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Let $f\colon \mathbb{R} \to \mathbb{R}$ be a differentiable function. Let us define the "discrete derivative"

$$ \Delta f( x) = f(x + 1) - f( x ) $$

and recursively $\Delta ^n f\left( x \right) = \Delta \left( {\Delta ^{n - 1} f\left( x \right)} \right)$ for all $n\geq 1$.

Is there some simple relation between $\Delta ^n f$ and the standard derivative $f^{(n)}$? Can we write a Taylor series expansion (say about 0) of $f$ in terms of discrete derivatives?

Heitor Fontana
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Yes, using the Stirling numbers one has $$ \frac{1}{k!} \frac{d^k}{dx^k} f(x) = \sum_{n=k}^{\infty} \frac{s(n,k)}{n!} \Delta^{n} f(x) $$ and $$ \frac{1}{k!} \Delta^{k} f(x) = \sum_{n=k}^{\infty} \frac{S(n,k)}{n!} \frac{d^n}{dx^n} f(x)$$

Edit: The Newton Series are defined as $$ f(x) = \sum_{k=0}^{\infty} \binom{x-a}{k} \: \Delta^k f(a) $$ some examples include: $$ \sum_{k=0}^{\infty} (-1)^k \binom{n}{2k} = 2^{\frac{n}{2}} \: \cos(\frac{\pi n}{4}) \\ \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} \binom{n}{k} = H_n $$ where $H_n$ are the Harmonic numbers.

  • I didn't know these formulas. Do you have a reference where I can find them established ? – Jean Marie Nov 14 '21 at 23:37
  • @JeanMarie I'm not sure of a reference that treats them thoroughly but I personally found them in the NIST Handbook of Mathematical Functions section 26.8 pg. 625 –  Nov 15 '21 at 00:14
  • Thank you very much. – Jean Marie Nov 15 '21 at 00:15
  • It should be underlined for the OP that these infinite summation formulas will be of limited practical interest... – Jean Marie Nov 15 '21 at 00:53
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    If OP wants a series based solely upon forward differences, see the Newton Series, which is the forward difference analogue to the Taylor series which Newton propounded in his Principia Mathemtica –  Nov 15 '21 at 02:04
  • @DecarbonatedOdes thank you, this is interesting! You're right that with infinite series becomes hard to handle in practical situations. Could you add some elaboration on the Newton Series applied to this case? what can we write using that then? – Heitor Fontana Nov 15 '21 at 11:38
  • very good (+1) you may be interested in the answer to this similar question – G Cab Nov 16 '21 at 17:35
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    @JeanMarie: I think the classic Calculus of Finite Differences is a most valuable reference. You can find e.g. the $n$-th differences using Stirling numbers in chapter IV: Stirling's numbers, $\S$ 56: Derivatives expressed by differences. – Markus Scheuer Nov 16 '21 at 19:29
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    @epi163sqrt thank you very much for the reference. – Jean Marie Nov 16 '21 at 19:31