Claudio Pita-Ruiz

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 24, 2018, Number 1, Pages 16–42

DOI: 10.7546/nntdm.2018.24.1.16-42

**Full paper (PDF, 261 Kb)**

## Details

### Authors and affiliations

Claudio Pita-Ruiz

*Facultad de Ingenierıa, Universidad Panamericana
Augusto Rodin 498, Mexico, Ciudad de Mexico, 03920, Mexico
*

### Abstract

We consider the sequence , product of the *rp*-th degree *n*-polynomial , where *a*, *b* ∈ ℂ, *a* ≠ 0, *r*, *p* ∈ ℕ, and the -th degree n-polynomial , where *α _{s}*,

*β*∈ ℂ,

_{s}*r*,

_{s}*p*∈ ℕ,

_{s}*s*= 2, …,

*l*. In the expansion of the polynomial in terms of the binomials , , the resulting coefficients are the generalized Eulerian numbers we consider in this work (the case

*P*(

*n*) = 1,

*a*= 1,

*b*= 0

*, r*= 1 corresponds to the standard Eulerian numbers). We obtain results on symmetries, recurrences, row sums, and alternating row sums, that generalize the corresponding well-known results for the standard Eulerian numbers. The main tool we use to obtain our results throughout the work, is the Z-transform of sequences.

### Keywords

- Generalized Eulerian numbers

### 2010 Mathematics Subject Classification

- 11B83

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## Cite this paper

Pita-Ruiz, C. (2018). On a generalization of Eulerian numbers. *Notes on Number Theory and Discrete Mathematics*, 24(1), 16-42, DOI: 10.7546/nntdm.2018.24.1.16-42.