Identities between polynomials related to Stirling and harmonic numbers
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arXiv
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| Formato: | Preprint |
| Publicado: |
2012
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| _version_ | 1866913403665121280 |
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| author | Kellner, Bernd C. |
| author_facet | Kellner, Bernd C. |
| contents | We consider two types of polynomials $F_n (x) = \sum_{ν=1}^n ν! S_2(n,ν) x^ν$ and $\hat{F}_n (x) = \sum_{ν=1}^n ν! S_2(n,ν) H_νx^ν$, where $S_2(n,ν)$ are the Stirling numbers of the second kind and $H_ν$ are the harmonic numbers. We show some properties and relations between these polynomials. Especially, the identity $\hat{F}_n (-\tfrac{1}{2}) = - (n-1)/2 \cdot F_{n-1} (-\tfrac{1}{2})$ is established for even $n$, where the values are connected with Genocchi numbers. For odd $n$ the value of $\hat{F}_n (-\tfrac{1}{2})$ is given by a convolution of these numbers. Subsequently, we discuss some of these convolutions, which are connected with Miki type convolutions of Bernoulli and Genocchi numbers, and derive some 2-adic valuations of them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1209_1018 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | Identities between polynomials related to Stirling and harmonic numbers Kellner, Bernd C. Number Theory 11B73 (Primary) 11B83, 11B68 (Secondary) We consider two types of polynomials $F_n (x) = \sum_{ν=1}^n ν! S_2(n,ν) x^ν$ and $\hat{F}_n (x) = \sum_{ν=1}^n ν! S_2(n,ν) H_νx^ν$, where $S_2(n,ν)$ are the Stirling numbers of the second kind and $H_ν$ are the harmonic numbers. We show some properties and relations between these polynomials. Especially, the identity $\hat{F}_n (-\tfrac{1}{2}) = - (n-1)/2 \cdot F_{n-1} (-\tfrac{1}{2})$ is established for even $n$, where the values are connected with Genocchi numbers. For odd $n$ the value of $\hat{F}_n (-\tfrac{1}{2})$ is given by a convolution of these numbers. Subsequently, we discuss some of these convolutions, which are connected with Miki type convolutions of Bernoulli and Genocchi numbers, and derive some 2-adic valuations of them. |
| title | Identities between polynomials related to Stirling and harmonic numbers |
| topic | Number Theory 11B73 (Primary) 11B83, 11B68 (Secondary) |
| url | https://arxiv.org/abs/1209.1018 |