Indefinite sum

 ( Mathematical Analysis ) 

In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by $\backslash sum\; \_x$ or $\backslash Delta^\{-1\}$,; reprinted by Dover Books, 1986 is the linear operator, inverse of the forward difference operator $\backslash Delta$. It relates to the forward difference operator as the indefinite integral relates to the derivative. Thus

$\backslash Delta\; \backslash sum\_x\; f(x)\; =\; f(x)\; \backslash ,\; .$

More explicitly, if $\backslash sum\_x\; f(x)\; =\; F(x)$, then

$F(x+1)\; -\; F(x)\; =\; f(x)\; \backslash ,\; .$

If F( x) is a solution of this functional equation for a given f( x), then so is F( x)+ C( x) for any periodic function C( x) with period 1. Therefore, each indefinite sum actually represents a family of functions. However, due to the Carlson's theorem, the solution equal to its Newton series expansion is unique up to an additive constant C. This unique solution can be represented by formal power series form of the antidifference operator: $\backslash Delta^\{-1\}=\backslash frac1\{e^D-1\}$.

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Fundamental theorem of discrete calculus Indefinite sums can be used to calculate definite sums with the formula:"Handbook of discrete and combinatorial mathematics", Kenneth H. Rosen, John G. Michaels, CRC Press, 1999,

$\backslash sum\_\{k=a\}^b\; f(k)=\backslash Delta^\{-1\}f(b+1)-\backslash Delta^\{-1\}f(a)$

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Definitions

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Laplace summation formula The Laplace summation formula allows the indefinite sum to be written as the indefinite integral plus correction terms obtained from iterating the difference operator, although it was originally developed for the reverse process of writing an integral as an indefinite sum plus correction terms. As usual with indefinite sums and indefinite integrals, it is valid up to an arbitrary choice of the constant of integration. Using operator algebra avoids cluttering the formula with repeated copies of the function to be operated on:

$$\backslash sum\_x\; =\; \backslash int\{\}\; +\; \backslash frac\{1\}\{2\}\; -\; \backslash frac\{1\}\{12\}\backslash Delta\; +\; \backslash frac\{1\}\{24\}\backslash Delta^2\; -\; \backslash frac\{19\}\{720\}\backslash Delta^3\; +\; \backslash frac\{3\}\{160\}\backslash Delta^4\; -\; \backslash cdots$$

In this formula, for instance, the term $\backslash tfrac12$ represents an operator that divides the given function by two. The coefficients $+\backslash tfrac12,\; -\backslash tfrac1\{12\},$ etc., appearing in this formula are the Gregory coefficients, also called Laplace numbers. The coefficient in the term $\backslash Delta^\{n-1\}$ is

$$\backslash frac\{\backslash mathcal\{C\}\_n\}\{n!\}=\backslash int\_0^1\; \backslash binom\{x\}\{n\}\backslash ,dx$$

where the numerator $\backslash mathcal\{C\}\_n$ of the left hand side is called a Cauchy number of the first kind, although this name sometimes applies to the Gregory coefficients themselves.

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Newton's formula

$\backslash sum\_x\; f(x)=\backslash sum\_\{k=1\}^\backslash infty\; \backslash binom\{x\}k\; \backslash Delta^\{k-1\}\; f\backslash left\; (0\backslash right)+C=\backslash sum\_\{k=1\}^\{\backslash infty\}\backslash frac\{\backslash Delta^\{k-1\}f(0)\}\{k!\}(x)\_k+C$

where $(x)\_k=\backslash frac\{\backslash Gamma(x+1)\}\{\backslash Gamma(x-k+1)\}$ is the falling factorial.

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Faulhaber's formula

$\backslash sum\; \_x\; f(x)=\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{f^\{(n-1)\}\; (0)\}\{n!\}\; B\_n(x)\; +\; C\; \backslash ,\; ,$

Faulhaber's formula provides that the right-hand side of the equation converges.

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Mueller's formula If $\backslash lim\_\{x\backslash to\{+\backslash infty\}\}f(x)=0,$ then Markus Müller. How to Add a Non-Integer Number of Terms, and How to Produce Unusual Infinite Summations (note that he uses a slightly alternative definition of fractional sum in his work, i.e. inverse to backwards difference, hence 1 as the lower limit in his formula)

$\backslash sum\; \_x\; f(x)=\backslash sum\_\{n=0\}^\backslash infty\backslash left(f(n)-f(n+x)\backslash right)+\; C.$

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Euler–Maclaurin formula

$\backslash sum\; \_x\; f(x)=\; \backslash int\_0^x\; f(t)\; dt\; -\; \backslash frac12\; f(x)+\backslash sum\_\{k=1\}^\{\backslash infty\}\backslash frac\{B\_\{2k\}\}\{(2k)!\}f^\{(2k-1)\}(x)\; +\; C$

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Choice of the constant term Often the constant C in indefinite sum is fixed from the following condition.

