The differential binomial theorem is an identity for a sum of products of derivatives
weighted by binomial coefficients. For a
positive integer 
, it states that
 |
where 
is a complex number with 
(Iwasaki and Kawamuko 1999, Trott
2006). It suffices to take 
to be a nonvanishing analytic
function on a simply connected domain
and define all powers using the same analytic branch
of the logarithm of 
. For 
, the two summands are 
and 
.
Taking 
reduces the identity to
 |
This is the vanishing of the 
th finite difference of
a polynomial of polynomial
degree 
. The general identity can be derived from the Lagrange
inversion theorem.
