The Jacobi-Stirling numbers

George E. Andrews; Eric S. Egge; Wolfgang Gawronski; Lance L. Littlejohn

doi:10.1016/j.jcta.2012.08.006

The Jacobi-Stirling numbers

George E. Andrews
, Eric S. Egge
, Wolfgang Gawronski
, Lance L. Littlejohn

Mathematics

Research output: Contribution to journal › Article › peer-review

30 Scopus citations

Abstract

The Jacobi-Stirling numbers were discovered as a result of a problem involving the spectral theory of powers of the classical second-order 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.

Original language	English (US)
Pages (from-to)	288-303
Number of pages	16
Journal	Journal of Combinatorial Theory. Series A
Volume	120
Issue number	1
DOIs	https://doi.org/10.1016/j.jcta.2012.08.006
State	Published - 2013

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1016/j.jcta.2012.08.006

Cite this

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abstract = "The Jacobi-Stirling numbers were discovered as a result of a problem involving the spectral theory of powers of the classical second-order 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.",

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note = "Funding Information: E-mail addresses: [email protected] (G.E. Andrews), [email protected] (E.S. Egge), [email protected] (W. Gawronski), Lance\[email protected] (L.L. Littlejohn). 1 The first author is partially supported by National Security Agency Grant H98230-12-1-0205.",

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doi = "10.1016/j.jcta.2012.08.006",

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AU - Andrews, George E.

AU - Egge, Eric S.

AU - Gawronski, Wolfgang

AU - Littlejohn, Lance L.

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AB - The Jacobi-Stirling numbers were discovered as a result of a problem involving the spectral theory of powers of the classical second-order 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.

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The Jacobi-Stirling numbers

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