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Let s be a positive integer and let DifferenceDelta be this function with step = 1. Define DifferenceDelta^s to be an operator that applies DifferenceDelta s-times to any function f[x].

What Mathematica syntax will give me the result of (1 + DifferenceDelta)^s f[x]? It seems that Mathematica does not support operations with "pure" operators.

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Is this what you're looking for? Nest[DifferenceDelta[#, i] + 1 &, f[i], s] – C. E. Oct 25 '13 at 11:20
Thank you! That's what I want. – Kimist Oct 25 '13 at 11:23
Sorry, it should be Nest[DifferenceDelta[#, i] + # &, f[i], s] because 1 is a identity operator. Thanks for your help. – Kimist Oct 25 '13 at 11:36
Imo, your description of the problem is not consistent with the fact that you find Anon's answer well fitted to your needs. The only thing that agrees is the expected result. – Kuba Oct 25 '13 at 11:42
@Kuba Would you explain that further? You got me interested... what is the difference? – C. E. Oct 25 '13 at 11:45

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