In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by
∑
x
{\textstyle \sum _{x}}
Δ
−
1
{\displaystyle \Delta ^{-1}}
[1] [2] linear operator , inverse of the forward difference operator
Δ
{\displaystyle \Delta }
indefinite integral relates to the derivative . Thus
Δ
∑
x
f
(
x
)
=
f
(
x
)
.
{\displaystyle \Delta \sum _{x}f(x)=f(x)\,.}
More explicitly, if
∑
x
f
(
x
)
=
F
(
x
)
{\textstyle \sum _{x}f(x)=F(x)}
F
(
x
+
1
)
−
F
(
x
)
=
f
(
x
)
.
{\displaystyle 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:
Δ
−
1
=
1
e
D
−
1
{\displaystyle \Delta ^{-1}={\frac {1}{e^{D}-1}}}
Fundamental theorem of discrete calculus
edit
Indefinite sums can be used to calculate definite sums with the formula:[3]
∑
k
=
a
b
f
(
k
)
=
Δ
−
1
f
(
b
+
1
)
−
Δ
−
1
f
(
a
)
{\displaystyle \sum _{k=a}^{b}f(k)=\Delta ^{-1}f(b+1)-\Delta ^{-1}f(a)}
Choice of the constant term
edit
Often the constant C in indefinite sum is fixed from the following condition.
Let
F
(
x
)
=
∑
x
f
(
x
)
+
C
{\displaystyle F(x)=\sum _{x}f(x)+C}
Then the constant C is fixed from the condition
∫
0
1
F
(
x
)
d
x
=
0
{\displaystyle \int _{0}^{1}F(x)\,dx=0}
or
∫
1
2
F
(
x
)
d
x
=
0
{\displaystyle \int _{1}^{2}F(x)\,dx=0}
Alternatively, Ramanujan's sum can be used:
∑
x
≥
1
ℜ
f
(
x
)
=
−
f
(
0
)
−
F
(
0
)
{\displaystyle \sum _{x\geq 1}^{\Re }f(x)=-f(0)-F(0)}
or at 1
∑
x
≥
1
ℜ
f
(
x
)
=
−
F
(
1
)
{\displaystyle \sum _{x\geq 1}^{\Re }f(x)=-F(1)}
respectively[6] [7]
Indefinite summation by parts:
∑
x
f
(
x
)
Δ
g
(
x
)
=
f
(
x
)
g
(
x
)
−
∑
x
(
g
(
x
)
+
Δ
g
(
x
)
)
Δ
f
(
x
)
{\displaystyle \sum _{x}f(x)\Delta g(x)=f(x)g(x)-\sum _{x}(g(x)+\Delta g(x))\Delta f(x)}
∑
x
f
(
x
)
Δ
g
(
x
)
+
∑
x
g
(
x
)
Δ
f
(
x
)
=
f
(
x
)
g
(
x
)
−
∑
x
Δ
f
(
x
)
Δ
g
(
x
)
{\displaystyle \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:
∑
i
=
a
b
f
(
i
)
Δ
g
(
i
)
=
f
(
b
+
1
)
g
(
b
+
1
)
−
f
(
a
)
g
(
a
)
−
∑
i
=
a
b
g
(
i
+
1
)
Δ
f
(
i
)
{\displaystyle \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)}
Some authors use the phrase "indefinite sum" to describe a sum in which the numerical value of the upper limit is not given:
∑
k
=
1
n
f
(
k
)
.
{\displaystyle \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
)
{\displaystyle F(x+1)-F(x)=f(x+1)}
which is called the telescoping equation.[8] backward difference
∇
{\displaystyle \nabla }
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
∑
x
a
=
a
x
+
C
{\displaystyle \sum _{x}a=ax+C}
From which
a
{\displaystyle a}
x
0
{\displaystyle x^{0}}
∑
x
x
0
=
x
{\displaystyle \sum _{x}x^{0}=\ x}
For the sum below, remember
x
=
x
1
{\displaystyle x=x^{1}}
∑
x
x
=
x
(
x
+
1
)
2
+
C
{\displaystyle \sum _{x}x={\frac {x(x+1)}{2}}+C}
For positive integer exponents Faulhaber's formula can be used. For negative integer exponents,
∑
x
1
x
a
=
(
−
1
)
a
+
1
ψ
(
a
+
1
)
(
x
)
a
!
