for the Meijer $G$-function.

See also Fox's H-Function, G-Function, MacRobert's E-Function, Ramanujan g- and G-Functions

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1979.

Luke, Y. L. The Special Functions and Their Approximations, 2 vols. New York: Academic Press, 1969.

Mathai, A. M. A Handbook of Generalized Special Functions for Statistical and Physical Sciences. New York: Oxford University Press, 1993.

Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A.; Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, 1990.