Girish Mahajan (Editor)

Series multisection

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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

Contents

∑ n = − ∞ ∞ a n ⋅ x n

then its multisection is a power series of the form

∑ m = − ∞ ∞ a c m + d ⋅ x c m + d

where c, d are integers, with 0 ≤ d < c.

Multisection of analytic functions

A multisection of the series of an analytic function

F ( x ) = ∑ n = − ∞ ∞ a n ⋅ x n

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

∑ m = − ∞ ∞ a c m + d ⋅ x c m + d = 1 c ⋅ ∑ k = 0 c − 1 w − k d ⋅ F ( w k ⋅ x ) ,

where w = e 2 π i c is a primitive c-th root of unity.

Example

Multisection of a binomial expansion

( 1 + x ) q = ( q 0 ) x 0 + ( q 1 ) x + ( q 2 ) x 2 + ⋯

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

( q d ) + ( q d + c ) + ( q d + 2 c ) + ⋯ = 1 c ⋅ ∑ k = 0 c − 1 ( 2 cos ⁡ π k c ) q ⋅ cos ⁡ π ( q − 2 d ) k c .

References

Series multisection Wikipedia
(Text) CC BY-SA


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