# Appendix A Definition of H-Functions

The H-function is defined as [43]

HP,Qm,n(z|α1,A1⋯αP,APβ1,B1⋯βQ,BQ)=12πi∫C∏j=1mΓβj-Bjs∏j=1nΓ1-αj+Ajs∏j=m+1QΓ1-βj+Bjs∏j=n+1PΓαj-Ajsz-sds, | | (A.1) |

[p. 2474l, §2]

where C is a contour from c-i∞ to c+i∞ separating
the poles of Γβj-Bjs, j=1,…,m from those of
Γ1-αj+Ajs, 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

0<∑nj=1Aj-∑Pj=n+1Aj+∑mj=1Bj-∑Qj=m+1Bj=Ω | | (A.2) |

then integral converges absolutely
and defines the H-function

[p. 2475l, §0]

in the sector z<12Ωπ.
The H-function is also well defined when either

δ=∑Qj=1Bj-∑Pj=1Aj>0and0<z<∞ | | (A.3) |

or

δ=0and0<z<R≡∏Pj=1Aj-Aj∏Qj=1BjBj. | | (A.4) |

[p. 2475r, §0]

The H-function is a generalization of Meijerâs 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.