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

∑ n = − ∞ ∞ a n ⋅ z n {\displaystyle \sum _{n=-\infty }^{\infty }a_{n}\cdot z^{n}}

then its multisection is a power series of the form

∑ m = − ∞ ∞ a q m + p ⋅ z q m + p {\displaystyle \sum _{m=-\infty }^{\infty }a_{qm+p}\cdot z^{qm+p}}

where p, q are integers, with 0 ≤ p < q. Series multisection represents one of the common transformations of generating functions.

A multisection of the series of an analytic function

f ( z ) = ∑ n = 0 ∞ a n ⋅ z n {\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}\cdot z^{n}}

has a closed-form expression in terms of the function f ( x ) {\displaystyle f(x)} :

∑ m = 0 ∞ a q m + p ⋅ z q m + p = 1 q ⋅ ∑ k = 0 q − 1 ω − k p ⋅ f ( ω k ⋅ z ) , {\displaystyle \sum _{m=0}^{\infty }a_{qm+p}\cdot z^{qm+p}={\frac {1}{q}}\cdot \sum _{k=0}^{q-1}\omega ^{-kp}\cdot f(\omega ^{k}\cdot z),}

where ω = e 2 π i q {\displaystyle \omega =e^{\frac {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 is the projection of the representation of Z / q Z {\displaystyle \mathbb {Z} /q\mathbb {Z} } via ( g n ⋅ f ) ( x ) = f ( ω − q x ) {\displaystyle (g^{n}\cdot f)(x)=f(\omega ^{-}qx)} onto the isotype of the irreducible representation whose character is χ ( g ) = ω p {\displaystyle \chi (g)=\omega ^{p}} (and in this case since it's abelian, the action is just multiplying by the character) .

In general, the bisections of a series are the even and odd parts of the series.

Consider the geometric series

∑ n = 0 ∞ z n = 1 1 − z  for  | z | < 1. {\displaystyle \sum _{n=0}^{\infty }z^{n}={\frac {1}{1-z}}\quad {\text{ for }}|z|<1.}

By setting z → z q {\displaystyle z\rightarrow z^{q}} in the above series, its multisections are easily seen to be

∑ m = 0 ∞ z q m + p = z p 1 − z q  for  | z | < 1. {\displaystyle \sum _{m=0}^{\infty }z^{qm+p}={\frac {z^{p}}{1-z^{q}}}\quad {\text{ for }}|z|<1.}

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

∑ p = 0 q − 1 z p = 1 − z q 1 − z . {\displaystyle \sum _{p=0}^{q-1}z^{p}={\frac {1-z^{q}}{1-z}}.}

The exponential function

e z = ∑ n = 0 ∞ z n n ! {\displaystyle e^{z}=\sum _{n=0}^{\infty }{z^{n} \over n!}}

by means of the above formula for analytic functions separates into

∑ m = 0 ∞ z q m + p ( q m + p ) ! = 1 q ⋅ ∑ k = 0 q − 1 ω − k p e ω k z . {\displaystyle \sum _{m=0}^{\infty }{z^{qm+p} \over (qm+p)!}={\frac {1}{q}}\cdot \sum _{k=0}^{q-1}\omega ^{-kp}e^{\omega ^{k}z}.}

The bisections are trivially the hyperbolic functions:

∑ m = 0 ∞ z 2 m ( 2 m ) ! = 1 2 ( e z + e − z ) = cosh ⁡ z {\displaystyle \sum _{m=0}^{\infty }{z^{2m} \over (2m)!}={\frac {1}{2}}\left(e^{z}+e^{-z}\right)=\cosh {z}}

∑ m = 0 ∞ z 2 m + 1 ( 2 m + 1 ) ! = 1 2 ( e z − e − z ) = sinh ⁡ z . {\displaystyle \sum _{m=0}^{\infty }{z^{2m+1} \over (2m+1)!}={\frac {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

∑ m = 0 ∞ z q m + p ( q m + p ) ! = 1 q ⋅ ∑ k = 0 q − 1 e z cos ⁡ ( 2 π k / q ) cos ⁡ ( z sin ⁡ ( 2 π k q ) − 2 π k p q ) . {\displaystyle \sum _{m=0}^{\infty }{z^{qm+p} \over (qm+p)!}={\frac {1}{q}}\cdot \sum _{k=0}^{q-1}e^{z\cos(2\pi k/q)}\cos {\left(z\sin {\left({\frac {2\pi k}{q}}\right)}-{\frac {2\pi kp}{q}}\right)}.}

