The Jacobi-Stirling numbers

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

Research output: Contribution to journalArticlepeer-review

29 Scopus citations


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 languageEnglish (US)
Pages (from-to)288-303
Number of pages16
JournalJournal of Combinatorial Theory. Series A
Issue number1
StatePublished - 2013

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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