Université de Versailles-St Quentin, Laboratoire de Mathématiques, 45 avenue des États-Unis 78035 Versailles cedex, France
e-mail: divizio[at]math.cnrs.fr          Office: bâtiment Fermat, office 3305

with Alin Bostan and Kilian Raschel

#### 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.

DOI:

ArXiv: 2012.15292

HAL: hal-03091272