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I'm aware of Taylor's theorem for polynomials over $\mathbb{R}$. More generally though, if working with formal power series over a coefficient ring which contains $\mathbb{Q}$, why does Taylor's formula still hold?

Thank you.

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What exactly is the Taylor's formula you are alluding to in the context of formal power series? – Jonas Meyer Jan 13 '12 at 7:52
I’m guessing that you mean Newton’s series, $$f(x)=\sum_{k=0}^\infty\frac{\Delta^k[f](a)}{k!}(x-a)^{\underline k}=\sum_{k=0}^\infty\binom{x-a}k\Delta^k[f](a)\;.$$ In what sense do you mean holds? You might find the discussion in Graham, Knuth, & Patashnik, Concrete Mathematics, pp. 189-192, helpful. – Brian M. Scott Jan 13 '12 at 8:53

A certain formal Taylor's theorem comes up fairly often in the theory of Vertex Algebras. Haisheng Li and James Lepowsky's introduction to vertex algebras spends a whole chapter on "formal calculus" proving (among many other things) a formal Taylor's theorem.

There is a more general formal Taylor theorem (taking into account formal logarithms) in HLZ (part II in a series of papers on logarithmic intertwining operators). A student of James Lepowsky named Thomas Robinson has written a bunch of papers refining various techniques of formal calculus. In particular this paper of Robinson has a fairly general Taylor theorem appearing as Theorem 4.1.

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