In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by ∑ x {\textstyle \sum _{x}} or Δ − 1 {\displaystyle \Delta ^{-1}} ,[1] [2] is the linear operator , inverse of the forward difference operator Δ {\displaystyle \Delta } . It relates to the forward difference operator as the 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)} , then
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)} Definitions [ edit ] Laplace summation formula [ edit ] ∑ x f ( x ) = ∫ 0 x f ( t ) d t − ∑ k = 1 ∞ c k Δ k − 1 f ( x ) k ! + C {\displaystyle \sum _{x}f(x)=\int _{0}^{x}f(t)dt-\sum _{k=1}^{\infty }{\frac {c_{k}\Delta ^{k-1}f(x)}{k!}}+C} where c k = ∫ 0 1 Γ ( x + 1 ) Γ ( x − k + 1 ) d x {\displaystyle c_{k}=\int _{0}^{1}{\frac {\Gamma (x+1)}{\Gamma (x-k+1)}}dx} are the Cauchy numbers of the first kind, also known as the Bernoulli Numbers of the Second Kind.[4] [citation needed ] Newton's formula [ edit ] ∑ x f ( x ) = ∑ k = 1 ∞ ( x k ) Δ k − 1 [ f ] ( 0 ) + C = ∑ k = 1 ∞ Δ k − 1 [ f ] ( 0 ) k ! ( x ) k + C {\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} where ( x ) k = Γ ( x + 1 ) Γ ( x − k + 1 ) {\displaystyle (x)_{k}={\frac {\Gamma (x+1)}{\Gamma (x-k+1)}}} is the falling factorial . Faulhaber's formula [ edit ] ∑ x f ( x ) = ∑ n = 1 ∞ f ( n − 1 ) ( 0 ) n ! B n ( x ) + C , {\displaystyle \sum _{x}f(x)=\sum _{n=1}^{\infty }{\frac {f^{(n-1)}(0)}{n!}}B_{n}(x)+C\,,} Faulhaber's formula provides that the right-hand side of the equation converges.
Mueller's formula [ edit ] If lim x → + ∞ f ( x ) = 0 , {\displaystyle \lim _{x\to {+\infty }}f(x)=0,} then[5]
∑ x f ( x ) = ∑ n = 0 ∞ ( f ( n ) − f ( n + x ) ) + C . {\displaystyle \sum _{x}f(x)=\sum _{n=0}^{\infty }\left(f(n)-f(n+x)\right)+C.} Euler–Maclaurin formula [ edit ] ∑ x f ( x ) = ∫ 0 x f ( t ) d t − 1 2 f ( x ) + ∑ k = 1 ∞ B 2 k ( 2 k ) ! f ( 2 k − 1 ) ( x ) + C {\displaystyle \sum _{x}f(x)=\int _{0}^{x}f(t)dt-{\frac {1}{2}}f(x)+\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k)!}}f^{(2k-1)}(x)+C}
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]
Summation by parts [ edit ] 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)} Period rules [ edit ] If T {\displaystyle T} is a period of function f ( x ) {\displaystyle f(x)} then
∑ x f ( T x ) = x f ( T x ) + C {\displaystyle \sum _{x}f(Tx)=xf(Tx)+C} If T {\displaystyle T} is an antiperiod of function f ( x ) {\displaystyle f(x)} , that is f ( x + T ) = − f ( x ) {\displaystyle f(x+T)=-f(x)} then
∑ x f ( T x ) = − 1 2 f ( T x ) + C {\displaystyle \sum _{x}f(Tx)=-{\frac {1}{2}}f(Tx)+C} Alternative usage [ edit ] 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] It is the inverse of the backward difference ∇ {\displaystyle \nabla } operator. It is related to the forward antidifference operator using the fundamental theorem of discrete calculus described earlier.
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} ∑ x x = x 2 2 − x 2 + C {\displaystyle \sum _{x}x={\frac {x^{2}}{2}}-{\frac {x}{2}}+C} ∑ x x a = B a + 1 ( x ) a + 1 + C , a ∉ Z − {\displaystyle \sum _{x}x^{a}={\frac {B_{a+1}(x)}{a+1}}+C,\,a\notin \mathbb {Z} ^{-}} where B a ( x ) = − a ζ ( − a + 1 , x ) {\displaystyle B_{a}(x)=-a\zeta (-a+1,x)} , the generalized to real order Bernoulli polynomials . ∑ x x a = ( − 1 ) a − 1 ψ ( − a − 1 ) ( x ) Γ ( − a ) + C , a ∈ Z − {\displaystyle \sum _{x}x^{a}={\frac {(-1)^{a-1}\psi ^{(-a-1)}(x)}{\Gamma (-a)}}+C,\,a\in \mathbb {Z} ^{-}} where ψ ( n ) ( x ) {\displaystyle \psi ^{(n)}(x)} is the polygamma function . ∑ x 1 x = ψ ( x ) + C {\displaystyle \sum _{x}{\frac {1}{x}}=\psi (x)+C} where ψ ( x ) {\displaystyle \psi (x)} is the digamma function . ∑ 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 Γ ( x ) + C {\displaystyle \sum _{x}\log _{b}x=\log _{b}\Gamma (x)+C} ∑ x log b a x = log b ( a x − 1 Γ ( x ) ) + C {\displaystyle \sum _{x}\log _{b}ax=\log _{b}(a^{x-1}\Gamma (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)} is the 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)} is the 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)} is the normalized 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)} is the 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}} is the 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 ) See also [ edit ] References [ edit ] ^ 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 , Samuel Goldberg ^ "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 Further reading [ edit ] "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