Total positivity properties of Jacobi–Stirling numbers

Abstract

The Jacobi–Stirling numbers of the first and second kinds were first introduced in Everitt et al. (2007) [8] and they are a generalization of the Legendre–Stirling numbers. Quite remarkably, they share many similar properties with the classical Stirling numbers. In this paper we study total positivity properties of these numbers. In particular, we prove that the matrix whose entries are the Jacobi–Stirling numbers is totally positive and that each row and each column is a Pólya frequency sequence, except for the columns with (unsigned) numbers of the first kind.