Differential transcendence of Bell numbers and relatives: a Galois theoretic approach - Archive ouverte HAL
Journal Articles American Journal of Mathematics Year : 2024

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 $\mathbb{C}(\{t\})$ 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 $\mathbb{C}(\{t\})$, 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 linear recurrence with polynomial coefficients.
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Dates and versions

hal-03091272 , version 1 (30-12-2020)
hal-03091272 , version 2 (25-08-2023)
hal-03091272 , version 3 (27-04-2024)

Identifiers

  • HAL Id : hal-03091272 , version 3

Cite

Alin Bostan, Lucia Di Vizio, Kilian Raschel. Differential transcendence of Bell numbers and relatives: a Galois theoretic approach. American Journal of Mathematics, In press. ⟨hal-03091272v3⟩
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