Peter J. Shiue, Shen C. Huang and Eric Jameson

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 26, 2020, Number 4, Pages 113–121

DOI: 10.7546/nntdm.2020.26.4.113-121

**Full paper (PDF, 156 Kb)**

## Details

### Authors and affiliations

Peter J. Shiue

*Department of Mathematical Sciences, University of Nevada, Las Vegas,
Las Vegas, NV 89154-4020, USA
*

Shen C. Huang

*Department of Mathematical Sciences, University of Nevada, Las Vegas,*

Las Vegas, NV 89154-4020, USA

Las Vegas, NV 89154-4020, USA

Eric Jameson

*Department of Mathematical Sciences, University of Nevada, Las Vegas,*

Las Vegas, NV 89154-4020, USA

Las Vegas, NV 89154-4020, USA

### Abstract

This paper is concerned with sums of powers of arithmetic progressions of the form , where , is a non-negative integer, and and are complex numbers with . This paper gives an elementary proof to a theorem presented by Laissaoui and Rahmani [9] as well as an algorithm based on their formula. Additionally, this paper presents a simplification to Laissaoui and Rahmani’s formula that is better suited to computation, and a second algorithm based on this simplification. Both formulas use Stirling numbers of the second kind, which are the number of ways to partition labelled objects into nonempty unlabelled subsets [4]. An analysis of both algorithms is presented to show the theoretical time complexities. Finally, this paper conducts experiments with varying values of . The experimental results show the proposed algorithm remains around 10% faster as increases.

### Keywords

- Analysis of algorithms
- Arithmetic progressions
- Dynamic programming
- Stirling number of the second kind

### 2010 Mathematics Subject Classification

- 05A10
- 11B25
- 11B65
- 11B73
- 68N15
- 68Q25

### References

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- Bein, W. (2013). Advanced techniques for dynamic programming. In: Pardalos, P., M., Du, D.-Z., & Graham, R. L. (Editors), Handbook of Combinatorial Optimization, Chapter 2, Springer, pp. 41–92.
- Bounebirat, F., Laissaoui, D., & Rahmani, M. (2018). Several explicit formulae of sums and hyper-sums of powers of integers. Online Journal of Analytic Combinatorics, 13.
- Charalambides, C. A. (2002). Enumerative Combinatorics. Chapman & Hall / CRC Press, pp. 278–279.
- Chen, W. Y. C., Fu, A. M., & Zhang, I. F. (2009). Faulhaber’s theorem on power sums. Discrete Mathematics, 309 (10), 2974–2981.
- Howard, F.T. (1996). Sums of powers of integers via generating functions. Fibonacci Quarterly, 34 (3), 244–256.
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- Laissaoui, D., & Rahmani, R. (2017). An explicit formula for sums of powers of integers in terms of Stirling numbers. Journal of Integer Sequences, 20 (4), Aricle 17.4.8.
- Pita-Ruiz, C. (2018). On a generalization of Eulerian numbers.
*Notes on Number Theory and Discrete Mathematics*, 24 (1), 16–42. - Vassilev, P., & Vassilev-Missana, M. (2005). On the sum of equal powers of the first
*n*terms of an arbitrary arithmetic progression. I.*Notes on Number Theory and Discrete Mathematics*, 11 (3), 15–21.

## Related papers

- Vassilev, P., and Vassilev-Missana, M. (2005). On the sum of equal powers of the first
*n*terms of an arbitrary arithmetic progression. I.*Notes on Number Theory and Discrete Mathematics*, 11(3), 15-21. - Vassilev, P., and Vassilev-Missana, M. (2005). On the sum of equal powers of the first
*n*terms of an arbitrary arithmetic progression. II.*Notes on Number Theory and Discrete Mathematics*, 11(4), 25-28. - Pita-Ruiz, C. (2018). On a generalization of Eulerian numbers.
*Notes on Number Theory and Discrete Mathematics*, 24 (1), 16–42. - Shiue, P. J., Huang, S. C., & Reyes, J. E. (2021). Algorithms for computing sums of powers of arithmetic progressions by using Eulerian numbers.
*Notes on Number Theory and Discrete Mathematics*, 27(4), 140-148.

## Cite this paper

Shiue, P. J., Huang, S. C., & Jameson, E. (2020). On algorithms for computing the sums of powers of arithmetic progressions. *Notes on Number Theory and Discrete Mathematics*, 26 (4), 113-121, DOI: 10.7546/nntdm.2020.26.4.113-121.