Let 
be a partition, let 
denote a Jack
polynomial in the "C" normalization, characterized for each nonnegative
integer 
by 
, and define
the generalized Pochhammer symbol by
![(a)_kappa^((alpha))=product_(i=1)^(l(kappa))[a-(i-1)/alpha]_(kappa_i).](/images/equations/MultivariateHypergeometricFunction/NumberedEquation1.svg) |
Here 
is the number of nonzero parts of 
. The multivariate hypergeometric function is
 |
where 
, 
, the inner sum is over all partitions 
of 
, and the parameters are chosen so that no denominator
factor vanishes (Dumitriu et al. 2007). For 
, this definition reduces to the ordinary generalized
hypergeometric function. It also gives a hypergeometric
function of matrix argument when the 
are interpreted as the eigenvalues
of a matrix.
If the series does not terminate, it converges for every 
when 
. When 
, it is absolutely
convergent for 
, with convergence on the boundary depending
on the parameters. When 
, it diverges unless it terminates (Koev and Edelman
2006). Values outside the domain of the defining series may be obtained by analytic
continuation when such a continuation exists.
