Series Multisection

If

$$f(x)=f_0+f_1x+f_2x^2+\ldots+f_nx^n+\ldots,$$

then

$$S(n,j)=f_jx^j+f_{j+n}x^{j+n}+f_{j+2n}x^{j+2n}+\ldots$$

is given by

$$S(n,j)={1\over n} \sum_{t=0}^{n-1} w^{-jt} f(w^tx),$$

where $w=e^{2\pi i/n}$.

See also Series Reversion

References

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 210-214, 1985.