Let

$F(x)=\backslash sum\; \_x\; f(x)+C$

Then the constant C is fixed from the condition

$\backslash int\_0^1\; F(x)\; \backslash ,\; dx=0$

or

$\backslash int\_1^2\; F(x)\; \backslash ,\; dx=0$

Alternatively, Ramanujan's sum can be used:

$\backslash sum\_\{x\; \backslash ge\; 1\}^\{\backslash Re\}f(x)=-f(0)-F(0)$

or at 1

$\backslash sum\_\{x\; \backslash ge\; 1\}^\{\backslash Re\}f(x)=-F(1)$

respectivelyBruce C. Berndt, Ramanujan's Notebooks , Ramanujan's Theory of Divergent Series, Chapter 6, Springer-Verlag (ed.), (1939), pp. 133–149.Éric Delabaere, Ramanujan's Summation, Algorithms Seminar 2001–2002, F. Chyzak (ed.), INRIA, (2003), pp. 83–88.

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Summation by parts Indefinite summation by parts:

$\backslash sum\_x\; f(x)\backslash Delta\; g(x)=f(x)g(x)-\backslash sum\_x\; (g(x)+\backslash Delta\; g(x))\; \backslash Delta\; f(x)$

$\backslash sum\_x\; f(x)\backslash Delta\; g(x)+\backslash sum\_x\; g(x)\backslash Delta\; f(x)=f(x)g(x)-\backslash sum\_x\; \backslash Delta\; f(x)\backslash Delta\; g(x)$

Definite summation by parts:

$\backslash sum\_\{i=a\}^b\; f(i)\backslash Delta\; g(i)=f(b+1)g(b+1)-f(a)g(a)-\backslash sum\_\{i=a\}^b\; g(i+1)\backslash Delta\; f(i)$

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Period rules If $T$ is a period of function $f(x)$ then

$\backslash sum\; \_x\; f(Tx)=x\; f(Tx)\; +\; C$

If $T$ is an antiperiod of function $f(x)$, that is $f(x+T)=-f(x)$ then

$\backslash sum\; \_x\; f(Tx)=-\backslash frac12\; f(Tx)\; +\; C$

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Alternative usage Some authors use the phrase "indefinite sum" to describe a sum in which the numerical value of the upper limit is not given:

$\backslash sum\_\{k=1\}^n\; f(k).$

In this case a closed form expression F( k) for the sum is a solution of

$F(x+1)\; -\; F(x)\; =\; f(x+1)$

which is called the telescoping equation. Algorithms for Nonlinear Higher Order Difference Equations, Manuel Kauers It is the inverse of the backward difference $\backslash nabla$ operator. It is related to the forward antidifference operator using the fundamental theorem of discrete calculus described earlier.

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List of indefinite sums This is a list of indefinite sums of various functions. Not every function has an indefinite sum that can be expressed in terms of elementary functions.

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Antidifferences of rational functions

For positive integer exponents Faulhaber's formula can be used. Note that x in the result must be replaced with x-1 due to the offset caused by the indefinite sum being defined the inverse of the forward difference operator. For negative integer exponents,

$\backslash sum\; \_x\; \backslash frac\{1\}\{x^a\}\; =\; \backslash frac\{(-1)^\{a+1\}\backslash psi^\{(a+1)\}(x-1)\}\{a!\}+\; C,\backslash ,a\backslash in\backslash mathbb\{Z\}$

where $\backslash psi^\{(n)\}(x)$ is the polygamma function can be used.

More generally,

$\backslash sum\; \_x\; x^a\; =\; \backslash begin\{cases\}$

- \zeta(-a, x) +C, &\text{if } a\neq-1 \\ \psi(x)+C, &\text{if } a=-1 \end{cases}

where $\backslash zeta(s,a)$ is the Hurwitz zeta function and $\backslash psi(z)$ is the Digamma function. By considering this for negative a (indefinite sum over reciprocal powers), and adding 1 to x, this becomes the Generalized harmonic number. For further information, refer to Balanced polygamma function and Hurwitz zeta function#Special cases and generalizations. Further generalization comes from use of the Lerch transcendent:

$\backslash sum\_x\; \backslash frac\{z^\{x\}\}\{(x+a)^\{s\}\}\; =\; -\; z^\{x\}\; \backslash ,\; \backslash Phi(z,\; s,\; x\; +\; a\; )\; +\; C$

