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

# DiscreteShift

 DiscreteShift gives the discrete shift . DiscreteShiftgives the multiple shift . DiscreteShiftgives the multiple shift of step h. DiscreteShiftcomputes partial shifts with respect to i, j, ....
• All quantities that do not explicitly depend on the variables given are taken to have constant partial shift.
Shift with respect to i:
Shift with step h:
Multiple shifts with respect to i:
Enter using Esc shift Esc, and subscripts using Ctrl+_:
The shift with respect to i of scoped operators:
Shift with respect to i:
 Out[1]=
Shift with step h:
 Out[2]=

Multiple shifts with respect to i:
 Out[1]=
 Out[2]=

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

The shift with respect to i of scoped operators:
 Out[1]=
 Out[2]=
 Out[3]=
 Scope   (10)
Compute the first and second shift:
First and second shift with step h:
The first partial shifts with respect to i and j :
Higher partial shifts:
Partial shifts with steps r and s:
Elementary functions:
Integer functions:
Holonomic sequences satisfy a linear difference equation:
Sums:
Shifting inside the summation sign:
In this case i is not a free variable:
Products:
Differencing product limits:
Integrals:
Shifting integration limits:
Limits:
Here the i is not a free variable:
 Applications   (2)
Define a symbolic mean operator using DiscreteShift:
It also works with scoping constructs:
Use on special functions:
Use DiscreteShift to define derivatives:
DiscreteShift is a linear operator:
Product rule:
Quotient rule:
Chain rule:
DiscreteShift can be expressed in terms of DifferenceDelta:
DifferenceDelta can be expressed in terms of DiscreteShift:
DiscreteRatio can be expressed in terms of DiscreteShift:
Using ReplaceAll to implement DiscreteShift can be dangerous:
DiscreteShift understands scoping rules:
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