## Abstract

In this paper, we study further properties of a recently introduced generalized Eulerian number, denoted by A_{m;r}(n; k), which reduces to the classical Eulerian number when m = 1 and r = 0. Among our results is a generalization of an earlier symmetric Eulerian number identity of Chung, Graham and Knuth. Using the row generating function for A_{m;r}(n; k) for a fixed n, we introduce the r-Whitney-Euler-Frobenius fractions, which generalize the Euler-Frobenius fractions. Finally, we consider a further four-parameter combinatorial generalization of A_{m;r}(n; k) and find a formula for its exponential generating function in terms of the Lambert-W function.

Original language | English |
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Pages (from-to) | 378-398 |

Number of pages | 21 |

Journal | Applicable Analysis and Discrete Mathematics |

Volume | 13 |

Issue number | 2 |

DOIs | |

State | Published - 2019 |

### Bibliographical note

Funding Information:ments which improved the article. The research of JoséL. Ramírez was partially supported by Universidad Nacional de Colombia, Project No. 46240.

Publisher Copyright:

© 2019 University of Belgrade.

## Keywords

- Combinatorial identity
- Euler-Frobenius fraction
- Eulerian number
- R-Whitney-Eulerian number

## ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics