The Erdélyi multivariate Laguerre polynomials 
are the coefficients
in the generating function
 |
where 
. For 
, they reduce to the associated
Laguerre polynomials. When all arguments are equal, they satisfy
 |
These polynomials are distinct from the symmetric polynomials indexed by partitions described under
multivariate Laguerre polynomials.
For 
, their representation in terms of special
functions uses the Humbert function 
.
Erdélyi (1937) also proved a multivariate extension of the Hille-Hardy
formula, while Guo et al. (2026) use the Le
Roy function in a generating function
for a main-diagonal sequence
of this family.