Which generalizes the Generalized harmonic number. Additionally, the partial derivative is given by

$\backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; \backslash left(\; -z^\{x\}\; \backslash Phi\; \backslash left(\; z,\; s,\; x+a\; \backslash right)\; \backslash right)\; =\; z^\{x\}\; \backslash left(\; s\; \backslash Phi\; \backslash left(\; z,\; s+1,\; x+a\; \backslash right)\; -\; \backslash ln(z)\; \backslash Phi\; \backslash left(\; z,\; s,\; x+a\; \backslash right)\; \backslash right)$

$\backslash sum\; \_x\; B\_a(x)=(x-1)B\_a(x)-\backslash frac\{a\}\{a+1\}\; B\_\{a+1\}(x)+C$

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Antidifferences of exponential functions

$\backslash sum\; \_x\; a^x\; =\; \backslash frac\{a^\{x\}\}\{a-1\}\; +\; C$

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Antidifferences of logarithmic functions

$\backslash sum\; \_x\; \backslash log\_b\; x\; =\; \backslash log\_b\; (x!)\; +\; C$

$\backslash sum\; \_x\; \backslash log\_b\; ax\; =\; \backslash log\_b\; (x!a^\{x\})\; +\; C$

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Antidifferences of hyperbolic functions

$\backslash sum\; \_x\; \backslash sinh\; ax\; =\; \backslash frac\{1\}\{2\}\; \backslash operatorname\{csch\}\; \backslash left(\backslash frac\{a\}\{2\}\backslash right)\; \backslash cosh\; \backslash left(\backslash frac\{a\}\{2\}\; -\; a\; x\backslash right)\; +\; C$

$\backslash sum\; \_x\; \backslash cosh\; ax\; =\; \backslash frac\{1\}\{2\}\; \backslash operatorname\{csch\}\; \backslash left(\backslash frac\{a\}\{2\}\backslash right)\; \backslash sinh\; \backslash left(ax-\backslash frac\{a\}\{2\}\backslash right)\; +\; C$

$\backslash sum\; \_x\; \backslash tanh\; ax\; =\; \backslash frac1a\; \backslash psi\; \_\{e^a\}\backslash left(x-\backslash frac\{i\; \backslash pi\; \}\{2\; a\}\backslash right)+\backslash frac1a\; \backslash psi\; \_\{e^a\}\backslash left(x+\backslash frac\{i\; \backslash pi\; \}\{2\; a\}\backslash right)-x\; +\; C$

where $\backslash psi\_q(x)$ is the q-analog function.

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Antidifferences of trigonometric functions

$\backslash sum\; \_x\; \backslash sin\; ax\; =\; -\backslash frac\{1\}\{2\}\; \backslash csc\; \backslash left(\backslash frac\{a\}\{2\}\backslash right)\; \backslash cos\; \backslash left(\backslash frac\{a\}\{2\}-\; ax\; \backslash right)\; +\; C\; \backslash ,,\backslash ,\backslash ,a\backslash ne\; 2n\; \backslash pi$

$\backslash sum\; \_x\; \backslash cos\; ax\; =\; \backslash frac\{1\}\{2\}\; \backslash csc\; \backslash left(\backslash frac\{a\}\{2\}\backslash right)\; \backslash sin\; \backslash left(ax\; -\; \backslash frac\{a\}\{2\}\backslash right)\; +\; C\; \backslash ,,\backslash ,\backslash ,a\backslash ne\; 2n\; \backslash pi$

$\backslash sum\; \_x\; \backslash sin^2\; ax\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{4\}\; \backslash csc\; (a)\; \backslash sin\; (a-2ax)\; +\; C\; \backslash ,\; \backslash ,,\backslash ,\backslash ,a\backslash ne\; n\backslash pi$

$\backslash sum\; \_x\; \backslash cos^2\; ax\; =\; \backslash frac\{x\}\{2\}-\backslash frac\{1\}\{4\}\; \backslash csc\; (a)\; \backslash sin\; (a-2\; a\; x)\; +\; C\; \backslash ,\backslash ,,\backslash ,\backslash ,a\backslash ne\; n\backslash pi$

$\backslash sum\_x\; \backslash tan\; ax\; =\; i\; x-\backslash frac1a\; \backslash psi\; \_\{e^\{2\; i\; a\}\}\backslash left(x-\backslash frac\{\backslash pi\; \}\{2\; a\}\backslash right)\; +\; C\; \backslash ,,\backslash ,\backslash ,a\backslash ne\; \backslash frac\{n\backslash pi\}2$

where $\backslash psi\_q(x)$ is the q-analog function.

