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

TL;DR

This paper uses difference Galois theory to prove the differential transcendence of Bell numbers' generating functions, extending Klazar's 2003 result and supporting the Pak-Yeliussizov conjecture.

Contribution

It introduces a Galois theoretic approach to establish differential transcendence for a broad class of generating functions, including Bell numbers and Sheffer sequences.

Findings

Bell numbers' generating functions are differentially transcendental.

The approach applies to various sequences in Sheffer's class.

Supports the Pak-Yeliussizov conjecture.

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.