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, to appear in American Journal of Mathematics.  


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.


ArXiv: 2012.15292

HAL: hal-03091272