Articles of formal power series

$k]$ is UFD for $k$ field

As the title says, I’m trying to prove that $k[[X_1,\ldots,X_n]]$ is UFD for $k$ field. In Lang’s Algebra, there is a proof by induction on $n$. The base of induction is clear (we even have discrete valuation ring). Let $R_n = k[[X_1,\ldots,X_n]] \cong R_{n-1}[[X_n]]$ and assume that $R_{n-1}$ is UFD. Let $f\in R_n$, $f(X_1,\ldots,X_n)\neq 0$. […]

Short form of few series

Is there a short form for summation of following series? $$\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^kn!\alpha^{2n}((2y-1)^{2k+1}+1)}{2^{2n+1}(2n)!k!(n-k)!(2k+1)}$$ $$\sum\limits_{n=0}^\infty\dfrac{\alpha^{2n+1}(\cos^{-1}(2y-1)-\pi)}{2^{4n+3}n!(n+1)!}$$ $$\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(k!)^2\alpha^{2n+1}(2y-1)(1-(2y-1)^2)^{k+\frac{1}{2}}}{2^{4n-2k+3}n!(n+1)!(2k+1)!}$$ Even sum of any combination of above terms could help. This is result of some integral. I guess they should contain Bessel and Struve functions. In the case of $y=0.5$ it seems the sum of above terms to be $$ -\frac{\pi}{2}\left[I_3(\frac{\alpha}{2}) +\frac{3}{\frac{\alpha}{2}}I_2(\frac{\alpha}{2})- […]

If $R$ and $R]$ are isomorphic, then are they isomorphic to $R$ as well?

This question already has an answer here: Why don't we have an isomorphism between $R[x]$ and $ R[[x]]$? 1 answer

Can $k$ be dense in $k$? where $p_xq_y-p_yq_x \in k^*$.

Let $k$ be a field of characteristic zero. Let $p,q \in k[x,y]$ have invertible Jacobian, namely, $p_xq_y-p_yq_x \in k^*$ (partial derivatives). It can be shown that each of $k[x,y]$ and $k[p,q]$ is dense in $k[[x,y]]$, see this related question. Then in the induced topology, $k[p,q]$ is dense in $k[x,y]$. (More precisely, when considering $p,q \in […]

A UFD for which the related formal power series ring is not a UFD

I know that the proposition $$ A \text{ is a UFD } \Rightarrow A[[X]] \text{ is a UFD }$$ is false. Wikipedia states that if $B=K[x,y,z]/(x^2+y^3+z^7)$ then $A=B_{(x,y,z)}$ is a counterexample, but I can’t show that $A$ is a UFD and $A[[X]]$ isn’t. Are there other known counterexample?

The group of $k$-automorphisms of $k]$, $k$ is a field

Let $k$ be a field. Is the group of $k$-automorphisms of $k[[x,y]]$ known? ($k[[x,y]]$ is the ring of formal power series in two variables, see Wikipedia.) A somewhat relevant question is this question, which deals with $k[[x]]$, with $k$ any commutative ring. Thanks for any hints and comments.