Editing Normal order (section)

===Bosonic operator functions===
Normal ordering of bosonic operator functions <math>f(\hat n)</math>, with occupation number operator <math>\hat n=\hat b\vphantom{\hat n}^\dagger \hat b</math>, can be accomplished using [[Factorial power|(falling) factorial powers]] <math>\hat n^{\underline{k}}=\hat n(\hat n-1)\cdots(\hat n-k+1)</math> and [[Newton series]] instead of [[Taylor series]]: 
It is easy to show 
<ref name="Hucht">{{cite journal | last1=König | first1=Jürgen | last2=Hucht | first2=Alfred | title=Newton series expansion of bosonic operator functions | journal=SciPost Physics | publisher=Stichting SciPost | volume=10 | issue=1 | date=2021-01-13 | issn=2542-4653 | doi=10.21468/scipostphys.10.1.007 | page=007| arxiv=2008.11139 | bibcode=2021ScPP...10....7K | s2cid=221293056 | doi-access=free }}</ref> 
that factorial powers <math>\hat n^{\underline{k}}</math> are equal to normal-ordered (raw) [[Exponentiation|powers]] <math>\hat n^{k}</math> and are therefore normal ordered by construction,

: <math>
\hat{n}^{\underline{k}}
 = \hat b\vphantom{\hat n}^{\dagger k} \hat b\vphantom{\hat n}^k
 = {:\,}\hat n^k{\,:},
</math>

such that the Newton series expansion 

: <math>
\tilde f(\hat n) = \sum_{k=0}^\infty \Delta_n^k \tilde f(0) \, \frac{\hat n^{\underline{k}}}{k!}
</math>

of an operator function <math>\tilde f(\hat n)</math>, with <math>k</math>-th [[forward difference]] <math>\Delta_n^k \tilde f(0)</math> at <math>n=0</math>, is always normal ordered. Here, the [[Second_quantization#Action_on_Fock_states|eigenvalue equation]] <math>\hat n |n\rangle = n |n\rangle</math> relates <math>\hat n</math> and <math>n</math>.

As a consequence, the normal-ordered Taylor series of an arbitrary function <math>f(\hat n)</math> is equal to the Newton series of an associated function <math>\tilde f(\hat n)</math>, fulfilling

: <math> 
\tilde f(\hat n) = {:\,} f(\hat n) {\,:},
</math>

if the series coefficients of the Taylor series of <math>f(x)</math>, with continuous <math>x</math>, match the coefficients of the Newton series of <math>\tilde f(n)</math>, with integer <math>n</math>,

: <math>
\begin{align}
       f(x) &= \sum_{k=0}^\infty F_k \, \frac{x^k              }{k!}, \\
\tilde f(n) &= \sum_{k=0}^\infty F_k \, \frac{n^{\underline{k}}}{k!}, \\
F_k &= \partial_x^k f(0) = \Delta_n^k \tilde f(0),
\end{align}
</math>

with <math>k</math>-th [[partial derivative]] <math>\partial_x^k f(0)</math> at <math>x=0</math>.
The functions <math>f</math> and <math>\tilde f</math> are related through the so-called '''[[normal-order transform]]''' <math>\mathcal N[f]</math> according to

: <math>
\begin{align}
\tilde f(n) &= \mathcal N_x[f(x)](n) \\
 &= \frac{1}{\Gamma(-n)} \int_{-\infty}^0 \mathrm d x \, e^x \, f(x) \, (-x)^{-(n+1)} \\
 &= \frac{1}{\Gamma(-n)}\mathcal M_{-x}[e^{x} f(x)](-n),
\end{align}
</math>

which can be expressed in terms of the [[Mellin transform]] <math>\mathcal M</math>, see <ref name="Hucht"/> for details.