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I'm interested in Singularity Analysis after reading this Math.Stack post. In the "Analytic Combinatorics" book by P. Flajolet and R. Sedgewick, at p.390, the following transfer theorem (Theorem VI.3) is stated :

Let $f(z)$ be a $\Delta$-analytic function at $1$ obeying the following asymptotic behaviour near $z = 1$

$$f(z) = \mathcal{O} \left( (1-z)^{-\alpha} \log \left(\frac{1}{1-z} \right)^{\beta} \right)$$

Then the coefficients $f_n$ of the Taylor expansion of $f(z)$ at $z = 0$ satisfy

$$f_n = \mathcal{O} \left( n^{\alpha-1} \log(n)^{\beta} \right)$$

and there are similar results for little-o or sim-relations.

My question is the following : is the converse statement true ? Assuming that $f(z)$ is $\Delta$-analytic at $1$, does the asymptotic growth of the $f_n$ tell us what is the worst singularity we can expect near $z = 1$ ?

References concerning Singularity Analysis are welcomed !

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  • $\begingroup$ Look at the zig zag algorithm in chapter VI of the book (and more generally whole chapter VI) $\endgroup$
    – Conrad
    Dec 10, 2022 at 16:06
  • $\begingroup$ @Conrad I've seen it and it is not clear as to how it applies here. In the setup of the algorithm, the asymptotic behaviour of $f(z)$ and $g(z)$ near the singularity is already assumed, so that the behaviour of the Hadamard product $h(z) = f \odot g(z)$ is known a priori. If we had $f_n \sim n^{\alpha-1} \log(n)^{\beta}$, I would expect that $f(z) = F(z)$ modulo some analytic function, where $F(z)$ is here a known function for which $F_n \sim n^{\alpha-1} \log(n)^{\beta}$. But there is no proof of this in Chapter VI. And it gets trickier with the other relations. $\endgroup$
    – Desura
    Dec 10, 2022 at 17:12
  • $\begingroup$ I would suggest to look into the example of Poluya drunkard walk page 425 for how this is done since if you have a known function $F$ with asymptotic up to some order same as $f$, you get that $f-F-P$ is smaller than the order in a precise sense (where $P$ polynomial that appears from differences in few first coefficients), so by induction one reconstructs $f$ asymptotically in terms of powers of $1-z, -\log (1-z)$ the hard part is going from function to coefficients, the reverse is easy by simple majorization if coefficients small enough ( eg $\sum a_nz^n = o (-\log (1-z)$ if $a_n=o(n)$ etc $\endgroup$
    – Conrad
    Dec 10, 2022 at 17:35
  • $\begingroup$ Oops meant $a_n=o(1/n)$ in the above $\endgroup$
    – Conrad
    Dec 10, 2022 at 17:41
  • $\begingroup$ Note that you already know that $f$ has a singularity only at $1$ and is extendable on the unit circle except there, so coefficients show how to write $f$ in terms of known functions with small error at $1$ but do not tell you things outside the Delta analyticity domain, so in particular one can probably construct examples where $f$ has lots of singularities on the boundary of the extended domain - in that sense this notion of "worst" singularity is a bit unclear and you definitely cannot conclude that say $f$ extends to the plane minus the half line $[1, \infty)$ like elementary functions $\endgroup$
    – Conrad
    Dec 10, 2022 at 17:57

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