The -function is defined as [43]

(A.1) |

[p. 2474l, §2]

where is a contour from to separating
the poles of , from those of
, .
Empty products are interpreted as unity.
The integers satisfy and .
The coefficients and are positive
real numbers and the complex parameters , are such

[p. 2474r, §2]

that no poles in the integrand coincide.
If

(A.2) |

then integral converges absolutely and defines the -function

[p. 2475l, §0]

in the sector .
The -function is also well defined when either

(A.3) |

or

(A.4) |

[p. 2475r, §0]

The -function is a generalization of Meijerâs function
and contains many of the known special functions.
In particular Mittag-Leffler and generalized Mittag-Leffler
functions are special cases of the -function.