These can be seen as solutions to the linear differential equation f ( q ) ( z ) = f ( z ) {\displaystyle f^{(q)}(z)=f(z)} with boundary conditions f ( k ) ( 0 ) = δ k , p {\displaystyle f^{(k)}(0)=\delta _{k,p}} , using Kronecker delta notation. In particular, the trisections are

∑ m = 0 ∞ z 3 m ( 3 m ) ! = 1 3 ( e z + 2 e − z / 2 cos ⁡ 3 z 2 ) {\displaystyle \sum _{m=0}^{\infty }{z^{3m} \over (3m)!}={\frac {1}{3}}\left(e^{z}+2e^{-z/2}\cos {\frac {{\sqrt {3}}z}{2}}\right)}

∑ m = 0 ∞ z 3 m + 1 ( 3 m + 1 ) ! = 1 3 ( e z − 2 e − z / 2 cos ⁡ ( 3 z 2 + π 3 ) ) {\displaystyle \sum _{m=0}^{\infty }{z^{3m+1} \over (3m+1)!}={\frac {1}{3}}\left(e^{z}-2e^{-z/2}\cos {\left({\frac {{\sqrt {3}}z}{2}}+{\frac {\pi }{3}}\right)}\right)}

∑ m = 0 ∞ z 3 m + 2 ( 3 m + 2 ) ! = 1 3 ( e z − 2 e − z / 2 cos ⁡ ( 3 z 2 − π 3 ) ) , {\displaystyle \sum _{m=0}^{\infty }{z^{3m+2} \over (3m+2)!}={\frac {1}{3}}\left(e^{z}-2e^{-z/2}\cos {\left({\frac {{\sqrt {3}}z}{2}}-{\frac {\pi }{3}}\right)}\right),}

and the quadrisections are

∑ m = 0 ∞ z 4 m ( 4 m ) ! = 1 2 ( cosh ⁡ z + cos ⁡ z ) {\displaystyle \sum _{m=0}^{\infty }{z^{4m} \over (4m)!}={\frac {1}{2}}\left(\cosh {z}+\cos {z}\right)}

∑ m = 0 ∞ z 4 m + 1 ( 4 m + 1 ) ! = 1 2 ( sinh ⁡ z + sin ⁡ z ) {\displaystyle \sum _{m=0}^{\infty }{z^{4m+1} \over (4m+1)!}={\frac {1}{2}}\left(\sinh {z}+\sin {z}\right)}

∑ m = 0 ∞ z 4 m + 2 ( 4 m + 2 ) ! = 1 2 ( cosh ⁡ z − cos ⁡ z ) {\displaystyle \sum _{m=0}^{\infty }{z^{4m+2} \over (4m+2)!}={\frac {1}{2}}\left(\cosh {z}-\cos {z}\right)}

∑ m = 0 ∞ z 4 m + 3 ( 4 m + 3 ) ! = 1 2 ( sinh ⁡ z − sin ⁡ z ) . {\displaystyle \sum _{m=0}^{\infty }{z^{4m+3} \over (4m+3)!}={\frac {1}{2}}\left(\sinh {z}-\sin {z}\right).}

Multisection of a binomial expansion

( 1 + x ) n = ( n 0 ) x 0 + ( n 1 ) x + ( n 2 ) x 2 + ⋯ {\displaystyle (1+x)^{n}={n \choose 0}x^{0}+{n \choose 1}x+{n \choose 2}x^{2}+\cdots }

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

( n p ) + ( n p + q ) + ( n p + 2 q ) + ⋯ = 1 q ⋅ ∑ k = 0 q − 1 ( 2 cos ⁡ π k q ) n ⋅ cos ⁡ π ( n − 2 p ) k q . {\displaystyle {n \choose p}+{n \choose p+q}+{n \choose p+2q}+\cdots ={\frac {1}{q}}\cdot \sum _{k=0}^{q-1}\left(2\cos {\frac {\pi k}{q}}\right)^{n}\cdot \cos {\frac {\pi (n-2p)k}{q}}.}

Series multisection converts 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.

• Weisstein, Eric W. "Series Multisection". MathWorld.
• Somos, Michael A Multisection of q-Series, 2006.
• John Riordan (1968). Combinatorial identities. New York: John Wiley and Sons.