Appendix A Definition of H-Functions

The H-function is defined as [43]

[p. 2474l, §2]
where C is a contour from c-i⁢∞ to c+i⁢∞ separating the poles of Γ⁢βj-Bj⁢s, j=1,…,m from those of Γ⁢1-αj+Aj⁢s, j=1,…,n. Empty products are interpreted as unity. The integers m,n,P,Q satisfy 0≤m≤Q and 0≤n≤P. The coefficients Aj and Bj are positive real numbers and the complex parameters αj, βj are such

[p. 2474r, §2]
that no poles in the integrand coincide. If

then integral converges absolutely and defines the H-function

[p. 2475l, §0]
in the sector z<12⁢Ω⁢π. The H-function is also well defined when either

or

[p. 2475r, §0]
The H-function is a generalization of Meijer’s G function and contains many of the known special functions. In particular Mittag-Leffler and generalized Mittag-Leffler functions are special cases of the H-function.