The Jacobi–Stirling numbers

Abstract

The Jacobi-Stirling numbers were discovered as a result of a problem involving the spectral theory of powers of the classical secondorder Jacobi differential expression. Specifically, these numbers are the coefficients of integral composite powers of the Jacobi expression in Lagrangian symmetric form. Quite remarkably, they share many properties with the classical Stirling numbers of the second kind which are the coefficients of integral powers of the Laguerre differential expression. In this paper, we establish several properties of the Jacobi-Stirling numbers and its companions including combinatorial interpretations, thereby extending and supplementing known recent contributions to the literature.

‐differential equations for ‐classical polynomials and ‐Jacobi–Stirling numbers

“…The study required the introduction of the new set of q-Jacobi-Stirling numbers, which we have characterised and provided a combinatorial interpretation. Clearly, by allowing q → 1 we recover all the classical polynomial related results found in [2,7,8,11,26,27,30] from the viewpoint of both the algebra of these operators and the combinatorial interpretations of the connection coefficients.…”

“…In this paper, we obtain explicit expressions for any positive integer power of the q-differential operator L q given in (1.2) and the key to achieve this is via the introduction of a new set of numbers, that we call as q-Jacobi-Stirling numbers, to which we obtain several properties as well as a combinatorial interpretation. The results here obtained are the q-version to those in [26] (and also in [7], [8]), as well as to those in [2], [5], [11], [12], [31], [32], since we provide here combinatorial interpretations to the arisen coefficients and eigenvalues. This study has the merit of addressing all the q-classical polynomial sequences as whole in a coherent framework that brings together generalisations of the q-differential equation (1.2) and associated combinatorial interpretations.…”

“…The results here obtained are the q-version to those in [26] (and also in [7,8]), as well as to those in [2,6,11,12,30,32], since we provide here combinatorial interpretations to the arisen coefficients and eigenvalues. This study has the merit of addressing all the q-classical polynomial sequences as whole in a coherent framework that brings together generalisations of the q-differential equation (1.2) and associated combinatorial interpretations.…”

“…Motivated by this goal, we introduce, develop and give a combinatorial interpretation to the so-called q-Jacobi-Stirling numbers. Among other things, this work consists of a q-version of the studies in [2], [5], [7], [8], [11], [12], [26], while assembling the information in a coherent framework. To our best knowledge the results here presented are new in the theory.…”

“…where α represents a complex number. Since their introduction, several properties of the Jacobi-Stirling numbers and its companions including combinatorial interpretations have been established, see [2], [5], [11], [12], [24], [32].…”