+
C
,
a
∈
Z
{\displaystyle \sum _{x}{\frac {1}{x^{a}}}={\frac {(-1)^{a+1}\psi ^{(a+1)}(x)}{a!}}+C,\,a\in \mathbb {Z} }
where
ψ
(
n
)
(
x
)
{\displaystyle \psi ^{(n)}(x)}
polygamma function can be used.
More generally,
∑
x
x
a
=
{
−
ζ
(
−
a
,
x
+
1
)
+
C
1
,
if
a
≠
−
1
ψ
(
x
+
1
)
+
C
2
,
if
a
=
−
1
{\displaystyle \sum _{x}x^{a}={\begin{cases}-\zeta (-a,x+1)+C_{1},&{\text{if }}a\neq -1\\\psi (x+1)+C_{2},&{\text{if }}a=-1\end{cases}}}
where
ζ
(
s
,
a
)
{\displaystyle \zeta (s,a)}
Hurwitz zeta function and
ψ
(
z
)
{\displaystyle \psi (z)}
Digamma function .
C
1
{\displaystyle C_{1}}
C
2
{\displaystyle C_{2}}
ζ
(
−
a
)
{\displaystyle \zeta (-a)}
ζ
(
s
)
{\displaystyle \zeta (s)}
Riemann zeta function ) and the Euler–Mascheroni constant respectively. By replacing the variable
a
{\displaystyle a}
−
a
{\displaystyle -a}
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
∂
∂
a
ζ
(
s
,
a
)
=
−
s
ζ
(
s
+
1
,
a
)
{\displaystyle {\frac {\partial }{\partial a}}\zeta (s,a)=-s\zeta (s+1,a)}
∑
x
x
a
=
∫
0
x
−
a
ζ
(
1
−
a
,
u
+
1
)
d
u
+
C
,
if
a
≠
−
1
{\displaystyle \sum _{x}x^{a}=\int _{0}^{x}-a\zeta (1-a,u+1)du+C,{\text{ if }}a\neq -1}
∑
x
B
a
(
x
)
=
(
x
−
1
)
B
a
(
x
)
−
a
a
+
1
B
a
+
1
(
x
)
+
C
{\displaystyle \sum _{x}B_{a}(x)=(x-1)B_{a}(x)-{\frac {a}{a+1}}B_{a+1}(x)+C}
Antidifferences of exponential functions
edit
∑
x
a
x
=
a
x
a
−
1
+
C
{\displaystyle \sum _{x}a^{x}={\frac {a^{x}}{a-1}}+C}
Particularly,
∑
x
2
x
=
2
x
+
C
{\displaystyle \sum _{x}2^{x}=2^{x}+C}
Antidifferences of logarithmic functions
edit
∑
x
log
b
x
=
log
b
(
x
!
)
+
C
{\displaystyle \sum _{x}\log _{b}x=\log _{b}(x!)+C}
∑
x
log
b
a
x
=
log
b
(
x
!
a
x
)
+
C
{\displaystyle \sum _{x}\log _{b}ax=\log _{b}(x!a^{x})+C}
Antidifferences of hyperbolic functions
edit
∑
x
sinh
a
x
=
1
2
csch
(
a
2
)
cosh
(
a
2
−
a
x
)
+
C
{\displaystyle \sum _{x}\sinh ax={\frac {1}{2}}\operatorname {csch} \left({\frac {a}{2}}\right)\cosh \left({\frac {a}{2}}-ax\right)+C}
∑
x
cosh
a
x
=
1
2
csch
(
a
2
)
sinh
(
a
x
−
a
2
)
+
C
{\displaystyle \sum _{x}\cosh ax={\frac {1}{2}}\operatorname {csch} \left({\frac {a}{2}}\right)\sinh \left(ax-{\frac {a}{2}}\right)+C}
∑
x
tanh
a
x
=
1
a
ψ
e
a
(
x
−
i
π
2
a
)
+
1
a
ψ
e
a
(
x
+
i
π
2
a
)
−
x
+
C
{\displaystyle \sum _{x}\tanh ax={\frac {1}{a}}\psi _{e^{a}}\left(x-{\frac {i\pi }{2a}}\right)+{\frac {1}{a}}\psi _{e^{a}}\left(x+{\frac {i\pi }{2a}}\right)-x+C}
where
ψ
q
(
x
)
{\displaystyle \psi _{q}(x)}
q-digamma function. Antidifferences of trigonometric functions
edit
∑
x
sin
a
x
=
−
1
2
csc
(
a
2
)
cos
(
a
2
−
a
x
)
+
C
,
a
≠
2
n
π
