### 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) |
---|---|

Pages (from-to) | 1255-1272 |

Number of pages | 18 |

Journal | Discrete Mathematics |

Volume | 311 |

Issue number | 14 |

DOIs | |

State | Published - Jul 28 2011 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

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

}

*Discrete Mathematics*, vol. 311, no. 14, pp. 1255-1272. https://doi.org/10.1016/j.disc.2011.02.028

**The Legendre-Stirling numbers.** / Andrews, George E.; Gawronski, W.; Littlejohn, L. L.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The Legendre-Stirling numbers

AU - Andrews, George E.

AU - Gawronski, W.

AU - Littlejohn, L. L.

PY - 2011/7/28

Y1 - 2011/7/28

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=79955589510&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79955589510&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2011.02.028

DO - 10.1016/j.disc.2011.02.028

M3 - Article

VL - 311

SP - 1255

EP - 1272

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 14

ER -