Formal Power Series as Initial Objects?

Question

We often apply formal power series in places where it seems, at face value, somewhat suspect to do so. I'm primarily interested in why these formal manipulations work so broadly.

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A prime example comes from Concrete Mathematics, page 470-471. Here, $(\Delta f)(x) = f(x+1) - f(x)$, and $Df = f'$

We can express $\Delta$ in terms of $D$ using Taylor's formula as follows:

$f(x + \epsilon) = f(x) + \frac{f'(x)}{1!}\epsilon + \frac{f''(x)}{2!}\epsilon^2 + \cdots$

Setting $\epsilon = 1$ tells us that

$\Delta f(x) = \\ f(x+1) - f(x) = \\ f'(x)/1! + f''(x)/2! + f'''(x)/3! + \cdots = \\ (D/1! + D^2/2! + D^3/3! + \cdots)f(x) = \\ (e^D - 1)f(x) $

The authors continue, saying the inverse operator $\sum = 1/\Delta$ should thus be $1/(e^D - 1)$. (Here $\sum$ is meant as an operator, though the authors continue using $\sum$ in its traditional context as well, as in the following power series.)

We recognize $z/(e^z-1) = \sum B_k z^k/k!$ as a known power series, and conclude, somewhat surprisingly, that

$\sum = \frac{B_0}{D} + \frac{B_1}{1!} + \frac{B_2}{2!}D + \frac{B_3}{3!}D^2 + \cdots = \int + \sum \frac{B_k}{k!}D^{k-1}$

This is the asymptotic expansionfor the Euler Summation Formula.

This derivation seems like nonsense, except for the fact that it isn't. We get a reasonable result out the other side, and every step makes sense if you're willing to suspend your disbelief.

I have seen multiple other arguments just like this, where we flippantly go back and forth between functions and their series, even in places where a topology is not clearly visible to make sense of the infinite sums!

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One idea that I had comes from a topic in Knapp's Basic Algebra, the Permanence of Identities (page 212-214). The idea here is that equations which are true over $\mathbb{Z}[x_1,\ldots,x_n]$ ought to remain true over general rings when we substitute ring elements for the $x_i$. While Knapp doesn't dwell on it, I justified this to myself since $\mathbb{Z}[x_1, \ldots, x_n]$ is initial among rings with $n$ distinguished elements, and since ring homs preserve truth, we get that a formula $p = q$ in this polynomial ring implies $p(r_1,\ldots,r_n) = q(r_1,\ldots,r_n)$ is true of any $r_i$ in any (commutative) ring $R$.

By analogy, it seems reasonable that a ring of formal power series (perhaps with rational coefficients?) should be initial in a suitable category, and that the identities we derive by working formally will then be true in, say, rings of operators (which would justify the above argument, modulo convergence issues).

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Finally, then,

Does anybody have references for the soundness of power-series methods being applied in somewhat surprising settings? Additionally, can the argument I've given be made formal? Are there broad outlines for when these formal methods are permissible, and when (if ever) they lead us astray?

Thanks in advance ^_^

Answer 1

This sort of things can sometimes be carried out in the framework of strongly linear algebra.

Objects in this setting are fields of formal series, called Noetherian series, over some field $F$. Those fields can be equipped with a notion of summation which is not topological and makes definite sense of the type of manipulations you did. See the Wikipedia article Hahn series for a definition of summation in case of Hahn series (those are a special case of Noetherian series).

Given two such fields $\mathbb{S}$ and $\mathbb{S}'$ and a function $\Phi: \mathbb{S} \rightarrow \mathbb{S}'$, say that $\Phi$ is a strongly linear operator if it is $F$-linear and commutes with sums. A good choice of morphisms $\mathbb{S} \rightarrow \mathbb{S}'$ is a that of strongly linear field morphisms.

Two examples to illustrate my point.

-If $\Phi: \mathbb{S} \rightarrow \mathbb{S}$ is a strongly linear operator which is contracting in a valuation theoretic sense, then the operator $\operatorname{id}_{\mathbb{S}}+\Phi$ has a reciprocal which can be seen as the sum $(\operatorname{id}_{\mathbb{S}}+\Phi)^{-1}=\sum \limits_{n \in \mathbb{N}} (-1)^n\ \Phi^{\circ n}$. Meaning that for $s \in \mathbb{S}$, the sum $t:=\sum \limits_{n \in \mathbb{N}} (-1)^n\ \Phi^{\circ n}(s)$ is defined with $s=t+\Phi(t)$ and vice versa.

