Series multisection explained

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 ^{inf ty} _{n =-inf ty} a _{n ⋅} z ^{n}

then its multisection is a power series of the form

\sum ^{inf ty} _{m =-inf ty} a _{qm+p} ⋅ z ^{qm+p}

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 ^{inf ty} _{n =0} a _{n ⋅} z ^{n}

has a closed-form expression in terms of the function

f(x) :\sum ^{inf ty} _{m =0} a _{qm+p} ⋅ z ^{qm+p} = (1)/(q) ⋅ \sum ^{q -1} _{k =0} \omega ^{-kp} ⋅ f(\omega ^{k ⋅} z),

where

\omega = e ^{(2 \pi i)/(q)} 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.^[1] 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 ^{inf ty} _{n =0} z ^{n = (1)/(1- z)} for |z| < 1 .

By setting

z → z ^{q} in the above series, its multisections are easily seen to be\sum ^{inf ty} _{m =0} z ^{qm+p} = (z ^{p})/(1- z ^{q}) for |z| < 1 .

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

\sum ^{q -1} _{p =0} z ^{p} = (1- z ^{q})/(1- z) .

Exponential function

The exponential function

e ^{inf ty} _{n =0} {z ^{n} \over n!}

by means of the above formula for analytic functions separates into

\sum ^{inf ty} _{m =0} {z ^{qm+p} \over (qm + p)!} = (1)/(q) ⋅ \sum ^{q -1} _{k =0} \omega ^{-kp} e ^{\omega ^{k} z} .

The bisections are trivially the hyperbolic functions:

\sum ^{inf ty} _{m =0} {z ^{2m} \over ( 2 m)!} = (1)/(2) \left(e ^{z+e} ^{-z} \right) = \cosh{z} \sum ^{inf ty} _{m =0} {z ^{2m+1} \over ( 2 m +1 )!} = (1)/(2) \left(e ^{z-e} ^{-z} \right) = \sinh{z}.

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 ^{inf ty} _{m =0} {z ^{qm+p} \over (qm + p)!} = (1)/(q) ⋅ \sum ^{q -1} _{k =0} e ^{z\cos(2\pi} \cos{\left(z\sin{\left( (2 \pi k)/(\right)} - q) (2 \pi kp)/(q) \right)}. f ^{(q)} (z) = f(z) with boundary conditions f ^{(k)} ( 0 ) = \delta _{k,p} , using Kronecker delta notation. In particular, the trisections are\sum ^{inf ty} _{m =0} {z ^{3m} \over ( 3 m)!} = (1)/(3) \left(e ^{z+2e} ^{-z/2} \cos{ (\sqrt{ 3)/(z}{ 2 }}\right)) \sum ^{inf ty} _{m =0} {z ^{3m+1} \over ( 3 m +1 )!} = (1)/(3) \left(e ^{z-e} ^{-z/2} \left(\cos{ (\sqrt{ 3)/(z}{ 2 }} - \sqrt{ 3 }\sin{ (\sqrt{ 3)/(z}{ 2 }}\right)\right))) \sum ^{inf ty} _{m =0} {z ^{3m+2} \over ( 3 m +2 )!} = (1)/(3) \left(e ^{z-e} ^{-z/2} \left(\cos{ (\sqrt{ 3)/(z}{ 2 }} + \sqrt{ 3 }\sin{ (\sqrt{ 3)/(z}{ 2 }}\right)\right),))

and the quadrisections are

\sum ^{inf ty} _{m =0} {z ^{4m} \over ( 4 m)!} = (1)/(2) \left(\cosh{z} + \cos{z}\right) \sum ^{inf ty} _{m =0} {z ^{4m+1} \over ( 4 m +1 )!} = (1)/(2) \left(\sinh{z} + \sin{z}\right) \sum ^{inf ty} _{m =0} {z ^{4m+2} \over ( 4 m +2 )!} = (1)/(2) \left(\cosh{z} - \cos{z}\right) \sum ^{inf ty} _{m =0} {z ^{4m+3} \over ( 4 m +3 )!} = (1)/(2) \left(\sinh{z} - \sin{z}\right).

Binomial series

Multisection of a binomial expansion

( 1+ x) ^{n} = {n\choose 0 } x ^{0} + {n\choose 1 } x + {n\choose 2 } x ^{2} + …

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

{n\choose p} + {n\choose p + q} + {n\choose p +2 q} + … = (1)/(q) ⋅ \sum ^{q -1} _{k =0} \left( 2 \cos (\pi k)/(q) \right ) ^{n ⋅} \cos (\pi(n -2 p)k)/(q) .

References

• • Somos, Michael A Multisection of q-Series, 2006.
• Book: John Riordan . Combinatorial identities . John Riordan (mathematician). John Wiley and Sons . New York . 1968.

Notes and References

1. Simpson . Thomas . 1757 . CIII. The invention of a general method for determining the sum of every 2d, 3d, 4th, or 5th, &c. term of a series, taken in order; the sum of the whole series being known . Philosophical Transactions of the Royal Society of London . 51 . 757–759 . 10.1098/rstl.1757.0104. free .