Compute the first and second difference:

Compute the first and second difference with step h:

The first partial difference :

Higher partial difference :

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:

DifferenceDelta on FactorialPower has the same effect as D on Power:

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 exponentials:

Rational exponentials:

Rational exponentials stay rational exponentials:

Differences of LerchPhi times exponential are rational exponentials:

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:

Holonomic sequences of order 2:

The difference of GammaRegularized with respect to i is a hypergeometric term:

Similarly for BetaRegularized:

The difference for MarcumQ is expressed in terms of BesselI:

Sums:

Differencing under the summation sign:

Differencing with respect to summation limits:

Product:

Differencing with respect to product limits:

Integrals:

Differencing integration limits:

Limits:

Here the i variable is scoped and not free:

Verify the answer for an indefinite sum:

Construct an exact difference form:

The indefinite sum may differ by a constant:

Use DifferenceDelta to define difference equations:

Define a symbolic Mean operator for sequences through DifferenceDelta:

Use it for any special sequences:

Define a backward difference operator:

Use it for any special sequences and operators:

Define a symmetric difference operator:

Use for any special functions and operators:

Define a factorial power series:

The factorial series is exact for polynomials when the order is larger than the degree:

The series is also a Newton series, which is computed by InterpolatingPolynomial:

Factorial power series approximate general functions:

The approximation gets better for higher degree:

Factorial power series interpolate exactly at a sequence of points:

Compare to power series that interpolate a sequence of derivatives exactly at a single point:

Define the n coefficient for a factorial power series:

The coefficient for FactorialPower[x,2]:

The coefficient for FactorialPower[x,n]:

The PDF of a discrete probability distribution can be computed from the CDF of the distribution by using DifferenceDelta:

Verify that the result agrees with the PDF: