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Series Multisection

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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

S(n,j)=1/nsum_(t=0)^(n-1)w^(-jt)f(w^tx),
(3)

where w=e^(2pii/n).

When applied to the generating function

(1+x)^n=sum_(k=0)^n(n; k)x^k
(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)
(5)

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 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 q-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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