The hypergeometric 
-function is a homogeneous
function obtained from the Lauricella functions
of type 
(Carlson 1963, 1970). For 
,
positive parameters 
, ..., 
,
positive arguments 
, ..., 
,
and a real number 
, put 
and define
 |
(1)
|
The Lauricella functions are continued analytically when their defining series do not converge. The 
-function is unchanged by simultaneous permutations
of the parameters 
and arguments 
. It is a symmetric
function of all arguments when all the 
are equal. Unequal weights do not in general allow permuting
the arguments alone.
For 
, an equivalent integral representation is
 |
(2)
|
where 
is the beta function. For 
, homogeneity
and diagonal normalization give
 |  |  |
(3)
|
 |  |  |
(4)
|
The Carlson elliptic integrals include the specializations
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
The 
identification uses analytic continuation
in 
, since 
is outside the range of the integral representation above
(NIST Digital Library of Mathematical Functions, §19.16).
."
J. Math. Anal. Appl. 7, 452-470, 1963.