indefinite and definite sums

An indefinite sum, like an indefinite integral, is an operator which acts on a function. 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 plane or along a contour with a singular point inside the contour, and so on.

\displaystyle\sum_{k}^{x-1}\varphi(k+z)=\int\limits_{0}^{x}\sum_{\nu=0}^{% \infty}\frac{B_{\nu}-A_{\nu}}{\nu!}\varphi^{(\nu)}(\xi+z)d\xi-\int\limits_{0}^% {x}\frac{A_{N}^{\prime}(z-\xi)}{N!}\varphi^{(N-1)}(z+\xi)d\xi-\\ \displaystyle-\int\limits_{0}^{x}\sum_{m=0}^{\infty}\varphi^{(N+m)}(z+\xi)A_{N% +m}(x-\xi)\hskip-54.0pt\rule[-10.0pt]{0.0pt}{23.0pt}_{k_{N+m}+1}^{k_{N+1+m}}% \hskip 25.0ptd\xi+H(x,z)=F(x,z,N)-f(x,z,N)-f\varepsilon(x,z,N)+H(x,z,N)

\displaystyle F(x,z,N)=F_{N}(x,z,N)+F_{N\varepsilon}(x,z,N).\text{\ I choose % that\ }|B_{\nu}-A_{\nu}|\leq(r(\nu))^{\nu},\\ \displaystyle\text{where\ }B_{\nu}\text{\ are Bernoulli numbers,}\\ \displaystyle|F_{N\varepsilon}(x,z,N)|=\left|\int\limits_{0}^{x}\sum_{\nu=N}^{% \infty}\frac{B_{\nu}-A_{\nu}}{\nu!}\varphi^{(\nu)}(z+z_{1})\right|\leqslant% \frac{|x|\left(r(N)\right)^{N}}{N!}\sup_{z+z_{1}\in G}\left|\varphi^{(n)}(z+z_% {1})\right|\\ \displaystyle|f\varepsilon(x,z,N)|=\left|\int\limits_{0}^{x}\sum_{m=0}^{\infty% }\varphi^{(N+m)}(z+\xi)A_{N+m}(x+\alpha-\xi)\hskip-75.0pt\rule[-10.0pt]{0.0pt}% {23.0pt}_{k_{N+m}+1}^{k_{N+1+m}}\quad\qquad\quad\ \,\right|\leqslant\frac{|x|% \left(r(N)\right)^{N}}{N!}\sup_{\zeta\in G}\left|\varphi^{(N)}(\zeta)\right|

where $G$ is the region of summation. In case of summation in complex plain $r(\nu)$ must be a positive constant, $r(\nu)=r_{z}=\max\left(r,e^{\ln r+\frac{1}{er}}\right)$ where $r$ is a positive value less or equal to the minimal radius of convergence of Tailor series of the function $\varphi(z)$ 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 $r(\nu)=\frac{1}{\ln\nu}$ or $r(\nu)=\frac{1}{\ln(\ln\nu)}$ , 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 $\varphi(z)$ is regarded as a function of real valued argument. The more close $r_{z}$ is to zero the more close the possible area of summation is to the hole area where $\varphi(z)$ is analytical.

$A_{\nu}=0,\ \nu=0,1,2,\dots,N-1,(N\geqslant 2),\ A_{2\nu}=0,\nu=0,1,2,\dots$

Periodical function with the period $1$

\displaystyle H(\alpha,z)=\\ \displaystyle=\hskip-2.0pt\int\limits^{x=0}\hskip-1.0pt\int\limits_{0}^{\alpha% }\hskip-2.0pt\left(\hskip-2.0pt\frac{A^{\prime\prime}_{N}(\xi+x)}{N!}\varphi^{% (N-1)}(z-x)\hskip-2.0pt+\hskip-2.0pt\sum_{m=0}^{\infty}\varphi^{(N+m)}(z-x)A_{% N+m}^{\prime}(\xi+x)\hskip-55.0pt\rule[-10.0pt]{0.0pt}{23.0pt}_{k_{N+m}+1}^{k_% {N+1+m}}\hskip 23.0pt\right)d\xi dx=\\ \displaystyle=h_{N}(\alpha,z)+h\varepsilon_{N}(\alpha,z),\\ \displaystyle\lim_{N\to\infty}\left|h\varepsilon_{N}(\alpha,z)\right|=\lim_{N% \to\infty}\left|\int\limits^{x=0}\int\limits_{0}^{\alpha}\left(\sum_{m=0}^{% \infty}\varphi^{(N+m)}(z-x)A_{N+m}^{\prime}(\xi+x)\hskip-55.0pt\rule[-10.0pt]{% 0.0pt}{23.0pt}_{k_{N+m}+1}^{k_{N+1+m}}\hskip 23.0pt\right)d\xi dx\right|% \leqslant\\ \displaystyle\leqslant\lim_{N\to\infty}\frac{|D\alpha|r^{N+1}}{(N+1)!}\sup_{% \zeta\in G}\left|\varphi^{(N+1)}(\zeta)\right|=0,

