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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, ….

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Basic Use  
Special Sequences  
Special Operators  
Applications  
Properties & Relations  
Possible Issues  
See Also
Related Guides
History
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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, ….

Details and Options

  • DiscreteShift[f,i] can be input as if. The character is entered using shift or \[DiscreteShift]. The variable i is entered as a subscript.
  • All quantities that do not explicitly depend on the variables given are taken to have constant partial shift.
  • DiscreteShift[f,i,j] can be input as i,jf. The character \[InvisibleComma], entered as ,, can be used instead of the ordinary comma.
  • DiscreteShift[f,{i,n,h}] can be input as {i,n,h}f.
  • DiscreteShift[f,…,Assumptions->assum] uses the assumptions assum in the course of computing discrete shifts.

Examples

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Basic Examples  (4)

Shift with respect to i:

Wolfram Language code: DiscreteShift[f[i], i]

Shift with step h:

Wolfram Language code: DiscreteShift[f[i], {i, h}]

Multiple shifts with respect to i:

Wolfram Language code: DiscreteShift[Sin[i ^ 3 2Pi / 7], {i, 10}]
Wolfram Language code: DiscreteShift[Fibonacci[i], {i, 5}]

Enter using shift, and subscripts using :

Wolfram Language code: Subscript[, {i, 5}]i!

The shift with respect to i of scoped operators:

Wolfram Language code: DiscreteShift[Sum[f[k], {k, i + a, i + b}], i]
Wolfram Language code: DiscreteShift[Limit[f[x], x -> i], i]
Wolfram Language code: DiscreteShift[Derivative[i][f][x], i]

Scope  (10)

Basic Use  (3)

Compute the first and second shift:

Wolfram Language code: DiscreteShift[f[i], i]
Wolfram Language code: DiscreteShift[f[i], {i, 2}]

First and second shift with step h:

Wolfram Language code: DiscreteShift[f[i], {i, 1, h}]
Wolfram Language code: DiscreteShift[f[i], {i, 2, h}]

The first partial shifts with respect to i and j :

Wolfram Language code: Subscript[, i, j]f[i, j]

Higher partial shifts:

Wolfram Language code: Subscript[, {i, 2}, j]f[i, j]

Partial shifts with steps r and s:

Wolfram Language code: Subscript[, {i, 1, r}, {j, 1, s}]f[i, j]

Special Sequences  (3)

Elementary functions:

Wolfram Language code: DiscreteShift[2 ^ i, i]
Wolfram Language code: DiscreteShift[i ^ 4 + 20 * i ^ 2 + 11i ^ 3, {i, 3}]
Wolfram Language code: DiscreteShift[(i ^ 2 + 5) / (i - 7), i]

Integer functions:

Wolfram Language code: DiscreteShift[Floor[i], i]
Wolfram Language code: DiscreteShift[FractionalPart[i], i]
Wolfram Language code: DiscreteShift[Round[i], {i, 3}]

Holonomic sequences satisfy a linear difference equation:

Wolfram Language code: DiscreteShift[Fibonacci[i], {i, 5}]
Wolfram Language code: DiscreteShift[Gamma[i], {i, 3}]
Wolfram Language code: DiscreteShift[CatalanNumber[i], {i, 3}]
Wolfram Language code: DiscreteShift[BesselJ[i, x], {i, 2}]

Special Operators  (4)

Sums:

Wolfram Language code: DiscreteShift[Sum[f[i], i], i]
Wolfram Language code: DiscreteShift[Sum[f[i], {i, n, n + a}], n]

Shifting inside the summation sign:

Wolfram Language code: DiscreteShift[Sum[f[i, j], i], j]

In this case i is not a free variable:

Wolfram Language code: DiscreteShift[Sum[f[i], {i, a, b}], i]

Products:

Wolfram Language code: DiscreteShift[Product[f[i], i], i]
Wolfram Language code: DiscreteShift[Product[f[i, j], i], j]

Differencing product limits:

Wolfram Language code: DiscreteShift[Product[f[i], {i, m, n}], m]
Wolfram Language code: DiscreteShift[Product[f[i], {i, m, n}], n]

