\[\operatorname{ce}_{m}\left(z,q\right)=\cos mz-\frac{q}{4}\left(\frac{1}{m+1}%
\cos(m+2)z-\frac{1}{m-1}\cos(m-2)z\right)+\frac{q^{2}}{32}\left(\frac{1}{(m+1)%
(m+2)}\cos(m+4)z+\frac{1}{(m-1)(m-2)}\cos(m-4)z-\frac{2(m^{2}+1)}{(m^{2}-1)^{2%
}}\cos mz\right)+\cdots.\]