indefinite sum

indefinite sum

Recall that the

finite difference

operator

Δ

defined on the set of

functions

ℝ

→

ℝ

is given by

Δ

⁢

f

⁢

(

x

)

:=

f

⁢

(

x

+

1

)

-

f

⁢

(

x

)

.

The difference operator can be thought of as the discrete version of the

derivative

operator sending a function to its derivative (if it exists). With the derivative operation, there corresponds an inverse operation called the

antiderivative

, which, given a function

f

, finds its antiderivative

F

so that the derivative of

F

gives

f

. There is also a discrete analog of this inverse operation, and it is called the

indefinite sum

.

The

indefinite sum

of a function

f

:

ℝ

→

ℝ

is the set of functions

{

F

:

ℝ

→

ℝ

∣

Δ

F

=

f

}

.

This set is often denoted by

Δ

-

1

⁢

f

or

Σ

⁢

f

, and any element in

Δ

-

1

⁢

f

is called an indefinite sum of

f

.

Remark

. Like the indefinite integral, the indefinite sum

Δ

-

1

is shift invariant. This means that for any

F

∈

Δ

-

1

⁢

f

, then

F

+

c

∈

Δ

-

1

⁢

f

for any

c

∈

ℝ

. But, unlike the indefinite integral, the indefinite sum is also invariant by a shift of a

periodic

real function

of period

1

. Conversely, the

difference

of two indefinite sums of a function

f

is a periodic real function of period

1

.

In the following discussion, we consider the indefinite sum of a function as a function.

Basic Properties

1.

Δ

⁢

Δ

-

1

⁢

f

=

f

, and

Δ

-

1

⁢

Δ

⁢

f

=

f

modulo a real function of period

1

.

2.

Modulo a

real number

, and treating

Δ

-

1

as an operator taking a function into a function, we see that

Δ

-

1

is linear, that is,

–

Δ

-

1

⁢

(

r

⁢

f

)

=

r

⁢

Δ

-

1

⁢

f

for any

r

∈

ℝ

, and

–

Δ

-

1

⁢

(

f

+

g

)

=

Δ

-

1

⁢

f

+

Δ

-

1

⁢

g

.

3.

If

F

⁢

(

x

)

=

Δ

-

1

⁢

f

⁢

(

x

)

, then

F

⁢

(

x

+

a

)

=

Δ

-

1

⁢

f

⁢

(

x

+

a

)

.

4.

If

F

=

Δ

-

1

⁢

f

, then we see that

F

⁢

(

a

+

1

)

-

F

⁢

(

a

)

=

f

⁢

(

a

)

,

F

⁢

(

a

+

2

)

-

F

⁢

(

a

+

1

)

=

f

⁢

(

a

+

1

)

,

⋮

F

⁢

(

x

)

-

F

⁢

(

x

-

1

)

=

f

⁢

(

x

-

1

)

.

where

x

-

a

is a

positive integer

.

Summing

these expressions, we get

F

⁢

(

x

)

-

F

⁢

(

a

)

=

∑

i

=

1

x

-

a

f

⁢

(

a

+

i

-

1

)

.

This is the discrete version of the

fundamental theorem of calculus

.

Below is a table of some basic functions and their indefinite sums (

C

is a real-valued

periodic function

with period

1

):

f

⁢

(

x

)

Δ

-

1

⁢

f

⁢

(

x

)

Comment

r

∈

ℝ

r

⁢

x

+

C

x

x

⁢

(

x

-

1

)

2

+

C

x

2

x

⁢

(

x

-

1

)

⁢

(

2

⁢

x

-

1

)

6

+

C

x

3

x

2

⁢

(

x

-

1

)

2

4

+

C

x

n

T

n

⁢

(

x

)

+

C

See this link (

http://planetmath.org/SumOfPowers

) for detail

a

x

a

x

a

-

1

+

C

a

≠

1

(

x

)

n

(

x

)

n

n

+

1

+

C

(

x

)

n

is the

falling factorial

of degree

n

(

x

n

)

(

x

n

+

1

)

+

C

(

x

n

)

:=

(

x

)

n

n

!

1

x

ψ

⁢

(

x

)

+

C

ψ

⁢

(

x

)

is the

digamma function

ln

⁡

x

ln

⁡

Γ

⁢

(

x

)

+

C

Γ

⁢

(

x

)

is the

gamma function

sin

⁡

x

-

cos

⁡

(

x

-

1

/

2

)

2

⁢

sin

⁡

(

1

/

2

)

+

C

cos

⁡

x

sin

⁡

(

x

-

1

/

2

)

2

⁢

sin

⁡

(

1

/

2

)

+

C

References

1

⁢

C

.

J

⁢

o

⁢

r

⁢

d

⁢

a

⁢

n

.

Calculus of Finite Differences

,

t

⁢

h

⁢

i

⁢

r

⁢

d

⁢

e

⁢

d

⁢

i

⁢

t

⁢

i

⁢

o

⁢

n

.

C

⁢

h

⁢

e

⁢

l

⁢

s

⁢

e

⁢

a

,

N

⁢

e

⁢

w

⁢

Y

⁢

o

⁢

r

⁢

k

⁢

(

1965

)

⁢

Title

indefinite sum

Canonical name

IndefiniteSum

Date of creation

2013-03-22 17:35:14

Last modified on

2013-03-22 17:35:14

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

20

Author

CWoo (3771)

Entry type

Definition

Classification

msc 39A99

Related topic

FiniteDifference

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