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

Alin Bostan, Lucia Di Vizio, Kilian Raschel

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.

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 of meromorphic functions at . 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 , 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.

Paper Structure

This paper contains 23 sections, 27 theorems, 81 equations, 2 tables.

Table of Contents

  1. Introduction
  2. From exponential generating functions to difference equations: setting and examples
  3. From exponential generating functions to tau-equations
  4. Other combinatorial examples
  5. Tangent numbers
  6. Alternating permutations
  7. Springer numbers
  8. Barred preferential arrangements
  9. Various other sequences
  10. Karande--Thakare polynomials
  11. Frobenius--Carlitz rational functions
  12. Generalized Bernoulli, Euler and Carlitz polynomials
  13. Glaisher's sequences
  14. Three more challenging examples
  15. Fishburn-Stoimenow numbers
  16. ...and 8 more sections

Key Result

Theorem 2

Let $f\in\mathbb{C}((t))$ be a Laurent series satisfying a linear functional equation of the form where $\alpha_i\in\mathbb{C}(t)$, not all zero, and $\tau$ is one of the following operators: Then either $f\in\mathbb{C}(t^{1/r})$ for some positive integer $r$, or $f$ is D-transcendental over $\mathbb{C}(t)$. Moreover, in the case of the first operator, $r$ is necessarily equal to $1$.

Theorems & Definitions (61)

  • Conjecture 1: Pak18
  • Theorem 2: Adamczewski-Dreyfus-Hardouin
  • Theorem 3
  • Corollary 4
  • Lemma 5
  • proof
  • Example 6
  • Proposition 7
  • proof
  • Remark 8
  • ...and 51 more