Indefinite sum

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In mathematics the indefinite sum operator (also known as the antidifference operator), denoted by \sum _x \, or \Delta^{-1} \,,[1][2][3] is the linear operator, inverse of the forward difference operator \Delta \,. It relates to the forward difference operator as the indefinite integral relates to the derivative. Thus

\Delta \sum_x f(x) = f(x) \, .

More explicitly, if \sum_x f(x) = F(x) \,, then

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

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 the solution equal to its Newton series expansion is unique up to an additive constant C.

Fundamental theorem of discrete calculus[edit]

Indefinite sums can be used to calculate definite sums with the formula:[4]

\sum_{k=a}^b f(k)=\Delta^{-1}f(b+1)-\Delta^{-1}f(a)

Definitions[edit]

Laplace summation formula[edit]

\sum _x f(x)=\int_0^x f(t) dt +\sum_{k=1}^\infty \frac{c_k\Delta^{k-1}f(x)}{k!} + C
where c_k=\int_0^1 \frac{\Gamma(x+1)}{\Gamma(x-k+1)}dx are the Bernoulli numbers of the second kind.[5]

Newton's formula[edit]

\sum_x f(x)=-\sum_{k=1}^{\infty}\frac{\Delta^{k-1}f(x)}{k!}(-x)_k+C
where (x)_k=\frac{\Gamma(x+1)}{\Gamma(x-k+1)} is the falling factorial.

Faulhaber's formula[edit]

\sum _x f(x)= \sum_{n=1}^{\infty} \frac{f^{(n-1)} (0)}{n!} B_n(x) + C \, ,

provided that the right-hand side of the equation converges.

Mueller's formula[edit]

If \lim_{x\to{+\infty}}f(x)=0, then[6]

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

Euler–Maclaurin formula[edit]

\sum _x f(x)= \int_0^x f(t) dt - \frac12 f(x)+\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}f^{(2k-1)}(x) + C

Connection to the Ramanujan summation[edit]

Often the constant C in indefinite sum is fixed from the following equation:

\int_1^2 \sum _x f(x) dx=0

or

\int_0^1 \sum _x f(x) dx=0

In this case, where

F(x)=\sum _x f(x) \,

then Ramanjuan's sum is defined as

\sum_{x \ge 1}^{\Re}f(x)=F(0)\,

or

\sum_{x \ge 1}^{\Re}f(x)=F(1)\,[7][8]

Summation by parts[edit]

Main article: Summation by parts

Indefinite summation by parts:

\sum_x f(x)\Delta g(x)=f(x)g(x)-\sum_x (g(x)+\Delta g(x)) \Delta f(x) \,
\sum_x f(x)\Delta g(x)+\sum_x g(x)\Delta f(x)=f(x)g(x)-\sum_x \Delta f(x)\Delta g(x) \,

Definite summation by parts:

\sum_{i=a}^b f(i)\Delta g(i)=f(b+1)g(b+1)-f(a)g(a)-\sum_{i=a}^b g(i+1)\Delta f(i)

Period rules[edit]

If T \, is a period of function f(x)\, then

\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

\sum _x f(Tx)=-\frac12 f(Tx) + C\,

Alternative usage[edit]

Some authors use the phrase "indefinite sum" to describe a sum in which the numerical value of the upper limit is not given. e.g.

\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.[9] It is inverse to backward difference \nabla operator.

It is related to the forward antidifference operator using the fundamental theorem of discrete calculus described earlier.

List of indefinite sums[edit]

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.

Antidifferences of rational functions[edit]

\sum _x a = ax + C \,
\sum _x x = \frac{x^2}{2}-\frac{x}{2} + C
\sum _x x^a = \frac{B_{a+1}(x)}{a+1} + C,\,a\notin \mathbb{Z}^-
where B_a(x)=-a\zeta(-a+1,x)\,, the generalized to real order Bernoulli polynomials.
\sum _x x^a = \frac{(-1)^{a-1}\psi^{(-a-1)}(x)}{\Gamma(-a)}+ C,\,a\in\mathbb{Z}^-
where \psi^{(n)}(x) is the polygamma function.
\sum _x \frac1x = \psi(x) + C
where \psi(x) is the digamma function.

Antidifferences of exponential functions[edit]

\sum _x a^x = \frac{a^x}{a-1} + C \,

Particularly,

\sum _x 2^x = 2^x + C \,

Antidifferences of logarithmic functions[edit]

\sum _x \log_b x = \log_b \Gamma (x) + C \,
\sum _x \log_b ax = \log_b (a^{x-1}\Gamma (x)) + C \,

Antidifferences of hyperbolic functions[edit]

\sum _x \sinh ax = \frac{1}{2} \operatorname{csch} \left(\frac{a}{2}\right) \cosh \left(\frac{a}{2} - a x\right) + C  \,
\sum _x \cosh ax = \frac{1}{2} \coth \left(\frac{a}{2}\right) \sinh ax -\frac{1}{2} \cosh ax + C  \,
\sum _x \tanh ax = \frac1a \psi _{e^a}\left(x-\frac{i \pi }{2 a}\right)+\frac1a \psi _{e^a}\left(x+\frac{i \pi }{2 a}\right)-x + C
where \psi_q(x) is the q-digamma function.

