Series Multisection

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

: $\sum_^\infty a_n\cdot z^n$

then its multisection is a power series of the form

: $\sum_^\infty a_\cdot z^$

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

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

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

: $\sum_^\infty a_\cdot z^ = \frac\cdot \sum_^ \omega^\cdot f(\omega^k\cdot z),$

where $\omega = e^$ 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 expression is especially useful in that it can convert 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''.

Examples

Bisection

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

Geometric series

Consider the geometric series

: $\sum_^ z^n=\frac \quad\text, z, < 1.$

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

: $\sum_^ z^ = \frac \quad\text, z, < 1.$

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

: $\sum_^ z^p = \frac.$

Exponential function

The exponential function

: $e^z=\sum_^$

by means of the above formula for analytic functions separates into

: $\sum_^\infty = \frac\cdot \sum_^ \omega^ e^.$

The bisections are trivially the hyperbolic functions:

: $\sum_^\infty = \frac\left(e^z+e^\right) = \cosh$
: $\sum_^\infty = \frac\left(e^z-e^\right) = \sinh.$

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

: $\sum_^\infty = \frac\cdot \sum_^ e^\cos.$

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

: $\sum_^\infty = \frac\left(e^z+2e^\cos\right)$
: $\sum_^\infty = \frac\left(e^z-e^\left(\cos-\sqrt\sin\right)\right)$
: $\sum_^\infty = \frac\left(e^z-e^\left(\cos+\sqrt\sin\right)\right),$

and the quadrisections are

: $\sum_^\infty = \frac\left(\cosh+\cos\right)$
: $\sum_^\infty = \frac\left(\sinh+\sin\right)$
: $\sum_^\infty = \frac\left(\cosh-\cos\right)$
: $\sum_^\infty = \frac\left(\sinh-\sin\right).$

Binomial series

Multisection of a binomial expansion

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

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

: $+ + + \cdots = \frac\cdot \sum_^ \left(2 \cos\frac\right )^n\cdot \cos \frac.$

References

*
*Somos, Michae
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

Algebra
Combinatorics
Mathematical analysis
Complex analysis
Mathematical series