On recurrence relations for Bernoulli and Euler numbers (Q2777721)
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scientific article; zbMATH DE number 1717650
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On recurrence relations for Bernoulli and Euler numbers |
scientific article; zbMATH DE number 1717650 |
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On recurrence relations for Bernoulli and Euler numbers (English)
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24 November 2002
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Bernoulli numbers
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Euler numbers
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Genocchi numbers
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For any integers \(n\geq 1\), \(m\geq 0\), the authors prove the identities NEWLINE\[NEWLINE \sum\limits_{k=\max (n-m,0)}^{2n}\binom {m+n+1}{m-n+k}\binom {2m+k+1}kB_k=0, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \sum\limits_{k=\max (n-m,0)}^{2n}\binom {m+n+1}{m-n+k}\binom {2m+k+1}k\frac{B_k}{2^k}=(-1)^n\frac{m+1}{2^{2n+1}}\binom {m+n+1}n, NEWLINE\]NEWLINE where \(B_k\) are the Bernoulli numbers. Similar relations are found for the Euler and Genocchi numbers.
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