indefinite sum
Recall that the finite difference operator defined on the set of functions is given by
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 , finds its antiderivative so that the derivative of gives . There is also a discrete analog of this inverse operation, and it is called the indefinite sum.
The indefinite sum of a function is the set of functions
This set is often denoted by or , and any element in is called an indefinite sum of .
Remark. Like the indefinite integral, the indefinite sum is shift invariant. This means that for any , then for any . But, unlike the indefinite integral, the indefinite sum is also invariant by a shift of a periodic real function of period . Conversely, the difference of two indefinite sums of a function is a periodic real function of period .
In the following discussion, we consider the indefinite sum of a function as a function.
Basic Properties
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, and modulo a real function of period .
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Modulo a real number, and treating as an operator taking a function into a function, we see that is linear, that is,
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for any , and
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If , then .
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If , then we see that
where is a positive integer. Summing these expressions, we get
This is the discrete version of the fundamental theorem of calculus.
Below is a table of some basic functions and their indefinite sums ( is a real-valued periodic function with period ):
Comment | ||
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See this link (http://planetmath.org/SumOfPowers) for detail | ||
is the falling factorial of degree | ||
is the digamma function | ||
is the gamma function | ||