A power series $f \in K[[x_1,...,x_n]]$ is called D-finite if all partial derivations of f lie in a finite-dimensional vector space over $K(x_1,...,x_n)$. For one variable this is equivalent to: f satisfies a linear differential equation with polynomial coefficients.
Let f and g be power series in one variable $x$ and let f be D-finite. If $g \cdot x^k = f$ for some k does it then follows(and if so why?) that g is D-finite? The same question with several variables: $f,g \in K[[x_1,...,x_n]]$ and $g \cdot x_n^k = f$, does it follow that g is D-finite?