{\displaystyle \sum _{x}\sin ax=-{\frac {1}{2}}\csc \left({\frac {a}{2}}\right)\cos \left({\frac {a}{2}}-ax\right)+C\,,\,\,a\neq 2n\pi }
∑
x
cos
a
x
=
1
2
csc
(
a
2
)
sin
(
a
x
−
a
2
)
+
C
,
a
≠
2
n
π
{\displaystyle \sum _{x}\cos ax={\frac {1}{2}}\csc \left({\frac {a}{2}}\right)\sin \left(ax-{\frac {a}{2}}\right)+C\,,\,\,a\neq 2n\pi }
∑
x
sin
2
a
x
=
x
2
+
1
4
csc
(
a
)
sin
(
a
−
2
a
x
)
+
C
,
a
≠
n
π
{\displaystyle \sum _{x}\sin ^{2}ax={\frac {x}{2}}+{\frac {1}{4}}\csc(a)\sin(a-2ax)+C\,\,,\,\,a\neq n\pi }
∑
x
cos
2
a
x
=
x
2
−
1
4
csc
(
a
)
sin
(
a
−
2
a
x
)
+
C
,
a
≠
n
π
{\displaystyle \sum _{x}\cos ^{2}ax={\frac {x}{2}}-{\frac {1}{4}}\csc(a)\sin(a-2ax)+C\,\,,\,\,a\neq n\pi }
∑
x
tan
a
x
=
i
x
−
1
a
ψ
e
2
i
a
(
x
−
π
2
a
)
+
C
,
a
≠
n
π
2
{\displaystyle \sum _{x}\tan ax=ix-{\frac {1}{a}}\psi _{e^{2ia}}\left(x-{\frac {\pi }{2a}}\right)+C\,,\,\,a\neq {\frac {n\pi }{2}}}
where
ψ
q
(
x
)
{\displaystyle \psi _{q}(x)}
q-digamma function.
∑
x
tan
x
=
i
x
−
ψ
e
2
i
(
x
+
π
2
)
+
C
=
−
∑
k
=
1
∞
(
ψ
(
k
π
−
π
2
+
1
−
x
)
+
ψ
(
k
π
−
π
2
+
x
)
−
ψ
(
k
π
−
π
2
+
1
)
−
ψ
(
k
π
−
π
2
)
)
+
C
{\displaystyle \sum _{x}\tan x=ix-\psi _{e^{2i}}\left(x+{\frac {\pi }{2}}\right)+C=-\sum _{k=1}^{\infty }\left(\psi \left(k\pi -{\frac {\pi }{2}}+1-x\right)+\psi \left(k\pi -{\frac {\pi }{2}}+x\right)-\psi \left(k\pi -{\frac {\pi }{2}}+1\right)-\psi \left(k\pi -{\frac {\pi }{2}}\right)\right)+C}
∑
x
cot
a
x
=
−
i
x
−
i
ψ
e
2
i
a
(
x
)
a
+
C
,
a
≠
n
π
2
{\displaystyle \sum _{x}\cot ax=-ix-{\frac {i\psi _{e^{2ia}}(x)}{a}}+C\,,\,\,a\neq {\frac {n\pi }{2}}}
∑
x
sinc
x
=
sinc
(
x
−
1
)
(
1
2
+
(
x
−
1
)
(
ln
(
2
)
+
ψ
(
x
−
1
2
)
+
ψ
(
1
−
x
2
)
2
−
ψ
(
x
−
1
)
+
ψ
(
1
−
x
)
2
)
)
+
C
{\displaystyle \sum _{x}\operatorname {sinc} x=\operatorname {sinc} (x-1)\left({\frac {1}{2}}+(x-1)\left(\ln(2)+{\frac {\psi ({\frac {x-1}{2}})+\psi ({\frac {1-x}{2}})}{2}}-{\frac {\psi (x-1)+\psi (1-x)}{2}}\right)\right)+C}
where
sinc
(
x
)
{\displaystyle \operatorname {sinc} (x)}
sinc function . Antidifferences of inverse hyperbolic functions
edit
∑
x
artanh
a
x
=
1
2
ln
(
Γ
(
x
+
1
a
)
Γ
(
x
−
1
a
)
)
+
C
{\displaystyle \sum _{x}\operatorname {artanh} \,ax={\frac {1}{2}}\ln \left({\frac {\Gamma \left(x+{\frac {1}{a}}\right)}{\Gamma \left(x-{\frac {1}{a}}\right)}}\right)+C}
Antidifferences of inverse trigonometric functions
edit
∑
x
arctan
a
x
=
i
2
ln
(
Γ
(
x
+
i
a
)
Γ
(
x
−
i
a
)
)
+
C
{\displaystyle \sum _{x}\arctan ax={\frac {i}{2}}\ln \left({\frac {\Gamma (x+{\frac {i}{a}})}{\Gamma (x-{\frac {i}{a}})}}\right)+C}
Antidifferences of special functions
edit
