[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 ).

[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) |

[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.