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Indefinite sum

In the calculus of finite differences, the indefinite sum (or antidifference operator), denoted by ∑ x {\textstyle \sum _{x}} or Δ − 1 {\displaystyle \Delta ^{-1}} , is the linear operator that inverts the forward difference operator Δ f ( x ) = f ( x + 1 ) − f ( x ) . {\displaystyle \Delta f(x)=f(x+1)-f(x).} That is, if ∑ x f ( x ) = F ( x )…

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Definition
Linear operator that inverts the finite difference: Δ − 1 {\displaystyle \Delta ^{-1}} : Δ F ( x ) = F ( x + 1 ) − F ( x ) = f ( x ) {\displaystyle \Delta F(x)=F(x+1)-F(x)=f(x)}…
Domain
Functions on integers; extended to real and complex arguments via analytic continuation
Notation
∑ x f ( x ) {\displaystyle \sum _{x}f(x)} , Δ − 1 f ( x ) {\displaystyle \Delta ^{-1}f(x)} (forward), ∇ − 1 f ( x ) {\displaystyle \nabla ^{-1}f(x)} (backward).
Other names
Antidifference, inverse finite difference
Related concepts
Indefinite integral, Bohr–Mollerup theorem, finite difference, summation, Bernoulli polynomials, Gamma function, Euler–Maclaurin formula, Abel–Plana formula
Uniqueness
Up to an arbitrary 1-periodic function; the Nørlund principal solution is unique up to a constant (minimal exponential type is imposed).

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Indefinite sum

Nodes70
Edges69
Triples129
Avg. degree1.97
Density0.028571
Components1

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Indefinite sum

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related to Further reading · 19
Indefinite sum → Academic Press, Allan, An Introduction, Applications, Difference Equations, Finite-Difference Equations And Simulations, Francis, Hildebrand, Indefinite, ISBN, Kelley, No, Peterson, Polyakov, Prenctice-Hall, Programmirovanie, Vol, Walter, XS
related to Complex analysis (exponential type) · 14
Indefinite sum → Among, Following, For, Hauptlösungen, He, However, Instead, Lösungen, Niels Erik Nørlund, Re, Specifically, The, Unter, Wachstum
related to Special functions · 14
Indefinite sum → Bernoulli, Delta, From, Gamma, Gauss Pi, Hurwitz, Lerch, Many, Nørlund, Pi, Sums, The, The Gamma, Under
related to Examples · 10
Indefinite sum → Abel, Bernoulli, Combined, Hurwitz, Laurent, Nørlund, Plana, Taylor, The, This
related to Integral mean condition · 9
Indefinite sum → Bernoulli, Delta, Euler, Faulhaber's, Following Nørlund's, From, In, Maclaurin, The
related to Quantum field theory and the Casimir effect · 9
Indefinite sum → Abel, Bessel, By, Casimir, For, In, Plana, The, The Abel
related to Relationship to indefinite products · 8
Indefinite sum → By, Conversely, Delta, In, It, Milne-Thomson, Niels Erik Nørlund, This
related to Numerical analysis and symbolic summation · 6
Indefinite sum → Gosper's, Karr's, Maclaurin, The, The Euler, The Gregory-Laplace
related to Uniqueness of the principal solution · 6
Indefinite sum → German, Hauptlösung, If, The, Therefore, To
has application · 5
Indefinite sum → Casimir, GNU Compiler Collection, Its, LLVM, The

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Important terminology

displaystyle sum solution indefinite function difference principal -1 formula delta analytic exponential type constant finite frac strip summation forward nørlund

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Indefinite sumDefinitionLinear operator that inverts the finite difference: Δ − 1 {\displaystyle \Delta ^{-1}} : Δ F ( x ) = F ( x + 1 ) − F ( x ) = f ( x ) {\displaystyle \Delta F(x)=F(x+1)-F(x)=f(x)}…1.00infobox
Indefinite sumDomainFunctions on integers; extended to real and complex arguments via analytic continuation1.00infobox
Indefinite sumNotation∑ x f ( x ) {\displaystyle \sum _{x}f(x)} , Δ − 1 f ( x ) {\displaystyle \Delta ^{-1}f(x)} (forward), ∇ − 1 f ( x ) {\displaystyle \nabla ^{-1}f(x)} (backward).1.00infobox
Indefinite sumOther namesAntidifference, inverse finite difference1.00infobox
Indefinite sumRelated conceptsIndefinite integral, Bohr–Mollerup theorem, finite difference, summation, Bernoulli polynomials, Gamma function, Euler–Maclaurin formula, Abel–Plana formula1.00infobox
Indefinite sumUniquenessUp to an arbitrary 1-periodic function; the Nørlund principal solution is unique up to a constant (minimal exponential type is imposed).1.00infobox
Indefinite sumVideo illustrationA 1‑periodic function C ( x ) {\displaystyle C(x)} vanishes under the forward difference ( Δ C ( x ) = 0 {\displaystyle \Delta C(x)=0} ).1.00infobox
LLVM or the GNU Compiler Collectioninstance ofscalar evolution in compilers0.80text
and various numericalinstance ofscalar evolution in compilers0.80text
symbolic computational methods.Special functionsMany standard transcendental functions are naturally defined as indefinite sums of elementary termsinstance ofscalar evolution in compilers0.80text
Indefinite sumhas applicationThe0.60section
Indefinite sumhas applicationIts0.60section

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