The 3-adic valuations of Stirling numbers of the first kind

Min Qiu; Zongbing Lin; Long Chen; Min Qiu; Zongbing Lin; Long Chen

doi:10.3934/math.20261025

AIMS Mathematics

2026, Volume 11, Issue 8: 25553-25600. doi: 10.3934/math.20261025

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Research article

The 3-adic valuations of Stirling numbers of the first kind

1.
School of Science, Xihua University, Chengdu 610039, China
2.
School of Mathematics and Computer Science, Panzhihua University, Panzhihua 617000, China

Received: 01 July 2026 Revised: 31 July 2026 Accepted: 07 August 2026 Published: 18 August 2026
MSC : 11A07, 11B73

Let $ v_3 $ denote the usual $ 3 $-adic valuation, and let $ s(n, k) $ be the unsigned Stirling number of the first kind. In this paper, for $ a\in\{1, 2\} $, we determine the values of $ v_3(s(a3^n, k)) $ for all $ 1\le k\le a3^n $. More precisely, for each admissible pair $ (m, k) $, we obtain an explicit formula for $ v_3(s(a3^n, a3^m-k)) $. This proves the case $ p = 3 $ of a conjecture of Hong and Qiu proposed in 2020. As applications, we derive formulas near the diagonal, comparison results for adjacent orders, sharp upper bounds for the families $ v_3(s(3^n, k)) $ and $ v_3(s(2\cdot3^n, k)) $, and partial confirmations of conjectures of Lengyel and of Leonetti and Sanna.
- 3-adic valuation,
- 3-adic analysis,
- Stirling number of the first kind,
- the $ m $-th Stirling number of the first kind,
- elementary symmetric function
Citation: Min Qiu, Zongbing Lin, Long Chen. The 3-adic valuations of Stirling numbers of the first kind[J]. AIMS Mathematics, 2026, 11(8): 25553-25600. doi: 10.3934/math.20261025

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Abstract

Let $ v_3 $ denote the usual $ 3 $-adic valuation, and let $ s(n, k) $ be the unsigned Stirling number of the first kind. In this paper, for $ a\in\{1, 2\} $, we determine the values of $ v_3(s(a3^n, k)) $ for all $ 1\le k\le a3^n $. More precisely, for each admissible pair $ (m, k) $, we obtain an explicit formula for $ v_3(s(a3^n, a3^m-k)) $. This proves the case $ p = 3 $ of a conjecture of Hong and Qiu proposed in 2020. As applications, we derive formulas near the diagonal, comparison results for adjacent orders, sharp upper bounds for the families $ v_3(s(3^n, k)) $ and $ v_3(s(2\cdot3^n, k)) $, and partial confirmations of conjectures of Lengyel and of Leonetti and Sanna.

References

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