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

: $\sum_{n=-\infty}^\infty a_n\cdot z^n$

then its multisection is a power series of the form

: $\sum_{m=-\infty}^\infty a_{qm+p}\cdot z^{qm+p}$

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

: $f(z) = \sum_{n=0}^\infty a_n\cdot z^n$

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

: $\sum_{m=0}^\infty a_{qm+p}\cdot z^{qm+p} = \frac{1}{q}\cdot \sum_{k=0}^{q-1} \omega^{-kp}\cdot f(\omega^k\cdot z),$

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.{{cite journal |last1=Simpson |first1=Thomas |date=1757 |title=CIII. The invention of a general method for determining the sum of every 2d, 3d, 4th, or 5th, &c. term of a series, taken in order; the sum of the whole series being known |journal=Philosophical Transactions of the Royal Society of London |volume=51 |pages=757–759 |doi=10.1098/rstl.1757.0104|doi-access=free }}

Examples

=Bisection=

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

=Geometric series=

Consider the geometric series

: $\sum_{n=0}^{\infty} z^n=\frac{1}{1-z} \quad\text{ for }|z| < 1.$

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

: $\sum_{m=0}^{\infty} z^{qm+p} = \frac{z^p}{1-z^q} \quad\text{ for }|z| < 1.$

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

: $e^z=\sum_{n=0}^{\infty} {z^n \over n!}$

by means of the above formula for analytic functions separates into

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

The bisections are trivially the hyperbolic functions:

: $\sum_{m=0}^\infty {z^{2m} \over (2m)!} = \frac{1}{2}\left(e^z+e^{-z}\right) = \cosh{z}$

: $\sum_{m=0}^\infty {z^{2m+1} \over (2m+1)!} = \frac{1}{2}\left(e^z-e^{-z}\right) = \sinh{z}.$

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

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

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

: $\sum_{m=0}^\infty {z^{3m} \over (3m)!} = \frac{1}{3}\left(e^z+2e^{-z/2}\cos{\frac{\sqrt{3}z}{2}}\right)$

: $\sum_{m=0}^\infty {z^{3m+1} \over (3m+1)!} = \frac{1}{3}\left(e^z-2e^{-z/2}\cos{\left(\frac{\sqrt{3}z}{2}+\frac{\pi}{3}\right)}\right)$

: $\sum_{m=0}^\infty {z^{3m+2} \over (3m+2)!} = \frac{1}{3}\left(e^z-2e^{-z/2}\cos{\left(\frac{\sqrt{3}z}{2}-\frac{\pi}{3}\right)}\right),$

and the quadrisections are

: $\sum_{m=0}^\infty {z^{4m} \over (4m)!} = \frac{1}{2}\left(\cosh{z}+\cos{z}\right)$

: $\sum_{m=0}^\infty {z^{4m+1} \over (4m+1)!} = \frac{1}{2}\left(\sinh{z}+\sin{z}\right)$

: $\sum_{m=0}^\infty {z^{4m+2} \over (4m+2)!} = \frac{1}{2}\left(\cosh{z}-\cos{z}\right)$

: $\sum_{m=0}^\infty {z^{4m+3} \over (4m+3)!} = \frac{1}{2}\left(\sinh{z}-\sin{z}\right).$

=Binomial series=

Multisection of a binomial expansion

: $(1+x)^n = {n\choose 0} x^0 + {n\choose 1} x + {n\choose 2} x^2 + \cdots$

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

: ${n\choose p} + {n\choose p+q} + {n\choose p+2q} + \cdots = \frac{1}{q}\cdot \sum_{k=0}^{q-1} \left(2 \cos\frac{\pi k}{q}\right )^n\cdot \cos \frac{\pi(n-2p)k}{q}.$

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 ==

{{MathWorld|urlname=SeriesMultisection|title=Series Multisection}}
Somos, Michael [https://grail.eecs.csuohio.edu/~somos/multiq.html A Multisection of q-Series], 2006.
{{cite book |author=John Riordan |title=Combinatorial identities |author-link=John Riordan (mathematician)|publisher=John Wiley and Sons |place=New York |year=1968}}

Category:Algebra

Category:Combinatorics

Category:Mathematical analysis

Category:Complex analysis

Category:Series (mathematics)