Central factorial numbers are Pólya frequency sequences

Abstract

The central factorial numbers are closely related to the central difference operator, and have received increasing attention from combinatorialists recently. Known properties of the central factorial numbers include their generating functions, recurrence relation and combinatorial interpretations. The purpose of this work is to adjoin to the list the analytical property that the generating functions of central factorial numbers of odd, and even, indices, are simply real-rooted, thus establishing the numbers being Pólya frequency sequences.