Indefinite sum explained

In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by \sum _x or

\Delta-1

,[1] 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\sumxf(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, 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:

\Delta-1=

1{e
D-1}
.

Fundamental theorem of discrete calculus

Indefinite sums can be used to calculate definite sums with the formula:[2]

b
\sum
k=a

f(k)=\Delta-1f(b+1)-\Delta-1f(a)

Definitions

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:

\sum_x = \int + \frac - \frac\Delta + \frac\Delta^2 - \frac\Delta^3 + \frac\Delta^4 - \cdots

In this formula, for instance, the term

\tfrac12

represents an operator that divides the given function by two. The coefficients

+\tfrac12

,

-\tfrac1{12}

, etc., appearing in this formula are the Gregory coefficients, also called Laplace numbers. The coefficient in the term

\Deltan-1

is

\frac=\int_0^1 \binom\,dx

where the numerator

l{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.

Newton's formula

\sumx

infty
f(x)=\sum
k=1

\binom{x}k\Deltak-1[f]\left

infty
(0\right)+C=\sum
k=1
\Deltak-1[f](0)
k!

(x)k+C

where

(x)
k=\Gamma(x+1)
\Gamma(x-k+1)
is the falling factorial.

Faulhaber's formula

See main article: Faulhaber's formula.

\sumxf(x)=

infty
\sum
n=1
f(n-1)(0)
n!

Bn(x)+C,

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

Mueller's formula

If

\limx\to{+infty

}f(x)=0, then[3]

\sumx

infty\left(f(n)-f(n+x)\right)+
f(x)=\sum
n=0

C.

Euler–Maclaurin formula

See main article: Euler–Maclaurin formula.

\sumxf(x)=

x
\int
0

f(t)dt-

12
f(x)+\sum
infty
k=1
B2k
(2k)!

f(2k-1)(x)+C

Choice of the constant term

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

Let

F(x)=\sumxf(x)+C

Then the constant C is fixed from the condition

1
\int
0

F(x)dx=0

or

2
\int
1

F(x)dx=0

Alternatively, Ramanujan's sum can be used:

\Re
\sum
x\ge1

f(x)=-f(0)-F(0)

or at 1

\Re
\sum
x\ge1

f(x)=-F(1)

respectively[4] [5]

Summation by parts

See main article: Summation by parts.

Indefinite summation by parts:

\sumxf(x)\Deltag(x)=f(x)g(x)-\sumx(g(x)+\Deltag(x))\Deltaf(x)

\sumxf(x)\Deltag(x)+\sumxg(x)\Deltaf(x)=f(x)g(x)-\sumx\Deltaf(x)\Deltag(x)

Definite summation by parts:

b
\sum
i=a

f(i)\Delta

b
g(i)=f(b+1)g(b+1)-f(a)g(a)-\sum
i=a

g(i+1)\Deltaf(i)

Period rules

If

T

is a period of function

f(x)

then

\sumxf(Tx)=xf(Tx)+C

If

T

is an antiperiod of function

f(x)

, that is

f(x+T)=-f(x)

then

\sumxf(Tx)=-

12
f(Tx)

+C

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:

n
\sum
k=1

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)

\nabla

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

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.

Antidifferences of rational functions

\sumxa=ax+C

From which

a

can be factored out, leaving 1, with the alternative form

x0

. From that, we have:

\sumxx0=x

For the sum below, remember

x=x1

\sumxx=

x(x+1)
2

+C

For positive integer exponents Faulhaber's formula can be used. For negative integer exponents,

\sumx

1
xa

=

(-1)a+1\psi(a+1)(x)
a!

+C,a\inZ

where

\psi(n)(x)

is the polygamma function can be used.

More generally,

\sumxxa=\begin{cases} -\zeta(-a,x+1)+C1,&ifa ≠ -1\\ \psi(x+1)+C2,&ifa=-1 \end{cases}

where

\zeta(s,a)

is the Hurwitz zeta function and

\psi(z)

is the Digamma function.

C1

and

C2

are constants which would normally be set to

\zeta(-a)

(where

\zeta(s)

is the Riemann zeta function) and the Euler–Mascheroni constant respectively. By replacing the variable

a

with

-a

, this becomes the Generalized harmonic number. For the relation between the Hurwitz zeta and Polygamma functions, refer to Balanced polygamma function and Hurwitz zeta function#Special cases and generalizations.

