Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

It is easy to express difference deltas in terms of delayed functions as follows:

$$\Delta^n [f](x)= \sum_{k=0}^n {n \choose k} (-1)^{n-k} f(x+k)$$

For example.

But what about the inverse process, is there a formula?

share|improve this question

closed as unclear what you're asking by Jonas Meyer, graydad, avid19, Solid Snake, Claude Leibovici Jun 14 at 5:27

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

2 Answers 2

There is: for every integer $n\geqslant0$, $$ f(x+n)=\sum_{k=0}^n{n\choose k}\Delta^k[f](x). $$

share|improve this answer
This is wrong, $y(x+2)=\Delta^2 y(x)+2\Delta y(x) +y(x)$ Where is the coefficient in your formula? –  Anixx Jun 25 '14 at 0:11
Thanks for the correction, but copying my answer and downvoting it is not a good behavior. –  Anixx Jun 25 '14 at 15:22
@Anixx Where did I copy your answer? I looked at your comment, pondered it, saw that indeed there was a problem with my answer, and corrected said answer. By the way, to downvote an answer to one of your questions AND unaccept it AND post an inflammatory comment on it AND post an answer of your own AND accept said answer, all in a few minutes, this is what I call the behaviour of an xxx-xxxx. –  Did Jun 25 '14 at 15:26
@Did FYI meta.math.stackexchange.com/questions/15092/… –  user61527 Jun 25 '14 at 15:32
Many users would consider continuing to comment on main after opening an accusatory meta thread to be rather close to harassment... –  user61527 Jun 26 '14 at 5:05
up vote 2 down vote accepted

The answer by Did is incorrect, the correct answer is

$$ f(x+n)=\sum_{k=0}^n {n\choose k} \Delta^k[f](x). $$

share|improve this answer
Can anyone explain the downvote? –  Anixx Jun 25 '14 at 15:18
Sure: I downvoted (I very rarely downvote answers to some question I also answered) because your behaviour is despicable. Satisfied? –  Did Jun 25 '14 at 15:27
+1 for your answer and -1 for did's answer. –  Zlatan der Zechpreller Jul 8 '14 at 19:18
@ZlatanderZechpreller Interesting take. Care to explain? –  Did Jul 8 '14 at 21:21
((It happens (and I just realized) that the user to which my last comment is addressed managed to get themselves suspended from the site for roughly 3 years. Thus, no answer to expect before May 1 '17 at 14:25.)) –  Did Aug 21 '14 at 23:37

Not the answer you're looking for? Browse other questions tagged or ask your own question.