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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.
Reference
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
series coefficient
SeriesCoefficient — find the n
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 ▪ ...
