The Humbert functions are seven confluent hypergeometric functions of two variables obtained as limits of the
Appell hypergeometric functions.
They are conventionally denoted 
, 
, 
, 
, 
, 
, and 
, and have defining series
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
Here 
is the Pochhammer symbol, and no denominator
parameter is a nonpositive
integer. The double series for 
, 
, 
, and 
converge for 
and finite 
, while those for 
, 
, and 
converge for all finite 
and 
. Setting one variable to zero gives a hypergeometric
function or a confluent
hypergeometric function of the first kind.
The function 
is the two-variable case of a confluent form of the Lauricella
functions, and it occurs in a finite representation of the Erdélyi
multivariate Laguerre polynomials (Erdélyi 1937, Guo et al. 2026).
