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