Differential transcendence of Bell numbers and relatives: A Galois theoretic approach

abstract: In 2003 Klazar proved that the ordinary generating function of the sequence of Bell numbers is differentially transcendental over the field $\C(\{t\})$ of meromorphic functions at $0$. We show that Klazar's result 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 about differential transcendence over $\C(\{t\})$, 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.