### Abstract

The Legendre-Stirling numbers are the coefficients in the integral Lagrangian symmetric powers of the classical Legendre second-order differential expression. In many ways, these numbers mimic the classical Stirling numbers of the second kind which play a similar role in the integral powers of the classical second-order Laguerre differential expression. In a recent paper, Andrews and Littlejohn gave a combinatorial interpretation of the Legendre-Stirling numbers. In this paper, we establish several properties of the Legendre-Stirling numbers; as with the Stirling numbers of the second kind, they have interesting generating functions and recurrence relations. Moreover, there are some surprising and intriguing results relating these numbers to some classical results in algebraic number theory.

Original language | English (US) |
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Pages (from-to) | 1255-1272 |

Number of pages | 18 |

Journal | Discrete Mathematics |

Volume | 311 |

Issue number | 14 |

DOIs | |

State | Published - Jul 28 2011 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*311*(14), 1255-1272. https://doi.org/10.1016/j.disc.2011.02.028