Newton series

Remarks

Example: for $f(x) = 2^x$ and $a = 0$, $\Delta^k f(0) = 1$ for all $k$, so the Newton series is $\sum_k \binom{x}{k}$. This is the binomial series of $(1 + z)^x$ at $z = 1$, which converges to $2^x$ for real $x > -1$. Non-example: $\sum_k \frac{x^k}{k!}$ is the Taylor series of $e^x$ at $0$, not its Newton series. The Newton series of $e^x$ at $0$ has coefficients $\Delta^k e^x |_{x=0} = (e - 1)^k$, and since $e - 1 > 1$ it diverges at every real $x$ that is not a natural number. At $x = a + n$ with $n \in \N$, only the terms with $k \le n$ are nonzero, and the sum is $f(a + n)$ by Newton's forward difference formula.