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# 4 H-Functions

## 4.1 Definition

[120.1.1] The -function of order and with parameters , , , and is defined for by the contour integral [55, 56, 57, 58, 59]

 (153)

where the integrand is

 (154)

[page 121, §0]    [121.0.1] In (153) and is not necessarily the principal value. [121.0.2] The integers must satisfy

 (155)

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

 (156)

where

 (157)

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

 (158a) (158b)
 (159a) (159b) (159c)
 (160a) (160b) (160c)

where . [121.0.5] Here denotes a contour in the complex plane starting at and ending at and separating the points in from those

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

 (161) (162) (163) (164)

was employed. [122.0.1] The -functions are analytic for and multivalued (single valued on the Riemann surface of ).

## 4.2 Basic Properties

[122.1.1] From the definition of the -functions follow some basic properties. [122.1.2] Let denote the symmetric group of elements, and let denote a permutation in . [122.1.3] Then the product structure of (154) implies that for all and

 (165)

where the parameter permutations

 (166)

have to be inserted on the right hand side. [122.1.4] If any of or vanishes the corresponding permutation is absent.

[122.2.1] The order reduction formula

 (167)

[page 123, §0]    holds for and , and similarly

 (168)

for and . [123.0.1] The formula

 (169)

holds for and . [123.0.2] Analogous formulae are readily found if a parameter pair or appears in one of the other groups.

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

 (170)

which allows to transform an -function with and to one with and . [123.1.2] For

 (171)

while for

 (172)

[page 124, §0]    holds.

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

 (173)

## 4.3 Integral Transformations

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

 (174)

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

 (175)

whenever the inequality

 (176)

is fulfilled.

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

 (177)

valid for , , and .

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

 (178)

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

 (179)

valid if . [125.1.2] The fractional Riemann-Liouville derivative is obtained from this formula by analytic continuation to .

## 4.4 Series Expansions

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

 (180a) where (180b)

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

 (181a) [page 126, §0]    where (181b)

whenever , is as in (158) or (160) and the poles in are simple.