Illinois Journal of Mathematics

Julia’s equation and differential transcendence

Matthias Aschenbrenner and Walter Bergweiler

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We show that the iterative logarithm of each non-linear entire function is differentially transcendental over the ring of entire functions, and we give a sufficient criterion for such an iterative logarithm to be differentially transcendental over the ring of convergent power series. Our results apply, in particular, to the exponential generating function of a sequence arising from work of Shadrin and Zvonkine on Hurwitz numbers.

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Illinois J. Math., Volume 59, Number 2 (2015), 277-294.

Received: 24 July 2013
Revised: 30 September 2015
First available in Project Euclid: 5 May 2016

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Zentralblatt MATH identifier

Primary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX] 34M15: Algebraic aspects (differential-algebraic, hypertranscendence, group- theoretical)


Aschenbrenner, Matthias; Bergweiler, Walter. Julia’s equation and differential transcendence. Illinois J. Math. 59 (2015), no. 2, 277--294. doi:10.1215/ijm/1462450701.

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