Discrete Calculus

With origins stretching back several centuries, discrete calculus is now an increasingly central methodology for many problems related to discrete systems and algorithms. Building on a large body of original research at Wolfram Research, Mathematica for the first time delivers a comprehensive system for discrete calculus.


Symbolic Operations

Sum, Product definite and indefinite sums and products

DifferenceDelta, DiscreteShift, DiscreteRatio discrete differences and ratios

SumConvergence test for convergence of a sum

ContinuedFractionK construct a continued fraction from a formula for terms

Difference Equations

RSolve symbolic solutions of recurrences and discrete functional equations

DifferenceRoot ▪ DifferenceRootReduce ▪ Casoratian

RecurrenceTable tables of values from recurrences and functional equations

Generating Functions & Transforms

GeneratingFunction construct the generating function from the n^(th) series coefficient

SeriesCoefficient find the n^(th) term from a generating function

ZTransform ▪ InverseZTransform ▪ ExponentialGeneratingFunction ▪ FourierSequenceTransform ▪ DiscreteConvolve ▪ DirichletConvolve

Special Sequences »

Fibonacci ▪ FactorialPower ▪ BernoulliB ▪ StirlingS1 ▪ HarmonicNumber ▪ PolyGamma ▪ Zeta ▪ QFactorial ▪ ...

LinearRecurrence generate a linear recurrence sequence from a kernel

DifferenceRoot symbolic representation of solutions to linear difference equations

Sequence Recognition

FindSequenceFunction try to find functional forms for sequences

FindLinearRecurrence ▪ FindGeneratingFunction

Sequence Visualization »

DiscretePlot plot discrete sequences specified by formulas

ListPlot plot sequences given as lists

DiscretePlot3D ▪ ListPlot3D ▪ ...

Operations on Explicit Lists »

Differences ▪ Ratios ▪ Accumulate ▪ Table ▪ ...

Numerical Discrete Calculus

NSum ▪ NProduct

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