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In 2003 Klazar proved that the ordinary generating function of the sequenceof Bell numbers is differentially transcendental over the field$\mathbb{C}(\{t\})$ of meromorphic functions at $0$. We show that Klazar'sresult is an instance of a general phenomenon that can be proven in a compactway using difference Galois theory. We present the main principles of thistheory in order to prove a general result about differential transcendence over$\mathbb{C}(\{t\})$, that we apply to many other (infinite classes of) examplesof generating functions, including as very special cases the ones consideredby~Klazar. Most of our examples belong to Sheffer's class, well studied notablyin umbral calculus. They all bring concrete evidence in support to thePak-Yeliussizov conjecture, according to which a sequence whose both ordinaryand exponential generating functions satisfy nonlinear differential equationswith polynomial coefficients necessarily satisfies a linear recurrence withpolynomial coefficients.

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