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, the Wolfram Language for the first time delivers a comprehensive system for discrete calculus.

ReferenceReference

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