Series multisection

Short description: In mathematics, series built from equally spaced terms of another series

In mathematics, a multisection of a power series is a new power series composed of equally spaced terms extracted unaltered from the original series. Formally, if one is given a power series

$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_n\cdot z^n$

then its multisection is a power series of the form

$\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_{qm+p}\cdot z^{qm+p}$

where p, q are integers, with 0 ≤ p < q. Series multisection represents one of the common transformations of generating functions.

Multisection of analytic functions

A multisection of the series of an analytic function

$f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n\cdot z^n$

has a closed-form expression in terms of the function $f(x)$:

$\sum _{m=0}^{\mathrm{\infty }}{a}_{qm+p}\cdot z^{qm+p}=\frac{1}{q}\cdot \sum _{k=0}^{q-1}{\omega }^{-kp}\cdot f({\omega }^k\cdot z),$

where $\omega =e^{\frac{2\pi i}{q}}$ is a primitive q-th root of unity. This expression is often called a root of unity filter. This solution was first discovered by Thomas Simpson.

This is the projection of the representation of $\mathbb{\mathbb{Z}}/q\mathbb{\mathbb{Z}}$ via $(g^n\cdot f)(x)=f({\omega }^{-}qx)$ onto the isotype of the irreducible representation whose character is $\chi (g)={\omega }^p$ (and in this case since it's abelian, the action is just multiplying by the character) .

Examples

Bisection

In general, the bisections of a series are the even and odd parts of the series.

Geometric series

Consider the geometric series

$\sum _{n=0}^{\mathrm{\infty }}{z}^n=\frac{1}{1-z}\text{ for }|z|<1.$

By setting $z\to z^q$ in the above series, its multisections are easily seen to be

$\sum _{m=0}^{\mathrm{\infty }}{z}^{qm+p}=\frac{z^p}{1-z^q}\text{ for }|z|<1.$

Remembering that the sum of the multisections must equal the original series, we recover the familiar identity

$\sum _{p=0}^{q-1}{z}^p=\frac{1-z^q}{1-z}.$

Exponential function

The exponential function

$e^z=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{n!}$

by means of the above formula for analytic functions separates into

$\sum _{m=0}^{\mathrm{\infty }}\frac{z^{qm+p}}{(qm+p)!}=\frac{1}{q}\cdot \sum _{k=0}^{q-1}{\omega }^{-kp}{e}^{{\omega }^{k}z}.$

The bisections are trivially the hyperbolic functions:

$\sum _{m=0}^{\mathrm{\infty }}\frac{z^{2m}}{(2m)!}=\frac{1}{2}(e^z+e^{-z})=\cosh z$

$\sum _{m=0}^{\mathrm{\infty }}\frac{z^{2m+1}}{(2m+1)!}=\frac{1}{2}(e^z-e^{-z})=\sinh z.$

Higher order multisections are found by noting that all such series must be real-valued along the real line. By taking the real part and using standard trigonometric identities, the formulas may be written in explicitly real form as

$\sum _{m=0}^{\mathrm{\infty }}\frac{z^{qm+p}}{(qm+p)!}=\frac{1}{q}\cdot \sum _{k=0}^{q-1}{e}^{z\cos (2\pi k/q)}\cos (z\sin \left(\frac{2\pi k}{q}\right)-\frac{2\pi kp}{q}).$

These can be seen as solutions to the linear differential equation $f^{(q)}(z)=f(z)$ with boundary conditions $f^{(k)}(0)={\delta }_{k,p}$, using Kronecker delta notation. In particular, the trisections are

$\sum _{m=0}^{\mathrm{\infty }}\frac{z^{3m}}{(3m)!}=\frac{1}{3}(e^z+2e^{-z/2}\cos \frac{\sqrt{3}z}{2})$

$\sum _{m=0}^{\mathrm{\infty }}\frac{z^{3m+1}}{(3m+1)!}=\frac{1}{3}(e^z-2e^{-z/2}\cos (\frac{\sqrt{3}z}{2}+\frac{\pi }{3}))$

$\sum _{m=0}^{\mathrm{\infty }}\frac{z^{3m+2}}{(3m+2)!}=\frac{1}{3}(e^z-2e^{-z/2}\cos (\frac{\sqrt{3}z}{2}-\frac{\pi }{3})),$

and the quadrisections are

$\sum _{m=0}^{\mathrm{\infty }}\frac{z^{4m}}{(4m)!}=\frac{1}{2}(\cosh z+\cos z)$

$\sum _{m=0}^{\mathrm{\infty }}\frac{z^{4m+1}}{(4m+1)!}=\frac{1}{2}(\sinh z+\sin z)$

$\sum _{m=0}^{\mathrm{\infty }}\frac{z^{4m+2}}{(4m+2)!}=\frac{1}{2}(\cosh z-\cos z)$

$\sum _{m=0}^{\mathrm{\infty }}\frac{z^{4m+3}}{(4m+3)!}=\frac{1}{2}(\sinh z-\sin z).$

Binomial series

Multisection of a binomial expansion

$(1+x{)}^n=(\genfrac{}{}{0ex}{}{n}{0})x^0+(\genfrac{}{}{0ex}{}{n}{1})x+(\genfrac{}{}{0ex}{}{n}{2})x^2+\cdots$

at x = 1 gives the following identity for the sum of binomial coefficients with step q:

$(\genfrac{}{}{0ex}{}{n}{p})+(\genfrac{}{}{0ex}{}{n}{p+q})+(\genfrac{}{}{0ex}{}{n}{p+2q})+\cdots =\frac{1}{q}\cdot \sum _{k=0}^{q-1}{(2\cos \frac{\pi k}{q})}^n\cdot \cos \frac{\pi (n-2p)k}{q}.$

Applications

Series multisection converts an infinite sum into a finite sum. It is used, for example, in a key step of a standard proof of Gauss's digamma theorem, which gives a closed-form solution to the digamma function evaluated at rational values p/q.

References

• Weisstein, Eric W.. "Series Multisection". http://mathworld.wolfram.com/SeriesMultisection.html. 
• Somos, Michael A Multisection of q-Series, 2006.
• John Riordan (1968). Combinatorial identities. New York: John Wiley and Sons.