[3.2.3.1] The general H-function is defined as the inverse
Mellin transform [24]
| 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+Ajsz-sds | | (A.1) |
where the contour C runs from c-i∞ to c+i∞ separating
the poles of Γ(bj+Bjs),(j=1,…,m) from those of
Γ(1-aj-Ajs),(j=1,…,n).
[3.2.3.2] Empty products are interpreted as
unity.
[3.2.3.3] The integers m,n,P,Q satisfy 0≤m≤Q and 0≤n≤P.
[3.2.3.4] The coefficients Aj and Bj are positive real numbers and the complex
parameters aj,bj are such that no poles in the integrand coincide.
[3.2.3.5] If
| Ω=∑j=1nAj-∑j=n+1PAj+∑j=1mBj-∑j=m+1QBj>0 | | (A.2) |
[page 4, §0]
then the integral converges absolutely and defines the H-function in
the sector argz<Ωπ/2.
[4.1.0.1] The H-function is also well
defined when either
| δ=∑j=1QBj-∑j=1PAj>0 with 0<z<∞ | | (A.3) |
or
| δ=0 and 0<z<R=∏j=1PAj-Aj∏j=1QBjBj. | | (A.4) |
[4.1.0.2] For δ≥0 the H-function has the series representation
| HPQmn(z|a1,A1…aP,APb1,B1…bQ,BQ)=∑i=1m∑k=0∞∏j=1j≠imΓbj-bi+kBjBi∏j=1nΓ1-aj+bi+kAjBi∏j=m+1QΓ1-bj+bi+kBjBi∏j=n+1PΓaj-bi+kAjBi-1kzbi+k/Bik!Bi | | (A.5) |
provided that Bkbj+l≠Bjbk+s for j≠k,1≤j,k≤m
and l,s=0,1,…
[4.1.0.3] The H-function is a generalization of Meijers G-function and
many of the known special functions are special cases of it.