∑
x
ψ
(
x
)
=
(
x
−
1
)
ψ
(
x
)
−
x
+
C
{\displaystyle \sum _{x}\psi (x)=(x-1)\psi (x)-x+C}
∑
x
Γ
(
x
)
=
(
−
1
)
x
+
1
Γ
(
x
)
Γ
(
1
−
x
,
−
1
)
e
+
C
{\displaystyle \sum _{x}\Gamma (x)=(-1)^{x+1}\Gamma (x){\frac {\Gamma (1-x,-1)}{e}}+C}
where
Γ
(
s
,
x
)
{\displaystyle \Gamma (s,x)}
incomplete gamma function .
∑
x
(
x
)
a
=
(
x
)
a
+
1
a
+
1
+
C
{\displaystyle \sum _{x}(x)_{a}={\frac {(x)_{a+1}}{a+1}}+C}
where
(
x
)
a
{\displaystyle (x)_{a}}
falling factorial .
∑
x
sexp
a
(
x
)
=
ln
a
(
sexp
a
(
x
)
)
′
(
ln
a
)
x
+
C
{\displaystyle \sum _{x}\operatorname {sexp} _{a}(x)=\ln _{a}{\frac {(\operatorname {sexp} _{a}(x))'}{(\ln a)^{x}}}+C}
(see super-exponential function )
^ On Computing Closed Forms for Indefinite Summations. Yiu-Kwong Man. J. Symbolic Computation (1993), 16, 355-376 [permanent dead link ^ "If Y is a function whose first difference is the function y , then Y is called an indefinite sum of y and denoted Δ−1 y " Introduction to Difference Equations
^ "Handbook of discrete and combinatorial mathematics", Kenneth H. Rosen, John G. Michaels, CRC Press, 1999, ISBN 0-8493-0149-1
^ Bernoulli numbers of the second kind on Mathworld ^ Markus Müller. How to Add a Non-Integer Number of Terms, and How to Produce Unusual Infinite Summations Archived 2011-06-17 at the Wayback Machine (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)^ Bruce C. Berndt, Ramanujan's Notebooks Archived 2006-10-12 at the Wayback Machine , Ramanujan's Theory of Divergent Series , Chapter 6, Springer-Verlag (ed.), (1939), pp. 133–149.
^ Éric Delabaere, Ramanujan's Summation , Algorithms Seminar 2001–2002 , F. Chyzak (ed.), INRIA, (2003), pp. 83–88.
^ Algorithms for Nonlinear Higher Order Difference Equations , Manuel Kauers
"Difference Equations: An Introduction with Applications", Walter G. Kelley, Allan C. Peterson, Academic Press, 2001, ISBN 0-12-403330-X
Markus Müller. How to Add a Non-Integer Number of Terms, and How to Produce Unusual Infinite Summations Markus Mueller, Dierk Schleicher. Fractional Sums and Euler-like Identities S. P. Polyakov. Indefinite summation of rational functions with additional minimization of the summable part. Programmirovanie, 2008, Vol. 34, No. 2. "Finite-Difference Equations And Simulations", Francis B. Hildebrand, Prenctice-Hall, 1968 
![{\displaystyle \sum _{x}f(x)=\sum _{k=1}^{\infty }{\binom {x}{k}}\Delta ^{k-1}[f]\left(0\right)+C=\sum _{k=1}^{\infty }{\frac {\Delta ^{k-1}[f](0)}{k!}}(x)_{k}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6758a97991a0410ddfb47366eb53c4d7597ba5b5)
