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.
Symbolic Operations
Sum, Product — definite and indefinite sums and products
DiscreteShift — shift 
DifferenceDelta — difference 
DiscreteRatio — ratio 
etc
DifferenceQuotient — difference quotients 
etc.
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
RSolveValue — find an expression for the solution of a difference equation
Summation Transforms »
ZTransform ▪ GeneratingFunction ▪ FourierSequenceTransform ▪ ...
Sequence Limits
DiscreteLimit — limits of sequences including recurrence and number theory
DiscreteMinLimit, DiscreteMaxLimit — lower and upper limits
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 Properties »
FunctionPeriod ▪ FunctionSign ▪ FunctionMonotonicity ▪ ...
Asymptotics »
AsymptoticSum — asymptotic approximation to sum
AsymptoticRSolveValue — asymptotic approximation to difference equations
Sequence Recognition
FindSequenceFunction — try to find functional forms for sequences
FindLinearRecurrence ▪ FindGeneratingFunction
Continuous Calculus »
Integrate — definite and indefinite integrals
Sequence Visualization »
DiscretePlot — plot discrete sequences specified by formulas
DiscretePlot3D — plot 2D discrete sequences
ListPlot, ListStepPlot — plot sequences given as lists
Operations on Explicit Lists »
Differences ▪ Ratios ▪ Accumulate ▪ Table ▪ ...