Vol. 9, No. 2, 2020

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On the roots of the Poupard and Kreweras polynomials

Frédéric Chapoton and Guo-Niu Han

Vol. 9 (2020), No. 2, 163–172
DOI: 10.2140/moscow.2020.9.163
Abstract

The Poupard polynomials are polynomials in one variable with integer coefficients, with some close relationship to Bernoulli and tangent numbers. They also have a combinatorial interpretation. We prove that every Poupard polynomial has all its roots on the unit circle. We also obtain the same property for another sequence of polynomials introduced by Kreweras and related to Genocchi numbers. This is obtained through a general statement about some linear operators acting on palindromic polynomials.

Keywords
palindromic polynomial, unit circle, complex root, linear operator, Bernoulli number
Mathematical Subject Classification 2010
Primary: 26C10, 47B39
Secondary: 11B68, 39A70
Milestones
Received: 16 January 2020
Revised: 15 May 2020
Accepted: 29 May 2020
Published: 7 August 2020
Authors
Frédéric Chapoton
Institut de Recherche Mathématique Avancée
UMR 7501
Université de Strasbourg et CNRS
Strasbourg
France
Guo-Niu Han
Institut de Recherche Mathématique Avancée
UMR 7501
Université de Strasbourg et CNRS
Strasbourg
France