Skew power series rings with general commutation formula (Q1194320)

Let \(K\) be a skew field equipped with an automorphism \(\sigma\) and let \(X\) be some variable. A higher \(\sigma\)-derivation is a sequence \(S = (\delta_ n)_{n \geq 0}\) of additive maps from \(K\) into \(K\) such that one has \(\delta_ 0 = \sigma\) and \((Xk)l = X(kl)\) for every \(k,l\in K\) if we define: \(Xk = \delta_ 0(k)X + \delta_ 1(k)X^ 2 + \dots + \delta_ n(k)X^{n+1} + \dots\) for every \(k\in K\). This allows to equip the module \(K[[X]]\) of formal power series with multiplicities in \(K\) with a structure of ring that is denoted \(K[[X,\sigma,S]]\). This paper is a survey of the theory of higher \(\sigma\)-derivations and gives a very good overview of this subject. In a first part, the general methods used for building higher \(\sigma\)-derivations are presented. Different classification results for higher \(\sigma\)-derivations are also given. Finally rationality and subfield questions are addressed.