In real and complex analysis, one has a powerful tool, namely the Taylor expansion, which expands an analytic function into a power series. In algebra, one can define the derivative of a polynomial aswell; for , being a unitary ring, define . This satisfies the same rules as the usual derivative, for example linearity and the product rule . One can also define recursively by , for . Unfortunately, one looses certain properties; for example, if is of finite characteristic , the polynomial satisfies , but is not constant as (assuming is not the zero ring). In particular, the Identity Theorem does not work. This example also shows one problem with a possible Taylor expansion: for that, one needs to compute for ; but in , whence has no inverse in ! Hence, a Taylor expansion in the classical sense cannot be defined. A “fix” for this problem is offered by Hasse derivatives: they are defined to make both the Identity Theorem and Taylor expansions work again.
Let and . Define
The function is called the th Hasse derivative.
The Hasse derivative shares several properties with the usual derivative, but not all of them; for example, in general. But we have the following properties:
Let and , and .
 We have that is linear, i.e. and .
 We have ; in particular, .
 We have .

(Leibniz Rule) We have
more generally, for , we have
where the sum goes over all such tuples .

(Faà di Bruno's Formula) We have
where the sum goes over all tuples with ; here, is a multinomial coefficient having the value

(Taylor Formula) We have
 (Identity Theorem) If we have for all , then .
For that reason, one can define , so that we can write Taylor's formula in a more tempting form as
which almost equals the classical form.
Note that the Leibniz rule, Faà di Bruno's formula and Taylor's formula take simpler forms than their classical counterparts; this is due to the fact that the additional factorial terms or binomial coefficients already hide in the definition of the Hasse derivative.
 This follows from the definition of .

Write ; then

By 1., it is suffices to show this for , (for smaller , both sides will be zero). We have
and
But since
these terms are equal.

Note that if we fix , we get an linear function , . Hence, it suffices to show this for arbitrary and , . By the same argument, for , we get an linbear function , ; therefore, it suffices to consider , . But now,
Hence, it suffices to show . By reorganizing the binomial coefficients, one transforms this into the equality
which is Vandermonde's Identity and, hence, true.
The more general equation is shown by induction on . For , we have . Now, assume that the equation is true for all for one . Then, for any ,
by the Leibniz rule and by the induction hypothesis; here, the second sum goes over all such tuples . But this equals , what we had to show.

Again, by 1., it suffices to show this for , . Now, by the second part of 4.,
where the sum goes over all such . The formula we want is now obtained by sorting the summands by the different powers of appearing, .
Consider the map
there , . Now, if satisfies , then and . Now, for a fixed with and , the number equals the multinomial coefficient
whence we get that the above formula for equals
where the sum goes over all tuples with and .
Now note that and . Therefore, we get
If one now replaces by , we obtain the formula from the statement.

By 1., it suffices to show this for , . Now
by the Binomial Theorem, what we had to show.
 This follows directly from 6.
Comments.
Hi! Perhaps it may be useful to place here some facts about mixed partial Hasse derivatives of polynomials from $R[x_1,\ldots,x_n]$. As for me I worked with polynomials from $R[x_1,x_2,x_3]$ and had to spend a lot of time proving some needed results on its Hasse derivatives. In particular, one of the popular fact is Taylor Formula.
Hi, sorry for not replying earlier, I was not available the last weeks. This sounds interesting, and I will write up something about that. Thanks for the suggestion!
Maybe a question for you: are there other facts about (mixed partial) Hasse derivatives which you think are worth showing here?
I wrote a first article on partial Hasse derivatives, which can be found here.
I believe your Faa di Bruno formula has a small error. In particular, I believe it should be:
$\displaystyle D^{(k)} (f \circ g) = \sum \binom{c_1+\cdots+c_k}{ c_1, \dots, c_k} (D^{(c_1+\cdots+c_k)} f) \circ g \cdot \prod_{j=1}^k (D^{(j)} g)^{c_j}$
where the sum ranges over all $c_i$ where
$\sum_{i=1}^k i c_i = k$.
In particular, in the derivation above for $x^n$ you end with a $g^{c_0}$ term, but this term is not equal to $D^{c_0}(f)(g)$, but is rather equal to $D^{nc_0}(f)(g)/\binom{n}{c_0}$. When you do this substitution, you get what I list above.
Incidentally, if you look on Wikipedia for this formula (for normal partial derivatives) it agrees with what I put above, when you ignore the factorials.
Yes, you are right. Thank you very much!