Integrals:

Wolfram Language code: DiscreteShift[Integrate[f[i], i], i]
Wolfram Language code: DiscreteShift[Integrate[f[i, j], i], j]

Shifting integration limits:

Wolfram Language code: DiscreteShift[Integrate[f[i], {i, a, j}], j]

Limits:

Wolfram Language code: DiscreteShift[Limit[f[i], i -> j], j]

Here the i is not a free variable:

Wolfram Language code: DiscreteShift[Limit[f[i], i -> j], i]

Applications  (2)

Define a symbolic mean operator using DiscreteShift:

Wolfram Language code: DiscreteMean[f_, i_] := Simplify[(1/2)(Subscript[, i]f + f)]
Wolfram Language code: DiscreteMean[f[i], i]

It also works with scoping constructs:

Wolfram Language code: DiscreteMean[Sum[f[i], i], i]
Wolfram Language code: DiscreteMean[Limit[f[i], i -> j], i]

Use on special functions:

Wolfram Language code: DiscreteMean[Gamma[i], i]

Use DiscreteShift to define derivatives:

Wolfram Language code: Der[f_, i_] := Limit[(DiscreteShift[f, {i, 1, h}] - f) / h, h -> 0]
Wolfram Language code: Der[i ^ 2, i]
Wolfram Language code: Der[Sin[i], i]
Wolfram Language code: Der[E ^ i, i]

Properties & Relations  (3)

DiscreteShift is a linear operator:

Wolfram Language code: Subscript[, i](a f[i] + b g[i])

Product rule:

Wolfram Language code: Subscript[, i](f[i]g[i])

Quotient rule:

Wolfram Language code: Subscript[, i](f[i]/g[i])

Chain rule:

Wolfram Language code: Subscript[, i]f[g[i]]

DiscreteShift can be expressed in terms of DifferenceDelta:

Wolfram Language code: DiscreteShiftDelta[f_, {i_, n_}] := Underoverscript[∑, k = 0, n]Binomial[n, k]Subscript[, {i, k}]f
Wolfram Language code: DiscreteShiftDelta[f[i], {i, 2}]
Wolfram Language code: DiscreteShift[f[i], {i, 2}]
Wolfram Language code: Simplify[%% - %]

DifferenceDelta can be expressed in terms of DiscreteShift:

Wolfram Language code: DifferenceDeltaShift[f_, {i_, n_}] := Underoverscript[∑, k = 0, n]Binomial[n, k](-1)^n - kSubscript[, {i, k}]f
Wolfram Language code: DifferenceDeltaShift[f[i], {i, 2}]
Wolfram Language code: DifferenceDelta[f[i], {i, 2}]

DiscreteRatio can be expressed in terms of DiscreteShift:

Wolfram Language code: DiscreteShift[f[i], i] / f[i]
Wolfram Language code: DiscreteRatio[f[i], i]

Possible Issues  (1)

Using ReplaceAll to implement DiscreteShift can be dangerous:

Wolfram Language code: Integrate[f[x], x] /. x -> x + 1

DiscreteShift understands scoping rules:

Wolfram Language code: DiscreteShift[Integrate[f[x], x], x]
Wolfram Research (2008), DiscreteShift, Wolfram Language function, https://reference.wolfram.com/language/ref/DiscreteShift.html.

Text

Wolfram Research (2008), DiscreteShift, Wolfram Language function, https://reference.wolfram.com/language/ref/DiscreteShift.html.

CMS

Wolfram Language. 2008. "DiscreteShift." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DiscreteShift.html.

APA

Wolfram Language. (2008). DiscreteShift. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DiscreteShift.html

BibTeX

@misc{reference.wolfram_2026_discreteshift, author="Wolfram Research", title="{DiscreteShift}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/DiscreteShift.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_discreteshift, organization={Wolfram Research}, title={DiscreteShift}, year={2008}, url={https://reference.wolfram.com/language/ref/DiscreteShift.html}, note=[Accessed: 13-June-2026]}