BUILT-IN MATHEMATICA SYMBOL

DifferenceDelta

gives the discrete difference

.

DifferenceDelta

gives the multiple difference

.

DifferenceDelta

gives the multiple difference with step h.

DifferenceDelta

computes the partial difference with respect to

.

MORE INFORMATION

DifferenceDelta can be input as . The character is entered Esc diffd Esc or \[DifferenceDelta]. The variable i is entered as a subscript.

All quantities that do not explicitly depend on the variables given are taken to have zero partial difference.

DifferenceDelta can be input as . The character \[InvisibleComma], entered as Esc , Esc, can be used instead of the ordinary comma.

DifferenceDelta can be input as .

DifferenceDelta[f, ..., Assumptions->assum] uses the assumptions assum in the course of computing discrete differences.

EXAMPLES

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Basic Examples (4)

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:

In[1]:=

Out[1]=

Difference with step h:

In[2]:=

Out[2]=

The fifth difference with respect to i:

In[1]:=

Out[1]=

The second difference with respect to i and step h:

In[2]:=

Out[2]=

Enter

using Esc diffd Esc, and subscripts using Ctrl+_:

In[1]:=

Out[1]=

DifferenceDelta is the inverse operator to Sum:

In[1]:=

Out[1]=

In[2]:=

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:

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 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

:

Properties & Relations (6)

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: