The unimodality of Stirling numbers defined by normal ordering in the generalized Weyl algebra

Abstract

The Stirling numbers of the second kind can be viewed as normal or dering coefficients in the Weyl algebra. By modifying the commutation rule of the Weyl algebra, Mansour, Schork and Shattuck studied a two-parameter family of generalized Stirling numbers. This paper is devoted to completely characterize the unimodality of the generalized Stirling numbers by proving the strict log-concavity. As a consequence, we confirm a conjecture proposed by Mansour, Schork and Shattuck.