[76.1.1.2] The general H-function is defined as the inverse
Mellin transform [32]
| HPQmn(z|a1,A1…aP,APb1,B1…bQ,BQ)=12πi∫C∏j=1mΓbj-Bjs∏j=1nΓ1-aj+Ajs∏j=m+1QΓ1-bj+Bjs∏j=n+1PΓaj-Ajszsds | | (A.1) |
where the contour C runs from c-i∞ to c+i∞ separating
the poles of Γ(bj-Bj),(j=1,…,m) from those of
Γ(1-aj+Ajs),(j=1,…,n).
[76.1.1.3] Empty products are interpreted as
unity.
[76.1.1.4] The integers m,n,P,Q satisfy 0≤m≤Q and 0≤n≤P.
[76.1.1.5] The coefficients Aj and Bj are positive real numbers and the complex
parameters aj,bj are such that no poles in the integrand coincide.
[76.1.1.6] If
| Ω=∑j=1nAj-∑j=n+1PAj+∑j=1mBj-∑j=m+1QBj>0 | | (A.2) |
then the integral converges absolutely and defines the H-function in
the sector argz<Ωπ/2.
[76.2.0.1] The H-function is also well
defined when either
| δ=∑j=1QBj-∑j=1PAj>0with 0<z<∞ | | (A.3) |
or
| δ=0and 0<z<R=∏j=1PAj-Aj∏j=1QBjBj. | | (A.4) |
[76.3.0.1] The H-function is a generalization of Meijers G-function and
many of the known special functions are special cases of it.
