The name de Moivre polynomial is used for two unrelated families of polynomials, one in combinatorics and the other in the theory of algebraic equations.
In the combinatorial usage, the de Moivre polynomials are the ordinary partial Bell polynomials (O'Sullivan 2022). For nonnegative
integers 
and 
, they are defined by the formal
power series identity
 |
(1)
|
Thus 
is the coefficient of 
in the 
th power of a formal
power series with its 
coefficient equal to 0.
The boundary values are 
, 
for 
, and 
for 
. For 
, the relation to the exponential partial
Bell polynomial is
 |
(2)
|
For example, 
. Multiplication of
the defining formal power series gives the
recurrence relation
 |
(3)
|
where 
and 
(Comtet 1974, O'Sullivan 2022).
In the algebraic usage, a de Moivre polynomial is a shifted, scaled Chebyshev polynomial of the first kind. For an odd number 
, one normalization is
 |
(4)
|
where 
and 
are nonzero rational numbers, and 
has no rational square root. Here 
is the Chebyshev
polynomial of the first kind. The square roots
cancel, leaving rational coefficients,
and the equation 
can be solved using radicals
(Girstmair 2022). For 
, it becomes
 |
(5)
|
which has the form of de Moivre's quintic with 
and 
.
