The generalized Lambert 
-function with real parameters
and 
is a generally multivalued function obtained
by inverting
 |
(1)
|
where 
and 
are nonnegative integers. Here 
is excluded for every 
, and an empty product is
interpreted as 1. For a real value 
, the inverse values are the real
solutions of 
. This definition does not select a particular branch
(Mező and Baricz 2017).
For 
,
,
and 
,
the defining equation is 
, so the inverse values are the real
values of the ordinary Lambert W-function.
For 
,
the inverse is the natural logarithm 
for 
(Mező and Baricz 2017).
A special case with a quadratic polynomial is
 |
(2)
|
where 
,
,
,
and 
are real, with 
and 
. The substitution 
gives 
, so the inverse values with 
,
,
,
and 
yield 
.
This equation can have three distinct real roots. It
has at most three because the third derivative of
never vanishes, and Rolle's theorem would force
that derivative to vanish between four distinct real roots (Scott et al. 2014, pp. 43-44).
If 
,
,
and 
,
taking both square-root signs gives
 |
(3)
|
where 
and 
denotes a branch of the ordinary Lambert
W-function (Scott et al. 2013, p. 77; Maignan and Scott 2016,
p. 46). The plus sign gives a real root for 
.
The minus sign gives real roots iff 
,
using 
and 
with coincident values at 
.
Equations involving an exponential function and a rational function also motivated earlier work on generalized Lambert functions (Scott and Mann 2006). A generalization of the defining equation does not by itself provide an explicit formula in ordinary Lambert W-functions.
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45-60, 2016. 
Function." Trans. Amer. Math. Soc. 369,
7917-7934, 2017. 
Function." ACM Commun. Comput. Algebra 47, 75-83, 2013. 
Function." ACM Commun. Comput. Algebra 48,
42-56, 2014. 
Function." 6 Sep 2006.