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
then its multisection is a power series of the form
where p, q are integers, with 0 ≤ p < q.

Multisection of analytic functions

A multisection of the series of an analytic function
has a closed-form expression in terms of the function :
where is a primitive q-th root of unity. 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
By setting in the above series, its multisections are easily seen to be
Remembering that the sum of the multisections must equal the original series, we recover the familiar identity

Exponential function

The exponential function
by means of the above formula for analytic functions separates into
The bisections are trivially the hyperbolic functions:
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
These can be seen as solutions to the linear differential equation with boundary conditions, using Kronecker delta notation. In particular, the trisections are
and the quadrusections are

Binomial theorem

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