$\backslash sum\_x\; \backslash tan\; x=ix-\backslash psi\; \_\{e^\{2\; i\}\}\backslash left(x+\backslash frac\{\backslash pi\; \}\{2\}\backslash right)\; +\; C\; =\; -\backslash sum\; \_\{k=1\}^\{\backslash infty\; \}\; \backslash left(\backslash psi\; \backslash left(k\; \backslash pi\; -\backslash frac\{\backslash pi\; \}\{2\}+1-x\backslash right)+\backslash psi\; \backslash left(k\; \backslash pi\; -\backslash frac\{\backslash pi\; \}\{2\}+x\backslash right)-\backslash psi\; \backslash left(k\; \backslash pi\; -\backslash frac\{\backslash pi\; \}\{2\}+1\backslash right)-\backslash psi\; \backslash left(k\; \backslash pi\; -\backslash frac\{\backslash pi\; \}\{2\}\backslash right)\backslash right)\; +\; C$

$\backslash sum\_x\; \backslash cot\; ax\; =-i\; x-\backslash frac\{i\; \backslash psi\; \_\{e^\{2\; i\; a\}\}(x)\}\{a\}\; +\; C\; \backslash ,,\backslash ,\backslash ,a\backslash ne\; \backslash frac\{n\backslash pi\}2$

$\backslash sum\_x\; \backslash operatorname\{sinc\}\; x=\backslash operatorname\{sinc\}(x-1)\backslash left(\backslash frac\{1\}\{2\}+(x-1)\backslash left(\backslash ln(2)+\backslash frac\{\backslash psi\; (\backslash frac\{x-1\}\{2\})+\backslash psi\; (\backslash frac\{1-x\}\{2\})\}\{2\}-\backslash frac\{\backslash psi\; (x-1)+\backslash psi\; (1-x)\}\{2\}\backslash right)\backslash right)\; +\; C$

where $\backslash operatorname\{sinc\}\; (x)$ is the normalized sinc function.

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Antidifferences of inverse hyperbolic functions

$\backslash sum\_x\; \backslash operatorname\{artanh\}\backslash ,\; a\; x\; =\backslash frac\{1\}\{2\}\; \backslash ln\; \backslash left(\backslash frac\{\backslash Gamma\; \backslash left(x+\backslash frac\{1\}\{a\}\backslash right)\}\{\backslash Gamma\; \backslash left(x-\backslash frac\{1\}\{a\}\backslash right)\}\backslash right)\; +\; C$

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Antidifferences of inverse trigonometric functions

$\backslash sum\_x\; \backslash arctan\; a\; x\; =\; \backslash frac\{i\}\{2\}\; \backslash ln\; \backslash left(\backslash frac\{\backslash Gamma\; (x+\backslash frac\; ia)\}\{\; \backslash Gamma\; (x-\backslash frac\; ia)\}\backslash right)+C$

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Antidifferences of special functions

$\backslash sum\; \_x\; \backslash psi(x)=(x-1)\; \backslash psi(x)-x+C$

$\backslash sum\; \_x\; \backslash Gamma(x)=(-1)^\{x+1\}\backslash Gamma(x)\backslash frac\{\backslash Gamma(1-x,-1)\}e+C$

where $\backslash Gamma(s,x)$ is the incomplete gamma function.

$\backslash sum\; \_x\; (x)\_a\; =\; \backslash frac\{(x)\_\{a+1\}\}\{a+1\}+C$

where $(x)\_a$ is the falling factorial.

$\backslash sum\; \_x\; \backslash operatorname\{sexp\}\_a\; (x)\; =\; \backslash ln\_a\; \backslash frac\{(\backslash operatorname\{sexp\}\_a\; (x))\text{'}\}\{(\backslash ln\; a)^x\}\; +\; C$

(see super-exponential function)

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See also

• Indefinite product
• Time scale calculus
• List of derivatives and integrals in alternative calculi

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Further reading

• "Difference Equations: An Introduction with Applications", Walter G. Kelley, Allan C. Peterson, Academic Press, 2001,
• Markus Müller. How to Add a Non-Integer Number of Terms, and How to Produce Unusual Infinite Summations
• Markus Mueller, Dierk Schleicher. Fractional Sums and Euler-like Identities
• S. P. Polyakov. Indefinite summation of rational functions with additional minimization of the summable part. Programmirovanie, 2008, Vol. 34, No. 2.
• "Finite-Difference Equations And Simulations", Francis B. Hildebrand, Prenctice-Hall, 1968

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