This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# DifferenceDelta

 DifferenceDelta gives the discrete difference . DifferenceDeltagives the multiple difference . DifferenceDeltagives the multiple difference with step h. DifferenceDeltacomputes the partial difference with respect to .
• All quantities that do not explicitly depend on the variables given are taken to have zero partial difference.
Difference with respect to i:
Difference with step h:
The fifth difference with respect to i:
The second difference with respect to i and step h:
Enter using Esc diffd Esc, and subscripts using Ctrl+_:
DifferenceDelta is the inverse operator to Sum:
Difference with respect to i:
 Out[1]=
Difference with step h:
 Out[2]=

The fifth difference with respect to i:
 Out[1]=
The second difference with respect to i and step h:
 Out[2]=

Enter using Esc diffd Esc, and subscripts using Ctrl+_:
 Out[1]=

DifferenceDelta is the inverse operator to Sum:
 Out[1]=
 Out[2]=
 Scope   (17)
Compute the first and second difference:
Compute the first and second difference with step h:
Partial differences with steps r and s:
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 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:
 Applications   (7)
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 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 which interpolate a sequence of derivatives exactly at a single point:
Define the n coefficient for a factorial power series:
The coefficient for FactorialPower:
The coefficient for FactorialPower:
DifferenceDelta is a linear operator:
Product rule:
Quotient rule:
DifferenceDelta satisfies a Leibniz product rule:
DifferenceDelta is the inverse operation of Sum:
DifferenceDelta can be expressed in terms of DiscreteShift:
DiscreteShift can be expressed in terms of DifferenceDelta:
DifferenceDelta is the discrete analog of D:
Use Differences to compute differences of list elements:
Higher differences:
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