From this, using

\partial
\partiala

\zeta(s,a)=-s\zeta(s+1,a)

, another form can be obtained:

\sumxxa=

x
\int
0

-a\zeta(1-a,u+1)du+C,ifa ≠ -1

\sumxBa(x)=(x-1)B

a(x)-a
a+1

Ba+1(x)+C

Antidifferences of exponential functions

\sumxax=

ax
a-1

+C

Particularly,

\sumx2x=2x+C

Antidifferences of logarithmic functions

\sumxlogbx=logb(x!)+C

\sumxlogbax=logb(x!ax)+C

Antidifferences of hyperbolic functions

\sumx\sinhax=

1
2

\operatorname{csch}\left(

a
2

\right)\cosh\left(

a
2

-ax\right)+C

\sumx\coshax=

1
2

\operatorname{csch}\left(

a
2

\right)\sinh\left(ax-

a
2

\right)+C

\sumx\tanhax=

1a
\psi
\left(x-
ea
i\pi\right)+
2a
1a
\psi
\left(x+
ea
i\pi
2a

\right)-x+C

where

\psiq(x)

is the q-digamma function.

Antidifferences of trigonometric functions

\sumx\sinax=-

1
2

\csc\left(

a
2

\right)\cos\left(

a
2

-ax\right)+C,a\ne2n\pi

\sumx\cosax=

1
2

\csc\left(

a
2

\right)\sin\left(ax-

a
2

\right)+C,a\ne2n\pi

\sumx\sin2ax=

x
2

+

1
4

\csc(a)\sin(a-2ax)+C,a\nen\pi

\sumx\cos2ax=

x-
2
1
4

\csc(a)\sin(a-2ax)+C,a\nen\pi

\sumx\tanax=ix-

1a
\psi
\left(x-
e2
\pi
2a

\right)+C,a\ne

n\pi
2

where

\psiq(x)

is the q-digamma function.

\sumx\tanx=ix-\psi

\left(x+
e2
\pi
2

\right)+C=-\sum

infty
k=1

\left(\psi\left(k\pi-

\pi
2

+1-x\right)+\psi\left(k\pi-

\pi
2

+x\right)-\psi\left(k\pi-

\pi
2

+1\right)-\psi\left(k\pi-

\pi
2

\right)\right)+C

\sumx\cotax=-ix-

i\psi
e2
(x)
a

+C,a\ne

n\pi
2

\sumx\operatorname{sinc}x=\operatorname{sinc}(x-1)\left(

1+(x-1)\left(ln(2)+
2
\psi
(x-1
2
)+\psi
(1-x
2
)
-
2
\psi(x-1)+\psi(1-x)
2

\right)\right)+C

where

\operatorname{sinc}(x)

is the normalized sinc function.

Antidifferences of inverse hyperbolic functions

\sumx\operatorname{artanh}ax=

1
2

ln\left(

\Gamma
\left(x+1
a
\right)
\Gamma
\left(x-1
a
\right)

\right)+C

Antidifferences of inverse trigonometric functions

\sumx\arctanax=

i
2

ln\left(

\Gamma
(x+ia)
\Gamma
(x-ia)

\right)+C

Antidifferences of special functions

\sumx\psi(x)=(x-1)\psi(x)-x+C

\sumx\Gamma(x)=(-1)x+1\Gamma(x)

\Gamma(1-x,-1)
e+C

where

\Gamma(s,x)

is the incomplete gamma function.

\sumx(x)a=

(x)a+1
a+1

+C

where

(x)a

is the falling factorial.

\sumx\operatorname{sexp}a(x)=lna

(\operatorname{sexp
a

(x))'}{(lna)x}+C

(see super-exponential function)

See also

Further reading

Notes and References

  1. reprinted by Dover Books, 1986
  2. "Handbook of discrete and combinatorial mathematics", Kenneth H. Rosen, John G. Michaels, CRC Press, 1999,
  3. http://www.math.tu-berlin.de/~mueller/HowToAdd.pdf Markus Müller. How to Add a Non-Integer Number of Terms, and How to Produce Unusual Infinite Summations
  4. Bruce C. Berndt, Ramanujan's Notebooks, Ramanujan's Theory of Divergent Series, Chapter 6, Springer-Verlag (ed.), (1939), pp. 133 - 149.
  5. Éric Delabaere, Ramanujan's Summation, Algorithms Seminar 2001–2002, F. Chyzak (ed.), INRIA, (2003), pp. 83–88.
  6. http://www.risc.uni-linz.ac.at/people/mkauers/publications/kauers05c.pdf Algorithms for Nonlinear Higher Order Difference Equations