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

n=anzn

then its multisection is a power series of the form

m=aqm+pzqm+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)=n=0anzn

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

m=0aqm+pzqm+p=1qk=0q1ωkpf(ωkz),

where ω=e2πiq 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

n=0zn=11z for |z|<1.

By setting zzq in the above series, its multisections are easily seen to be

m=0zqm+p=zp1zq for |z|<1.

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

p=0q1zp=1zq1z.

Exponential function

The exponential function

ez=n=0znn!

by means of the above formula for analytic functions separates into

m=0zqm+p(qm+p)!=1qk=0q1ωkpeωkz.

The bisections are trivially the hyperbolic functions:

m=0z2m(2m)!=12(ez+ez)=coshz
m=0z2m+1(2m+1)!=12(ezez)=sinhz.

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=0zqm+p(qm+p)!=1qk=0q1ezcos(2πk/q)cos(zsin(2πkq)2πkpq).

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

m=0z3m(3m)!=13(ez+2ez/2cos3z2)
m=0z3m+1(3m+1)!=13(ez2ez/2cos(3z2+π3))
m=0z3m+2(3m+2)!=13(ez2ez/2cos(3z2π3)),

and the quadrisections are

m=0z4m(4m)!=12(coshz+cosz)
m=0z4m+1(4m+1)!=12(sinhz+sinz)
m=0z4m+2(4m+2)!=12(coshzcosz)
m=0z4m+3(4m+3)!=12(sinhzsinz).

Binomial series

Multisection of a binomial expansion

(1+x)n=(n0)x0+(n1)x+(n2)x2+

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

(np)+(np+q)+(np+2q)+=1qk=0q1(2cosπkq)ncosπ(n2p)kq.

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. doi:10.1098/rstl.1757.0104.