A Recursive Formula for Sums of Values of Degenerate Falling Factorials

Authors

  • Dae San Kim Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea
  • Taekyun Kim Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea
  • Jongkyum Kwon Department of Mathematics Education, Gyeongsang National University, Jinju, 52828, Republic of Korea
  • Hyunseok Lee Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea

DOI:

https://doi.org/10.37256/cm.6120255881

Keywords:

sums of values of degenerate falling factorials, uniform distribution

Abstract

The classical Faulhaber's formula expresses the sum of a fixed positive integer powers of the first n positive integers in terms of Bernoulli polynomial. As a degenerate version of this, we may consider sums of values of degenerate falling factorials, which reduce to aforementioned sum as λ tends to 0. The aim of this note is to derive a recursive formula for sums of values of degenerate falling factorials by using probabilistic method. In this manner, we obtain a new recursive formula for such sums, which involves the (signed) Stirling numbers of the first kind.

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Published

2025-02-17

Issue

Contemporary Mathematics, Volume 6 Issue 1 (2025), 1-1379

Section

Research Article

License

Copyright (c) 2025 Taekyun Kim, et al.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

1.
A Recursive Formula for Sums of Values of Degenerate Falling Factorials. Contemp. Math. [Internet]. 2025 Feb. 17 [cited 2025 Dec. 24];6(1):1138-49. Available from: https://ojs353.mebyme.cn/index.php/CM/article/view/5881