Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. The elements of this group are of the form
where 
, 
, 
, 
are real
numbers, and
. Vilenkin (1968, Chapter 8) constructs irreducible
representations of this group, in which the diagonal matrices correspond
to operators of multiplication by an exponential function. The other group elements
correspond to integral operators whose kernels can be expressed in terms
of Whittaker functions. This identification can be used to obtain various
properties of the Whittaker functions, including recurrence relations
and derivatives.
