Series multisection represents one of the common transformations of generating functions.
This expression is often called a root of unity filter.
This solution was first discovered by Thomas Simpson.
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.
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 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
In particular, the trisections are and the quadrisections are Multisection of a binomial expansion at x = 1 gives the following identity for the sum of binomial coefficients with step q: