An Ince polynomial is a trigonometric polynomial that solves the Ince equation with a nonnegative
integer degree parameter 
, real 
, and a suitable value of the
spectral parameter 
. Its harmonics have frequencies at most 
and of the same parity as 
(Arscott 1967, NIST DLMF). The even
and odd families are denoted by 
and 
, respectively. Their
finite Fourier series have the forms
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
contains the nonnegative harmonic indices. The order 
ranges over 
for the even family and over its positive elements for the
odd family. Thus 
for either family.
To compute the coefficients, write the Ince equation as 
, where 
. For either basis
function 
or 
,
![Lphi_k(z)=k^2phi_k(z)+xi/2[(p+k)phi_(k-2)(z)+(p-k)phi_(k+2)(z)].](/images/equations/IncePolynomial/NumberedEquation1.svg) |
(3)
|
The coefficient of the higher harmonic vanishes at 
, so each finite Fourier space is invariant. The allowed
values of 
and the Fourier coefficients follow from the eigenvalues
and eigenvectors of the two finite matrices representing
on these spaces (Arscott 1967, NIST
DLMF).
