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# Differential transcendence of Bell numbers and relatives: a Galois theoretic approach

Abstract : We show that Klazar's results on the differential transcendence of the ordinary generating function of the Bell numbers over the field $\mathbb{C}(\{t\})$ of meromorphic functions at $0$ is an instance of a general phenomenon that can be proven in a compact way using difference Galois theory. We present the main principles of this theory in order to prove a general result of differential transcendence over $\mathbb{C}(\{t\})$, that we apply to many other (infinite classes of) examples of generating functions, including as very special cases the ones considered by~Klazar. Most of our examples belong to Sheffer's class, well studied notably in umbral calculus. They all bring concrete evidence in support to the Pak-Yeliussizov conjecture according to which {a sequence whose both ordinary and exponential generating functions satisfy nonlinear differential equations with polynomial coefficients necessarily satisfies a \emph{linear} recurrence with polynomial coefficients}.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03091272
Contributor : Alin Bostan Connect in order to contact the contributor
Submitted on : Wednesday, December 30, 2020 - 7:02:39 PM
Last modification on : Tuesday, October 25, 2022 - 4:24:08 PM
Long-term archiving on: : Wednesday, March 31, 2021 - 6:52:52 PM

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klazar-galois.pdf
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• HAL Id : hal-03091272, version 1

### Citation

Alin Bostan, Lucia Di Vizio, Kilian Raschel. Differential transcendence of Bell numbers and relatives: a Galois theoretic approach. {date}. ⟨hal-03091272⟩

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