Discrete Calculus
Discrete calculus is the calculus of sequences, a.k.a. discrete time signals. Discrete calculus is the foundation for continuous calculus and used to derive numerical algorithms for it. It is the calculus used for discrete-time signal processing, discrete-time control systems and digital image processing. It is also a calculus used for combinatorics, discrete probability, finance and analysis of algorithms. The Wolfram Language provides extensive coverage of discrete calculus and its applications, including a large library of special sequences and sequence guessing from data; state-of-the-art solvers for sums, difference equations and many more; and full support for its many application domains, from signal processing and control systems to probability and number theory.
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 — find formulas for sequences from data
FindLinearRecurrence ▪ FindGeneratingFunction
Sequence Visualization »
DiscretePlot — plot 1D sequences 
given by formulas
DiscretePlot3D — plot 2D sequences 
given by formulas
ListPlot, ListStepPlot — plot sequences given as lists
Sums & Differences
Sum (∑)— symbolic definite and indefinite sums, single and multiple...
NSum — numeric approximate sum
AsymptoticSum — asymptotic approximate sum
DifferenceDelta ()— difference 
, inverse of sum
DifferenceQuotient — difference quotients 
, approximate derivative
SumConvergence — test for convergence of a sum
TruncateSum — truncate the sums in an expression
Products & Ratios
Product (∏) — symbolic definite and indefinite products, single and multiple...
NProduct — numeric approximate product
AsymptoticProduct — asymptotic approximate product
DiscreteRatio () — ratio 
, inverse of product
Difference Equations
RSolve, RSolveValue — symbolic solutions of recurrences and discrete functional equations
RecurrenceTable — numeric solutions of recurrences
AsymptoticRSolveValue — asymptotic solution of recurrences
DifferenceRoot ▪ DifferenceRootReduce ▪ Casoratian
Summation Transforms »
ZTransform ▪ GeneratingFunction ▪ FourierSequenceTransform ▪ ...
Sequence Limits
DiscreteLimit — limits of sequences including recurrence and number theory
DiscreteMinLimit, DiscreteMaxLimit — lower and upper limits
Sequence Properties »
FunctionPeriod ▪ FunctionSign ▪ FunctionMonotonicity ▪ ...
Sequence Optimization »
Minimize — find the global optimum of a sequence
Maximize ▪ MinValue ▪ MaxValue ▪ ArgMin ▪ ArgMax
Other Sequence Operations
DiscreteShift — shift 
ContinuedFractionK — construct a continued fraction from a formula for terms
Finite Sequence Operations »
Differences ▪ Ratios ▪ Accumulate ▪ Table ▪ ...