[120.1.1] The -function of order
and with parameters
,
,
, and
is defined for
by the contour integral
[55, 56, 57, 58, 59]
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(153) |
where the integrand is
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(154) |
[page 121, §0]
[121.0.1] In (153) and
is not necessarily the principal value.
[121.0.2] The integers
must satisfy
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(155) |
and empty products are interpreted as being unity. [121.0.3] The parameters are restricted by the condition
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(156) |
where
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||
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(157) |
are the poles of the numerator in (154). [121.0.4] The integral converges if one of the following conditions holds [59]
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(158a) | |
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(158b) |
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(159a) | |
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(159b) | |
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(159c) |
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(160a) | |
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(160b) | |
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(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
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(161) | |
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(162) | |
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(163) | |
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(164) |
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
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(165) |
where the parameter permutations
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||
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||
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(166) | |
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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
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(167) |
[page 123, §0]
holds for and
, and similarly
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(168) |
for and
.
[123.0.1] The formula
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(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
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(170) |
which allows to transform an -function with
and
to one
with
and
.
[123.1.2] For
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(171) |
while for
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(172) |
[page 124, §0] holds.
[124.1.1] For with conditions (159) the integrand is analytic
and thus
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(173) |
[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
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(174) |
by the Mellin inversion theorem [60]. [124.2.2] Therefore
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(175) |
whenever the inequality
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(176) |
is fulfilled.