If $ F:V\to W$ is a smooth at $ a\in V$ function between finite-dimensional vector spaces over $ \mathbb{R}$ , then we have $ $ F(x) = \sum_{k=0}^N\frac{1}{k!}(D^kF)(a)[(x-a)^{\otimes k}]+\text{remainder}, $ $ where $ (D^kF)(a):\mathrm{Sym}^k(V)\to W$ are $ \mathbb{R}$ -linear maps. Algebraically, this is a statement about writing tuples of homogeneous polynomials of same degree as a multilinear map, whose coefficients are the respective mixed partial derivatives (at $ a$ ).

Now, let $ R$ be a commutative ring, in which $ k!$ are invertible. I would like to be able to reference the analogous statement about the formal Taylor expansion at $ 0$ of tuples of formal power series $ F:=(F_1,\dots,F_n)\in R[[X_1,\dots,X_m]]^n$ (or equivalently, tuples of polynomials). This seems like a completely standard fact, but I have not been able to pinpoint it in the literature.

I am also curious, is there a workaround statement in the case that $ 1/k!$ don’t exist in $ R$ (e.g. if $ R$ has positive characteristic)?