Compute the first and second difference:

Compute the first and second difference with step h:

Partial differences with steps r and s:

DifferenceDelta threads over lists:

Polynomial functions:

Each difference will lower the degree by one:

FactorialPower is typically more convenient than

Power for discrete operations:

You can always convert to a

Power representation through

FunctionExpand:

Rational functions:

Differences of rational functions will stay as rational functions:

Negative powers of

FactorialPower are rational functions:

Their differences are particularly simple:

Differences of

PolyGamma are rational functions:

PolyGamma in discrete calculus plays a role similar to

Log in continuous calculus:

HarmonicNumber and

Zeta also produce rational function differences:

Exponential functions:

Differences of exponentials stay exponentials:

In general the

n difference:

Binary powers

play the same role for

DifferenceDelta that

does for

D:

Polynomial exponentials:

Polynomial exponentials stay polynomial exponential:

Rational exponentials:

Rational exponentials stay rational exponential:

Differences of

LerchPhi times exponential are rational exponential:

Trigonometric and hyperbolic functions:

Differences of trigonometric functions stay trigonometric:

Hypergeometric terms:

A general hypergeometric term is defined by having a rational

DiscreteRatio:

The difference of hypergeometric will produce a rational function times a hypergeometric term:

The difference of a q-hypergeometric term is a q-rational multiple of the input.

Holonomic sequences:

Sums:

Differencing under the summation sign:

Differencing with respect to summation limits:

Differencing with respect to product limits:

Integrals:

Differencing integration limits:

Limits:

Here the i variable is scoped and not free: