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DiscreteShift

DiscreteShift[f, i]
gives the discrete shift TemplateBox[{{f, (, i, )}, i}, DiscreteShift2]=f(i+1).
DiscreteShift[f, {i, n}]
gives the multiple shift .
DiscreteShift[f, {i, n, h}]
gives the multiple shift of step h.
DiscreteShift[f, i, j, ...]
computes 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.
EXAMPLESCLOSE ALL
Basic Examples   (4)
Shift with respect to i:
Shift with step h:
Multiple shifts with respect to i:
Enter DiscreteShift using Esc shift Esc, and subscripts using Ctrl+_:
The shift with respect to i of scoped operators:
Shift with respect to i:
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Shift with step h:
In[2]:=
Click for copyable input
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Multiple shifts with respect to i:
In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=
 
Enter DiscreteShift using Esc shift Esc, and subscripts using Ctrl+_:
In[1]:=
Click for copyable input
Out[1]=
 
The shift with respect to i of scoped operators:
In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=
In[3]:=
Click for copyable input
Out[3]=
Scope   (10)
Compute the first and second shift:
First and second shift with step h:
The first partial shifts wrt i and j _(i,j)f(i,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:
Properties & Relations   (3)
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:
Possible Issues   (1)
Using ReplaceAll to implement DiscreteShift can be dangerous:
DiscreteShift understands scoping rules:
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