Reading ``Finite Difference Equations'' by Levy and
Lessman. Interesting book. Opens my eyes about a lot of things related
to the continuous and discrete cases. The one that struck me today is
$$2^x = (1 + 1)^x = \sum_n \binom{x}{n} = \sum_n {x^{\nf}
\over n!}.$$ Here $x^{\nf} = x(x-1)\ldots(x-n+1)$, the
falling powers of $x$. This is a notation due to Knuth; the book uses
$(x)_n$ which I find less clear. The expression akin to the Taylor
expansion is
$$f(x_0 + xh) = \sum_n {x^{\nf} \over n!} \Delta^n f(x_0),$$ where
$\Delta^n f(x_0)$ is the $n$ times iterated difference operation
$\Delta f(x) = f(x+h) - f(x)$. The sum is finite when $f$ is a
polynomial, and is otherwize defined if $f$ is analytic. The
relationships between the difference operator $\Delta$, the shift
operator $E=1+\Delta$, and the differentiation $D$ are expressed as
$$E=e^{hD}$$
and
$$\log(1+\Delta) = \log E = hD,$$ where the exponential and logarithm
signify the usual power series of the operators.