On the p-adic properties of Stirling numbers of the first kind
On the p-adic properties of Stirling numbers of the first kind
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关于第一类斯特林数的 p 进数性质
DOI：
10.1007/s10474-020-01037-2
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发表时间：
2019-08
影响因子：
0.9
通讯作者：
Qiu M.
中科院分区：
文献类型：
--
作者：
Hong S. F.;Qiu M.
Let n, k and a be positive integers. The Stirling numbers of the first kind, denoted by s ( n, k ), count the number of permutations of n elements with k disjoint cycles. Let p be a prime. Lengyel, Komatsu and Young, Leonetti and Sanna, Adelberg, Hong and Qiu made some progress in the study of the p -adic valuations of s ( n, k ). In this paper, by using Washington’s congruence on the generalized harmonic number and the n -th Bernoulli number B n and the properties of m -th Stirling numbers of the first kind obtained recently by the authors, we arrive at an exact expression or a lower bound on v p ( s ( ap, k )) with a and k being integers such that $$1\le a\le p-1$$ 1 ≤ a ≤ p - 1 and $$1\le k\le ap$$ 1 ≤ k ≤ a p . This infers that for any regular prime $$p\ge 7$$ p ≥ 7 and for arbitrary integers a and k with $$5\le a\le p-1$$ 5 ≤ a ≤ p - 1 and $$a-2\le k\le ap-1$$ a - 2 ≤ k ≤ a p - 1 , one has $$v_{p}(H(ap-1,k)) < -\frac{\log{(ap-1)}}{2\log p}$$ v p ( H ( a p - 1 , k ) ) < - log ( a p - 1 ) 2 log p with $$H(ap-1, k)$$ H ( a p - 1 , k ) being the k -th elementary symmetric function of $$1, \frac{1}{2}, \ldots , \frac{1}{ap-1}$$ 1 , 1 2 , … , 1 a p - 1 . This gives a partial support to a conjecture of Leonetti and Sanna. We also present results on $$v_p(s(ap^{n},ap^{n}-k))$$ v p ( s ( a p n , a p n - k ) ) from which one can derive that under certain condition, for any prime $$p\ge 5$$ p ≥ 5 , any odd number $$k\ge 3$$ k ≥ 3 and any sufficiently large integer n , if $$(a,p)=1$$ ( a , p ) = 1 , then $$v_p(s(ap^{n+1},ap^{n+1}-k))=v_p(s(ap^{n},ap^{n}-k))+2$$ v p ( s ( a p n + 1 , a p n + 1 - k ) ) = v p ( s ( a p n , a p n - k ) ) + 2 . It confirms partially Lengyel’s conjecture.
Let n, k and a be positive integers. The Stirling numbers of the first kind, denoted by s ( n, k ), count the number of permutations of n elements with k disjoint cycles. Let p be a prime. Lengyel, Komatsu and Young, Leonetti and Sanna, Adelberg, Hong and Qiu made some progress in the study of the p -adic valuations of s ( n, k ). In this paper, by using Washington’s congruence on the generalized harmonic number and the n -th Bernoulli number B n and the properties of m -th Stirling numbers of the first kind obtained recently by the authors, we arrive at an exact expression or a lower bound on v p ( s ( ap, k )) with a and k being integers such that $$1\le a\le p-1$$ 1 ≤ a ≤ p - 1 and $$1\le k\le ap$$ 1 ≤ k ≤ a p . This infers that for any regular prime $$p\ge 7$$ p ≥ 7 and for arbitrary integers a and k with $$5\le a\le p-1$$ 5 ≤ a ≤ p - 1 and $$a-2\le k\le ap-1$$ a - 2 ≤ k ≤ a p - 1 , one has $$v_{p}(H(ap-1,k)) < -\frac{\log{(ap-1)}}{2\log p}$$ v p ( H ( a p - 1 , k ) ) < - log ( a p - 1 ) 2 log p with $$H(ap-1, k)$$ H ( a p - 1 , k ) being the k -th elementary symmetric function of $$1, \frac{1}{2}, \ldots , \frac{1}{ap-1}$$ 1 , 1 2 , … , 1 a p - 1 . This gives a partial support to a conjecture of Leonetti and Sanna. We also present results on $$v_p(s(ap^{n},ap^{n}-k))$$ v p ( s ( a p n , a p n - k ) ) from which one can derive that under certain condition, for any prime $$p\ge 5$$ p ≥ 5 , any odd number $$k\ge 3$$ k ≥ 3 and any sufficiently large integer n , if $$(a,p)=1$$ ( a , p ) = 1 , then $$v_p(s(ap^{n+1},ap^{n+1}-k))=v_p(s(ap^{n},ap^{n}-k))+2$$ v p ( s ( a p n + 1 , a p n + 1 - k ) ) = v p ( s ( a p n , a p n - k ) ) + 2 . It confirms partially Lengyel’s conjecture.
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影响因子：
0.7
作者：
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影响因子：
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DOI：
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发表时间：
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期刊：
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影响因子：
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期刊：
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影响因子：
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