Series multisection
Encyclopedia
In mathematics, a multisection of a power series is a new power series composed of equally-spaced terms extracted unaltered from the original. Formally, if one is given
then a multisection is a power series of the form
where c, d are integers, with 0 ≤ d < c.
has a closed-form expression
in terms of the function
:
where
is a primitive c-th root of unity.
at x = 1 gives the following identity for the sum of binomial coefficients with step c:
then a multisection is a power series of the form
where c, d are integers, with 0 ≤ d < c.
Multisection of analytic functions
A multisection of the series of an analytic functionAnalytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
has a closed-form expression
Closed-form expression
In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a bounded number of certain "well-known" functions...
in terms of the function
:
where
is a primitive c-th root of unity.Example
Multisection of a binomial
at x = 1 gives the following identity for the sum of binomial coefficients with step c:
