If
![f(x)=f_0+f_1x+f_2x^2+...+f_nx^n+...,](/images/equations/SeriesMultisection/NumberedEquation1.svg) |
(1)
|
then
![S(n,j)=f_jx^j+f_(j+n)x^(j+n)+f_(j+2n)x^(j+2n)+...](/images/equations/SeriesMultisection/NumberedEquation2.svg) |
(2)
|
is given by
![S(n,j)=1/nsum_(t=0)^(n-1)w^(-jt)f(w^tx),](/images/equations/SeriesMultisection/NumberedEquation3.svg) |
(3)
|
where
.
When applied to the generating function
![(1+x)^n=sum_(k=0)^n(n; k)x^k](/images/equations/SeriesMultisection/NumberedEquation4.svg) |
(4)
|
it gives the identity
![sum_(m=0)^infty(n; t+sm)=1/ssum_(j=0)^(s-1)cos[(pi(n-2t)j)/s]2^ncos^n((pij)/s)](/images/equations/SeriesMultisection/NumberedEquation5.svg) |
(5)
|
with integers
(and where the sum can be taken only up to
).
Other multisection examples are given by Somos (2006).
See also
Multisection,
Series,
Series Reversion
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References
Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 210-214, 1985.Somos,
M. "A Multisection of
-Series." Mar 31, 2006. http://cis.csuohio.edu/~somos/multiq.html.Referenced
on Wolfram|Alpha
Series Multisection
Cite this as:
Weisstein, Eric W. "Series Multisection."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SeriesMultisection.html
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