In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by or
\Delta-1
\Delta
\Delta\sumxf(x)=f(x).
More explicitly, if , 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} |
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)
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:
In this formula, for instance, the term
\tfrac12
+\tfrac12
-\tfrac1{12}
\Deltan-1
where the numerator
l{C}n
\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) | ||||
|
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.
If
\limx\to{+infty
\sumx
infty\left(f(n)-f(n+x)\right)+ | |
f(x)=\sum | |
n=0 |
C.
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
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)
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)
If
T
f(x)
\sumxf(Tx)=xf(Tx)+C
If
T
f(x)
f(x+T)=-f(x)
\sumxf(Tx)=-
12 | |
f(Tx) |
+C
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
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.
\sumxa=ax+C
From which
a
x0
\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)
More generally,
\sumxxa=\begin{cases} -\zeta(-a,x+1)+C1,&ifa ≠ -1\\ \psi(x+1)+C2,&ifa=-1 \end{cases}
where
\zeta(s,a)
\psi(z)
C1
C2
\zeta(-a)
\zeta(s)
a
-a
From this, using
\partial | |
\partiala |
\zeta(s,a)=-s\zeta(s+1,a)
\sumxxa=
x | |
\int | |
0 |
-a\zeta(1-a,u+1)du+C,ifa ≠ -1
\sumxBa(x)=(x-1)B
|
Ba+1(x)+C
\sumxax=
ax | |
a-1 |
+C
Particularly,
\sumx2x=2x+C
\sumxlogbx=logb(x!)+C
\sumxlogbax=logb(x!ax)+C
\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)
\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)
\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-
| |||||||
a |
+C,a\ne
n\pi | |
2 |
\sumx\operatorname{sinc}x=\operatorname{sinc}(x-1)\left(
1 | +(x-1)\left(ln(2)+ | |
2 |
| - | |||||||||
2 |
\psi(x-1)+\psi(1-x) | |
2 |
\right)\right)+C
where
\operatorname{sinc}(x)
\sumx\operatorname{artanh}ax=
1 | |
2 |
ln\left(
| ||||||
|
\right)+C
\sumx\arctanax=
i | |
2 |
ln\left(
| ||||||
|
\right)+C
\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)
\sumx(x)a=
(x)a+1 | |
a+1 |
+C
where
(x)a
\sumx\operatorname{sexp}a(x)=lna
(\operatorname{sexp | |
a |
(x))'}{(lna)x}+C
(see super-exponential function)