4 H-Functions

[120.1.1] The H-function of order m,n,p,q∈N4 and with parameters Ai∈R+(i=1,…,p), Bi∈R+(i=1,…,q), ai∈C(i=1,…,p), and bi∈C(i=1,…,q) is defined for z∈C,z≠0 by the contour integral [55, 56, 57, 58, 59]

where the integrand is

[page 121, §0]    [121.0.1] In (153) z-s=exp⁡-s⁢log⁡z-i⁢arg⁡z and arg⁡z is not necessarily the principal value. [121.0.2] The integers m,n,p,q must satisfy

and empty products are interpreted as being unity. [121.0.3] The parameters are restricted by the condition

where

are the poles of the numerator in (154). [121.0.4] The integral converges if one of the following conditions holds [59]

where γ1<γ2. [121.0.5] Here L⁢z1,z2;G1,G2 denotes a contour in the complex plane starting at z1 and ending at z2 and separating the points in G1 from those

[page 122, §0]    in G2, and the notation

was employed. [122.0.1] The H-functions are analytic for z≠0 and multivalued (single valued on the Riemann surface of log⁡z).

[122.1.1] From the definition of the H-functions follow some basic properties. [122.1.2] Let Sn(n≥1) denote the symmetric group of n elements, and let πn denote a permutation in Sn. [122.1.3] Then the product structure of (154) implies that for all πn∈Sn,πm∈Sm,πp-n∈Sp-n and πq-m∈Sq-m

where the parameter permutations

have to be inserted on the right hand side. [122.1.4] If any of n,m,p-n or q-m vanishes the corresponding permutation is absent.

[122.2.1] The order reduction formula

[page 123, §0]    holds for n≥1 and q>m, and similarly

for m≥1 and p>n. [123.0.1] The formula

holds for n≥1 and Re⁢1-a>0. [123.0.2] Analogous formulae are readily found if a parameter pair a,0 or b,0 appears in one of the other groups.

[123.1.1] A change of variables in (153) shows

which allows to transform an H-function with D>0 and arg⁡z to one with D<0 and arg⁡1/z. [123.1.2] For γ>0

while for γ∈R

[page 124, §0]    holds.

[124.1.1] For m=0 with conditions (159) the integrand is analytic and thus

[124.2.1] The definition of an H-function in eq. (153) becomes an inverse Mellin transform if L is chosen parallel to the imaginary axis inside the strip

by the Mellin inversion theorem [60]. [124.2.2] Therefore

whenever the inequality

is fulfilled.

[124.3.1] The Laplace transform of an H-function is obtained from eq. (175) by using eq. (78). [124.3.2] One finds

valid for Re⁢s>0, C>0, arg⁡z<12⁢C⁢π and min1≤j≤m⁡Re⁢bj/Bj>-1.

[page 125, §1]   
[125.1.1] The definite integral found in [59, 2.25.2.2.]

contains as a special case the fractional Riemann-Liouville integral [58, (2.7.13)]

valid if min1≤j≤m⁡Re⁢bj/Bj>0. [125.1.2] The fractional Riemann-Liouville derivative is obtained from this formula by analytic continuation to α<0.

[125.2.1] The H-functions may be represented as the series [56, 57, 58, 59]

whenever D≥0, L is as in (158) or (159) and the poles in Pb are simple. [125.2.2] Similarly

whenever D≤0, L is as in (158) or (160) and the poles in Pa are simple.