On the internet, most theorems about lacunary function only give the sufficient conditions. For example, Ostrowski-Hadamard Gap Theorem concerns the asymptotic length of null Taylor coefficients, while a theorem of Carlson states that a Taylor series with integral coefficients is either rational or lacunary.

However, it seems like that a Taylor series not satisfying the conditions in these theorems may also be lacunary. Are there any theorems about the necessary conditions of lacunary functions?

  • 2
    $\begingroup$ Incidentally, Carlson's theorem also assumes the radius of convergence is 1. For instance, $\sum {2n\choose n}x^n$ is neither rational nor lacunary. $\endgroup$ Mar 8, 2020 at 14:10

1 Answer 1


Edit. I added two references to the work of Miloš Kössler I found recently: he, by using the variable transformation $z\mapsto \zeta + \eta e^{-i\theta} \zeta^2$, $0< \eta \le {3/ 4}$, gives two necessary and sufficient condition for the power series \eqref{1} to have the point $e^{i\theta}$ as a singular boundary point. These condition, which do not rely on the one of Fabry described below, are briefly described in the final notes.

Terminology: by singular boundary point I mean a boundary point across which you cannot analytically continue an analytic function defined on the interior of the give domain, while this is always possible at a regular boundary point.

A necessary and sufficient condition. I am not aware of any necessary condition for the "lacunarity". However, a conceptually simple necessary and sufficient condition for a boundary point to be a singular point is known: and as a corollary, you can use it and check every boundary point of the domain of your function is singular or not. Without any restriction on generality, let's consider a power series $$ f(z)=\sum_{n=0}^{\infty} a_n z^n,\label{1}\tag{1} $$ whose radius of convergence $R_f$ is 1 and suppose we want to check if $z=1$ is singular or not.
If the expansion of $f(x)$ as a power series centered at any point on the real segment $]0,1[$ converges on a disk which includes $z=1$ then this point is regular, otherwise it is singular. This form of the condition is fully given by Markushevich ([2] chapter IX §IX.7 pp313-314) and Titchmarsh ([3] chapter 7, §7.23, p. 216): however, this latter author follows Landau in simplifying the calculations by introducing the following function $F(\zeta)$ ([3] chapter 7, §7.23, pp. 216-217, [1] chapter 5, §19, p. 76-77). Let $$ F(\zeta)=\frac{1}{1-\zeta}f\left(\frac{\zeta}{1-\zeta}\right)=\sum_{n=0}^{\infty} b_n z^n, $$ where $b_n=\sum_{m=0}^{n}\binom{n}{m} a_m$. Then $z=1$ is a singular boundary point for $f$ if and only if $$ R_F=\limsup_{n\to\infty}|b_n|^{-\frac{1}{n}}=\frac{1}{2}, \label{2}\tag{2} $$ and thus we get the following
Corollary. Let $f:\Bbb C\to\Bbb C$ be an analytic function whose power series expansion at $0\in\Bbb C$ is \eqref{1}. Then $f$ is lacunary on the open unit disk $\Bbb D$ (i.e. $\partial\Bbb D$ is the "natural boundary" for $f$) if and only if $$ \limsup_{n\to\infty} |a_n|^\frac{1}{n}=1\;\wedge\;\limsup_{n\to\infty}{\left|\sum_{m=0}^{n}\binom{n}{m} a_me^{im\theta}\right|^\frac{1}{n}}\!\!=2\quad\forall \theta\in[0,2\pi] $$


  • According to Landau ([1] chapter 5, §19, p. 76), this criterion is due to Fabry.
  • Reference [2] have been translated in English as Markushevich A.I. The theory of analytic functions: a brief course, Moscow: MIR: however, I do not have access to a copy of that book, therefore I refer to the Italian edition listed below.
  • Miloš Kössler, by using the variable transformation $z\mapsto \zeta + \eta e^{-i\theta} \zeta^2$, transforms \eqref{1} in the power series $F(\zeta)=\sum_{n=0}^{\infty} A_n(\eta, e^{i\theta}) \zeta^n$. Then proves that ([A1], p. 27) a necessary and sufficient condition for the point $e^{i\theta}$ to be a singular boundary point for \eqref{1} is that $$ \limsup_{n \to \infty} {|A_n(\eta, e^{i\theta})|^{1\over n}}=\frac{2\eta}{\sqrt{1+4\eta}-1}\qquad 0< \eta \le {3\over 4}\label{3}\tag{2b} $$ Obviously, if the above value is independent from $\theta\in [0,2\pi]$ then the all the boundary of the unit disk is made of singular points for \eqref{1}. Kössler's condition may be interesting due to the fact that while the coefficients $\{b_n\}_{n\in \Bbb N}$ above are customarily polynomials of order $n$ respect to $e^{i\theta}$, the coefficients $A_n$ are polinomials at most of degree $\big[{n \over2}\big]$ (respect to the same variable). In the paper [A2] (pp. 528-529), with the same title, he further simplifies condition \eqref{3} assuming directly $\eta= {3/4}$ and defining a sequence $\{B_n\}_{n\in \Bbb N}$ by which \eqref{3} becomes $$ \limsup_{n \to \infty} {|B_n(e^{i\theta})|^{1\over n}}= {3\over 4}.\label{4}\tag{2c} $$ Each $B_n$ is a polynomial whose (variable, depending on an arbitrarily small positive constant $\mu$) number of terms is howhever less than $\big[{n \over2}\big]$, thus again \eqref{4} could be easier to veryfy respect to conditions \eqref{2} and \eqref{3} for some power series $f$.


[1] Landau, Edmund; Gaier, Dieter, Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie. 3 erw. Auflage (German), Berlin-Heildelberg-New York: Springer-Verlag, pp. XI+201 (1986), ISBN: 3-540-16886-9, MR0869998, Zbl 0601.30001.

[2] Markushevich, Alekseĭ Ivanovich, Elementi di teoria delle funzioni analitiche. Translated from the Russian by Ernest Kozlov, (Italian) Nuova Biblioteca di Cultura, Serie Scientifica. Roma: Editori Riuniti; Moscow: Edizioni Mir. pp. 384 (1988), ISBN: 88-359-3284-X, MR1011460, Zbl 0694.30002.

[3] Titchmarsh, Edward Charles, The theory of functions 2nd ed., (English), Oxford: Oxford University Press, pp. X+454 (1939), JFM 65.0302.01, MR3728294, Zbl 0336.30001.

Addendum reference

[A1] M. Kössler, "Sur les singularités des séries entières" (French), Atti della Reale Accademia dei Lincei, Rendiconti, Classe di Scienze Fisiche Matematiche Naturali, Serie V, 32, No. 1, pp. 26-29 (1923), JFM 49.0236.01.

[A2] M. Kössler, "Sur les singularités des séries entières" (French), Atti della Reale Accademia dei Lincei, Rendiconti, Classe di Scienze Fisiche Matematiche Naturali, Serie V, 32, No. 1, pp. 528-531 (1923), JFM 49.0236.03.

  • $\begingroup$ To the downvoter: why the downvote? Is there a particular reason for which my answer in not useful? $\endgroup$ Aug 22, 2022 at 13:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.