n-th Derivative

Question

It can be proven the for a function $h(x)=f(x)g(x)$, letting $f^{(k)}(x)=\frac{d^k}{dx^k}f(x)$ and $g^{(k)}(x)=\frac{d^k}{dx^k}g(x)$ then the n-th derivative, for n is an integer is: $$\frac{d^n}{dx^n}f(x)g(x)=\sum^n_{k=0}\left(\begin{align*}n\\k\\\end{align*}\right)f^{(n-k)}g^{(k)}(x)$$ Interestingly assuming a form similar to a binomial expansion. According to this result, my question is can this formula be extended to $\{n\in R\}$ where $R$ is the set of real numbers? It seems to be tricky as the $\sum^n_{k=0}$ operator doesn't exactly accept non integer values for $n$. However we know that there definitely exist formulas for $f^{(k)}$, $\{k\in R\}$ where $f$ is a certain function...thanks everyone

Answer 1

Notation

In here the following notation is used:

$$\lfloor a\rfloor=\text{The largest integer smaller or equal to }a$$ $$[a]=a-\lfloor a\rfloor$$ $\Gamma(x)$ is the gamma function

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The general formula for the fractional derivative of $f(x)$ is

$$\frac{d^\alpha}{dx^\alpha}f(x)=\frac1{\Gamma(1-\alpha)}\frac{d}{dx}\int_0^x \frac{f(t)}{(x-t)^\alpha}\mathrm dt$$

where $0 < \alpha < 1$

So if we wish to define

$$\frac{d^n}{dx^n}f(x)g(x)$$

with $n\in\mathbb R$, we simply combine the rule above and the rule that for $n\in\mathbb N$

$$\frac{d^n}{dx^n}f(x)g(x)=\sum^n_{k=0}\binom nkf^{(n-k)}g^{(k)}(x)$$

Here's what we do:

We know that

$$\frac {d^{[n]}}{dx^{[n]}}\frac {d^{\lfloor n\rfloor}}{dx^{\lfloor n\rfloor}}f(x)=\frac{d^n}{dx^n}f(x)$$

So therefore $$\frac{d^n}{dx^n}f(x)g(x)=\frac {d^{[n]}}{dx^{[n]}}\left(\sum^n_{k=0}\binom nkf^{(n-k)}g^{(k)}(x)\right)$$ $$\frac{d^n}{dx^n}f(x)g(x)=\frac1{\Gamma(1-[n])}\frac{d}{dx}\int_0^x \dfrac{\displaystyle\sum^{\lfloor n\rfloor}_{k=0}\binom {\lfloor n\rfloor}kf^{(\lfloor n\rfloor-k)}g^{(k)}(t)}{(x-t)^{[n]}}\mathrm dt$$

Answer 2

In such cases there are inequalities, instead of identities, controlling various Sobolev norms of $(fg)^{(n)}$ by the corresponding norms of $f^{(n)}$ and $g$ plus the ones of $g^{(n)}$ and $f$, i.e., $$ \|D^a(fg)\|\le c\big(\|D^af\|\|g\|+\|f\|\|D^ag\|\big), $$ for some $C>0$ depending only on $a$ and the norm.

These are the Kato-Ponce inequalities.

See for examples here.