Antidifferences of trigonometric functions[edit]

\sum _x \sin ax = -\frac{1}{2} \csc \left(\frac{a}{2}\right) \cos \left(\frac{a}{2}- a x \right) + C \,,\,\,a\ne n \pi
\sum _x \cos ax = \frac{1}{2} \cot \left(\frac{a}{2}\right) \sin ax -\frac{1}{2} \cos ax + C \,,\,\,a\ne n \pi
\sum _x \sin^2 ax = \frac{x}{2} + \frac{1}{4} \csc (a) \sin (a-2 a x) + C \, \,,\,\,a\ne \frac{n\pi}2
\sum _x \cos^2 ax = \frac{x}{2}-\frac{1}{4} \csc (a) \sin (a-2 a x) + C  \,\,,\,\,a\ne \frac{n\pi}2
\sum_x \tan ax = i x-\frac1a \psi _{e^{2 i a}}\left(x-\frac{\pi }{2 a}\right) + C \,,\,\,a\ne \frac{n\pi}2
where \psi_q(x) is the q-digamma function.
\sum_x \tan x=ix-\psi _{e^{2 i}}\left(x+\frac{\pi }{2}\right) + C = -\sum _{k=1}^{\infty } \left(\psi \left(k \pi -\frac{\pi }{2}+1-z\right)+\psi \left(k \pi -\frac{\pi }{2}+z\right)-\psi \left(k \pi -\frac{\pi }{2}+1\right)-\psi \left(k \pi -\frac{\pi }{2}\right)\right) + C\,
\sum_x \cot ax =-i x-\frac{i \psi _{e^{2 i a}}(x)}{a} + C \,,\,\,a\ne \frac{n\pi}2

Antidifferences of inverse hyperbolic functions[edit]

\sum_x \operatorname{artanh}\, a x =\frac{1}{2} \ln \left(\frac{(-1)^x \Gamma \left(-\frac{1}{a}\right) \Gamma \left(x+\frac{1}{a}\right)}{\Gamma \left(\frac{1}{a}\right) \Gamma \left(x-\frac{1}{a}\right)}\right) + C

Antidifferences of inverse trigonometric functions[edit]

\sum_x \arctan a x = \frac{i}{2} \ln \left(\frac{(-1)^x \Gamma (\frac{-i}a) \Gamma (x+\frac ia)}{\Gamma (\frac ia) \Gamma (x-\frac ia)}\right)+C

Antidifferences of special functions[edit]

\sum _x \psi(x)=(x-1) \psi(x)-x+C \,
\sum _x \Gamma(x)=(-1)^{x+1}\Gamma(x)\frac{\Gamma(1-x,-1)}e+C
where \Gamma(s,x) is the incomplete gamma function.
\sum _x (x)_a = \frac{(x)_{a+1}}{a+1}+C
where (x)_a is the falling factorial.
\sum _x \operatorname{sexp}_a (x) = \ln_a \frac{(\operatorname{sexp}_a (x))'}{(\ln a)^x} + C \,
(see super-exponential function)

See also[edit]

References[edit]

  1. ^ Indefinite Sum at PlanetMath.org.
  2. ^ On Computing Closed Forms for Indefinite Summations. Yiu-Kwong Man. J. Symbolic Computation (1993), 16, 355-376
  3. ^ "If Y is a function whose first difference is the function y, then Y is called an indefinite sum of y and denoted Δ-1y" Introduction to Difference Equations, Samuel Goldberg
  4. ^ "Handbook of discrete and combinatorial mathematics", Kenneth H. Rosen, John G. Michaels, CRC Press, 1999, ISBN 0-8493-0149-1
  5. ^ Bernoulli numbers of the second kind on Mathworld
  6. ^ 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)
  7. ^ Bruce C. Berndt, Ramanujan's Notebooks, Ramanujan's Theory of Divergent Series, Chapter 6, Springer-Verlag (ed.), (1939), pp. 133–149.
  8. ^ Éric Delabaere, Ramanujan's Summation, Algorithms Seminar 2001–2002, F. Chyzak (ed.), INRIA, (2003), pp. 83–88.
  9. ^ Algorithms for Nonlinear Higher Order Difference Equations, Manuel Kauers

Further reading[edit]