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

A_{N}(\alpha)=\begin{cases}\displaystyle{2(-1)^{\lfloor\frac{N}{2}\rfloor+1}N!% \sum_{k=1}^{k_{N}}\frac{\cos 2\pi k\alpha}{(2\pi k)^{N-1}}},&\text{when $N$ % even}\\ \rule{0.0pt}{10.0pt}\\ \displaystyle{2(-1)^{\lfloor\frac{N}{2}\rfloor+1}N!\sum_{k=1}^{k_{N}}\frac{% \sin 2\pi k\alpha}{(2\pi k)^{N-1}}},&\text{when $N$ odd}\end{cases},

$A_{N}(0)=A_{N}$ , and

A_{N+m}(x)\hskip-38.0pt\rule[-10.0pt]{0.0pt}{23.0pt}_{k_{N+m}+1}^{k_{N+1+m}}% \hskip 7.0pt=\begin{cases}\displaystyle{2(-1)^{\lfloor\frac{N+m}{2}\rfloor+1}}% \sum_{k=k_{N+m}+1}^{k_{N+1+m}}\frac{\cos 2\pi kx}{(2\pi k)^{N+m}},&\text{when % $N+m$ even}\\ \rule{0.0pt}{10.0pt}\\ \displaystyle{2(-1)^{\lfloor\frac{N+m}{2}\rfloor+1}}\sum_{k=k_{N+m}+1}^{k_{N+1% +m}}\frac{\sin 2\pi kx}{(2\pi k)^{N+m}},&\text{when $N+m$ odd}\end{cases}

The floor of $x$ ( $x$ is real) $\lfloor x\rfloor$ is the largest integer less then $x$ .

From the condition $|B_{\nu}(x)-A_{\nu}(x)|\leqslant(r(\nu))^{\nu}=r^{\nu},(0\leqslant x\leqslant 1% \ )(B_{\nu}(x)$ are Bernoulli polynomials) I find out that

\displaystyle k_{\nu}=\lfloor\frac{\nu}{2\pi re}\rfloor(1-\delta_{\nu,1}),\nu=% 1,2,\dots\\ \displaystyle\text{where}\ \delta_{\nu,1}\ \text{is the Kronecker delta,}\ % \delta_{\nu,1}=1\ \text{when}\ \nu=1\text{and}\ 0\ \text{otherwise.}

The definite sum is defined as:

\sum\limits_{k=a}^{x-1}\varphi(k+z)=\sum\limits_{k}^{x-1}\varphi(k+z)-\sum% \limits_{k}^{a-1}\varphi(k+z)

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

\displaystyle\sum_{k=n_{1}}^{n_{2}-1}=\int\limits_{n_{1}}^{n_{2}}\left(\sum_{% \nu=0}^{\infty}\frac{B_{\nu}-A_{\nu}}{\nu!}\varphi^{(\nu-1)}(\xi+z)-\frac{A_{N% }^{\prime}(z-\xi)}{N!}\varphi^{(N-1)}(\xi+z)\right)d\xi+\varepsilon_{N},

where

\displaystyle|\varepsilon_{N}|\leqslant\left|\int\limits_{n_{1}}^{n_{2}}\left(% \sum_{m=0}^{\infty}\varphi^{(N+m)}(z+\xi)A_{N+m}(-\xi)\hskip-54.0pt\rule[-10.0% pt]{0.0pt}{23.0pt}_{k_{N+m}+1}^{k_{N+1+m}}\hskip 25.0ptd\right)\xi\right|% \leqslant\frac{|n_{2}-n_{1}|\left(r(N)\right)^{N}}{N!}\sup_{n_{1}\leqslant% \zeta\leqslant n_{2}}\left|\varphi^{(N)}(\zeta)\right|.\\ \displaystyle r(\nu)=r,\ r(\nu)=\frac{1}{ln\nu\ }\text{\ or\ }r(\nu)=\frac{1}{% ln(ln\nu)}.

Notes:

1. Complete details are provided through the link to the following http://www.oddmaths.info/indefinitesumweb site: http://www.oddmaths.info/indefinitesum.

2. The complete pdf of the entire article can be downloaded here from the http://planetmath.org/files/papers/554/Summation.pdfcomplete 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