where the integrand is
[page 121, §0] [121.0.1] In (153) and is not necessarily the principal value. [121.0.2] The integers must satisfy
and empty products are interpreted as being unity. [121.0.3] The parameters are restricted by the condition
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
was employed. [122.0.1] The -functions are analytic for and multivalued (single valued on the Riemann surface of ).
[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
where the parameter permutations
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
[page 123, §0] holds for and , and similarly
for and . [123.0.1] The formula
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
which allows to transform an -function with and to one with and . [123.1.2] For
[page 124, §0] holds.
[124.1.1] For with conditions (159) the integrand is analytic and thus
[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
by the Mellin inversion theorem . [124.2.2] Therefore
whenever the inequality
valid for , , and .