Series Multisection
If
| f(x)=f_0+f_1x+f_2x^2+...+f_nx^n+..., |
(1)
|
then
| S(n,j)=f_jx^j+f_(j+n)x^(j+n)+f_(j+2n)x^(j+2n)+... |
(2)
|
is given by
where w=e^(2pii/n).
When applied to the generating function
it gives the identity
![画像: sum_(m=0)^infty(n; t+sm)=1/ssum_(j=0)^(s-1)cos[(pi(n-2t)j)/s]2^ncos^n((pij)/s)](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/SeriesMultisection/NumberedEquation5.svg){kind=link}
with integers 0<=t<s (and where the sum can be taken only up to t+sm<=n).
Other multisection examples are given by Somos (2006).
See also
Multisection, Series, Series ReversionExplore with Wolfram|Alpha
WolframAlpha
More things to try:
References
Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 210-214, 1985.Somos, M. "A Multisection of q-Series." Mar 31, 2006. http://cis.csuohio.edu/~somos/multiq.html.Referenced on Wolfram|Alpha
Series MultisectionCite this as:
Weisstein, Eric W. "Series Multisection." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SeriesMultisection.html