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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.
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
RecurrenceTable tables of values from recurrences and functional equations
Generating Functions & Transforms
GeneratingFunction construct the generating function from the nth series coefficient
SeriesCoefficient find the nth term from a generating function
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
DiscretePlot plot discrete sequences specified by formulas
ListPlot plot sequences given as lists
Differences  ▪ Ratios  ▪ Accumulate  ▪ Table  ▪ ...
Numerical Discrete Calculus
NSum  ▪ NProduct