indefinite and definite sums

An indefinite sum, like an indefinite integral, is an operator which acts on a functionMathworldPlanetmath. In other words, it transforms a given function to another via a certain law. This article presents the so called Caves summation formula. The advantages of the formula in comparison with other summation methods are that it gives the indefinite sum for any analytical function, and that it also completely reduces summation to integration. One can do with the Caves summation formula everything that one can do with an integral. For example, one can take a sum along a path either in the complex planeMathworldPlanetmath or along a contour with a singular point inside the contour, and so on.

kx-1φ(k+z)=0xν=0Bν-Aνν!φ(ν)(ξ+z)dξ-0xAN(z-ξ)N!φ(N-1)(z+ξ)𝑑ξ--0xm=0φ(N+m)(z+ξ)AN+m(x-ξ)kN+m+1kN+1+m    dξ+H(x,z)=F(x,z,N)-f(x,z,N)-fε(x,z,N)+H(x,z,N)
F(x,z,N)=FN(x,z,N)+FNε(x,z,N). I choose that |Bν-Aν|(r(ν))ν,where Bν are Bernoulli numbers,|FNε(x,z,N)|=|0xν=NBν-Aνν!φ(ν)(z+z1)||x|(r(N))NN!supz+z1G|φ(n)(z+z1)||fε(x,z,N)|=|0xm=0φ(N+m)(z+ξ)AN+m(x+α-ξ)kN+m+1kN+1+m      ||x|(r(N))NN!supζG|φ(N)(ζ)|

where G is the region of summation. In case of summation in complex plain r(ν) must be a positive constant, r(ν)=rz=max(r,elnr+1er) where r is a positive value less or equal to the minimal radius of convergenceMathworldPlanetmath of Tailor series of the function φ(z) on the intersection of the area of summation G with the x-axis. In case of summation exclusively on a segment of the x-axis it is more convenient to choose r(ν)=1lnν or r(ν)=1ln(lnν), especially in a case when there is a singular point on the path of summation.The same for a path parallel to the x-axis when φ(z) is regarded as a function of real valued argument. The more close rz is to zero the more close the possible area of summation is to the hole area where φ(z) is analytical.


Periodical function with the period 1

H(α,z)==x=00α(AN′′(ξ+x)N!φ(N-1)(z-x)+m=0φ(N+m)(z-x)AN+m(ξ+x)kN+m+1kN+1+m   )dξdx==hN(α,z)+hεN(α,z),limN|hεN(α,z)|=limN|x=00α(m=0φ(N+m)(z-x)AN+m(ξ+x)kN+m+1kN+1+m   )dξdx|limN|Dα|rN+1(N+1)!supζG|φ(N+1)(ζ)|=0,

where D is the diameter of the area of summation and z is a parameterMathworldPlanetmath.

AN(α)={2(-1)N2+1N!k=1kNcos2πkα(2πk)N-1,when N even2(-1)N2+1N!k=1kNsin2πkα(2πk)N-1,when N odd,

AN(0)=AN, and

AN+m(x)kN+m+1kN+1+m={2(-1)N+m2+1k=kN+m+1kN+1+mcos2πkx(2πk)N+m,when N+m even2(-1)N+m2+1k=kN+m+1kN+1+msin2πkx(2πk)N+m,when N+m odd

The floor of x (x is real) x is the largest integer less then x.

From the condition |Bν(x)-Aν(x)|(r(ν))ν=rν,(0x1)(Bν(x) are Bernoulli polynomialsDlmfDlmfMathworld) I find out that

kν=ν2πre(1-δν,1),ν=1,2,whereδν,1is the Kronecker delta,δν,1=1whenν=1and 0otherwise.

The definite sum is defined as:


In the case of integer summation boundaries the summation formula can be simplified.



|εN||n1n2(m=0φ(N+m)(z+ξ)AN+m(-ξ)kN+m+1kN+1+m    d)ξ||n2-n1|(r(N))NN!supn1ζn2|φ(N)(ζ)|.r(ν)=r,r(ν)=1lnν or r(ν)=1ln(lnν).


1. Complete details are provided through the link to the following site:

2. The complete pdf of the entire article can be downloaded here from the article on “Summation” uploaded to the Papers section.

Title indefinite and definite sums
Canonical name IndefiniteAndDefiniteSums
Date of creation 2013-03-22 19:22:22
Last modified on 2013-03-22 19:22:22
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 80
Author bci1 (20947)
Entry type Topic
Classification msc 34A36
Classification msc 39B72
Classification msc 33E30
Classification msc 39A99
Related topic IndefiniteSum
Defines non-analytical function
Defines definite sum
Defines Caves summation formula