-In the field $\mathbb{L}$ of logarithmic hyperseries (which is a field of Noetherian series, see this article), there is a strongly linear field endomorphism $\mathbb{L} \rightarrow \mathbb{L}$ denoted $\circ_{x+1}$ which acts on a series $f$ as the pre-composition with the series $x+1$, where $x$ is seen as the identity series / function. So in $\mathbb{L}$, we can make sense of $\Delta$ as the strongly linear operator $\Delta:=\circ_{x+1}-\operatorname{id}_{\mathbb{L}}$. The field $\mathbb{L}$ is also equipped with a derivation $\partial$ which is a strongly linear operator $\mathbb{L} \rightarrow \mathbb{L}$ satisfying Leibniz's rule $\forall f,g \in \mathbb{L},\partial (f \ g) = f \ \partial g + \partial f \ g$. Because series in $\mathbb{L}$ have Taylor expansions, one actually has $\Delta = \sum \limits_{n \in \mathbb{N}^{>0}} \frac{\partial^n}{n!}$. A right inverse $\int$ to $\partial$ can also be found by using the method of the first paragraph.

You will find precise definitions and more information in the article Operators on generalized power series of Joris van der Hoeven. A good heuristic with those objects is that they are combinatorial realizations of Banach spaces.

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Notice that the ring $\mathbb{Z}[x_1,...,x_n]$ is only initial in the category of commutative rings with prescribed $n$-uples $(a_1,...,a_n)$. Likewise, in the category $\mathcal{C}$ of fields of Noetherian series over the field $F$ with prescribed $n$-uples, the field $\mathbb{F}_n:=F[[{\varepsilon_1}^{\mathbb{Z}},...,{\varepsilon_n}^{\mathbb{Z}}]]$ of formal Laurent series over $F$ with variables $\varepsilon_1,...,\varepsilon_n$ is initial. Indeed, given an object $(\mathbb{S},s_1,...,s_n)$ in this category and a series $f= \sum \limits_{(z_1,...,z_n) \in \mathbb{Z}^n} f_{z_1,...,z_n} {\varepsilon_1}^{z_1} \cdot \cdot \cdot {\varepsilon_n}^{z_n}$ in $\mathbb{F}_n$, the sum $f(s_1,...,s_n):= \sum \limits_{(z_1,...,z_n) \in \mathbb{Z}^n} f_{z_1,...,z_n} {s_1}^{z_1} \cdot \cdot \cdot {s_n}^{z_n}$ is well-defined in $\mathbb{S}$. The correspondence $f \mapsto f(s_1,...,s_n)$ is the unique morphism $(\mathbb{F}_n,\varepsilon_1,...,\varepsilon_n) \longrightarrow (\mathbb{S},s_1,...,s_n)$.

So if an equation in $\mathbb{S}$ can be reduced to an identity $f(s_1,...,s_n)=0$ for a certain $f \in \mathbb{F}_n$, then it hods in $\mathbb{S}$.

Since $\mathbb{F}_1$ itself (this is just the field of formal Laurent series over $F$) is equipped with a formal derivation $\partial$ and pre-compositions $\circ_g$ for certain series $g$, this implies that many general relations regarding derivation and composition can be derived in $\mathbb{F}_1$ and then deduced in more general settings.

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Now let me give counter example to the validity of certain formal identities in the same setting. Consider, in a well-chosen extension $\mathbb{T}$ of the field of logarithmic-exponential transseries, the series $\operatorname{e}^x$. We can choose $\mathbb{T}$ such that it is equipped with a pre-composition $\circ_{x+1}$ and a derivation $\partial$ as in the previous examples. We do not have $\Delta(\operatorname{e}^x)=(\operatorname{e}^{\partial}-1)(\operatorname{e}^x)$, simply because the sum $\sum \limits_{n \in \mathbb{N}^{>0}} \frac{\partial^n \operatorname{e}^x}{n!}$ does not exist.

Answer 2

There is some discussion of these matters in "A lifting theorem for formal power series" by Bender.

Also, thinking of a formal power series as a listing of a sequence of numbers or other commutative expressions/variables/indeterminates to which an 'infinite involution' can be applied order by order (a graded algebra) as in the three examples in the MathOverflow question "Examples of infinite dimensional involutions" makes it clear that definitions of the multiplicative and compositional inverses of a formal power series (and certain Laurent series) can be made that are consistent with convergent truncations (or other modifications) of an otherwise divergent formal power series.