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A few years ago, I asked a question on MSE about the existence of an infinite product representation of a functional square root of the sine function. No answers were given, though user conditionalMethod suggested in the comments that the lack of analyticity of such a function, potentially a quasi-analytic (Gevrey class) function could exist.

In the cross-posted MO version of the question I asked a few months later, Alexandre Eremenko explained that such a function cannot be defined on the whole complex plane. He went on to say that

Of course this answer is related to a true product, it does not exclude that there is some "formal" product representing this square root, whatever a "formal product" may mean.

I'd like to explore this idea of a formal product a bit further, and I wonder whether it can be defined in such a way so as to find, for instance, a formal product representation analogue of the formal power series for one of the functional square roots of the sine. By "formal" I mean in this case that we leave aside any notion of convergence, and it can be manipulated with the customary algebraic operations.

Let $(a_{n})_{n \geq 1}$ be a sequence of real numbers. One way to represent a function uniquely as a formal product, as explained by Will Jagy in this answer, is:

$$f(X) := \prod_{n=1}^{\infty} \left(1 + a_{n} X^{n} \right). \tag{1} $$

There might also be other types of formal product representations that uniquely correspond to a formal power series, though. For instance, I'm also interested in whether there are any formal products for the functional square root of the sine of the form

$$ g(X) := \prod_{n=1}^{\infty} \left(1 + b_{n} X^{k} \right) \tag{2} $$ for some fixed $k \in \mathbb{Z}_{\geq 1}$.

Questions

  1. Is there a theory of formal product representations of the form $(1)$ or $(2)$ for functions?
  2. Are there any other theories for formal power representations of functions and/or formal power series? Is there any literature on the subject?

In general, I'm interested in references that delve into either one, or both, of the above questions.

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    $\begingroup$ Your product (2) does not seem to have algebraic meaning since to compute the coefficient of $X^k$ you need to sum an infinite series. $\endgroup$
    – Vladimir Dotsenko
    Commented Feb 18, 2023 at 13:17
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    $\begingroup$ @MaxMuller the difference between power series (an analytic object) and formal power series (an algebraic object) is that in the formal context you are only permitted to perform algebraic operations - in particular, computing infinite sums of real numbers is not an option. So in the formal context, your formula (1) makes sense, and your formula (2) does not. $\endgroup$
    – Vladimir Dotsenko
    Commented Feb 18, 2023 at 14:48
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    $\begingroup$ (continued) In the analytic context, you talk about convergence in $\mathbb{R}$ for specific values of $x$. In the context of formal series, you look at convergence with respect to the topology given by powers of the maximal ideal, which forces the coefficients of a convergent sequence of power series to stabilize, so sums have to be finite. Until you have (much more) clarity about what you want, your question is too vague and too broad. $\endgroup$
    – Vladimir Dotsenko
    Commented Feb 18, 2023 at 15:51
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    $\begingroup$ @VladimirDotsenko One can make sense of a "mixture" by considering the topological $\mathbb R$-algebra $\mathbb R[\![T]\!]:=\varprojlim_n\mathbb R[T]/(T^n)$, which allows "archimedean" topology on the $\mathbb R$-direction, but non-archimedean on the (infinitesimal) $T$-direction. But the OP do have to clarify what they want. $\endgroup$
    – Z. M
    Commented Feb 18, 2023 at 17:21
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    $\begingroup$ The traditional way of translating between infinite series and infinite products is taking exponentials and logs. And infinite sums of formal power series are OK, as long as for any $n$ the leading $n$ terms stabilize after finitely many terms (depending on $n$). Exercise: translate that to an infinite product of formal power series. $\endgroup$
    – Igor Khavkine
    Commented Feb 19, 2023 at 8:34

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