The Legendre-Stirling numbers

Abstract. The Legendre-Stirling numbers were discovered in 2002 as a result of a problem involving the spectral theory of powers of the classical second-order Legendre di¤erential expression. Speci…cally, these numbers are the coe¢ cients of integral composite powers of the Legendre expression in Lagrangian symmetric form. Quite remarkably, they share many similar properties with the classical Stirling numbers of the second kind which, as shown in [9], are the coe¢ cients of integral powers of the Laguerre di¤erential expression. An open question, regarding the Legendre-Stirling numbers, has been to obtain a combinatorial interpretation of these numbers. In this paper, we provide such an interpretation.

• Type: Journal
• Year: 2011
• Where: DM
• Authors: George E. Andrews, Wolfgang Gawronski, Lance L. Littlejohn
• Keywords: Coe¢ Cients, Di¤erential Expression, DM 2011, Education, Legendre-Stirling Numbers