## Discrete-analytic functions

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I do not know if such concept already exists but lets consider functions which are equal to its Newton series.

We know that functions which are equal to their Taylor series are called analytic, so lets call functions that are equal to their Newton series "discrete analytic".

The formula is alalogious to Taylor series but uses finite differences instead of dirivatives, so for any discrete-analytic function:

$$f(x) = \sum_{k=0}^\infty \binom{x-a}k \Delta^k f\left (a\right)$$

It is known that for a functional equation $\Delta f=F\,$there are infinitely many solutions which differ by any 1-periodic function. But it appears that there is only one (up to a constant) discrete-analytic solution, i.e. all discrete-analytic solutions differ only by a constant term.

Thus I have the following questions:

• Do discrete-analytic functions express special properties on the complex plane?

• Is there a method to extend the notion of discrete analiticity to a range of functions for which Newton series does not converge (so to make it possible to choose the distinguished solution to the abovementioned equation)?

For the second part of the question, as I know there is at least one one similar attempt, the Mueller's formula:

If $$\lim_{x\to{+\infty}}\Delta f(x)=0$$ then $$f(x)=\sum_{n=0}^\infty\left(\Delta f(n)-\Delta f(n+x)\right)$$

although it seems not to be universal and I do not now whether it is always useful.

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Also posted to math.stackexchange.com/questions/53646/… – Gerry Myerson Jul 25 2011 at 12:51

## 2 Answers

An analytic characterization of functions represented by the Newton series is known - that is a class of functions analytic on some half-plane $\operatorname{Re}x>\lambda$ and satisfying there some estimates. See

Gelʹfond, A. O. Calculus of finite differences. Translated from the Russian. International Monographs on Advanced Mathematics and Physics. Hindustan Publishing Corp., Delhi, 1971 (or several earlier Russian language editions).

On the other hand, there is a more recent attempt to develop a kind of complex analysis based on more complicated difference operators. See

Dynnikov, I. A.; Novikov, S. P. Geometry of the triangle equation on two-manifolds. Mosc. Math. J. 3, No. 2, 419-438 (2003),

and the review in http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1046.39016&format=complete

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@Anatoly: why are you advising him only Russian editions? – Ilya Jul 25 2011 at 13:15

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@Gortaur: If you know other appropriate material, you are welcome. – Anatoly Kochubei Jul 25 2011 at 14:12
Thanks, I looked through the Russian edition, found on the Internet. There is a section that deals with "functions that can be represented as Newton series" (he did not give a special name to the class). The properties are mostly dealing with the maximum growth rate of such functions. Another section indeed gives the conditions when an analytic function can be represented exactly with a Newton series. The question about generalizations still remains though... – Anixx Jul 25 2011 at 20:28

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Also look at the book titled "Discrete calculus by analogy" wirtten by Izadi, Aliev, and Bagirev.

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