Summation formulas of hyperharmonic numbers with their generalizations. (English) Zbl 07773239
Summary: J. Spieß [Math. Comput. 55, No. 192, 839–863 (1990; Zbl 0724.05005)] gave some identities of harmonic numbers including the types \(\sum_{\ell =1}^n\ell^k H_\ell\), \(\sum_{\ell =1}^n\ell^k H_{n-\ell}\) and \(\sum_{\ell =1}^n\ell^k H_\ell H_{n-\ell}\). In this paper, we derive several formulas of hyperharmonic numbers including \(\sum_{\ell =0}^n \ell^p h_\ell^{(r)} h_{n-\ell}^{(s)}\) and \(\sum_{\ell =0}^n \ell^p\left(h_\ell^{(r)}\right)^2\). Some more formulas of generalized hyperharmonic numbers are also shown.
MSC:
| 11B65 | Binomial coefficients; factorials; \(q\)-identities |
| 11B73 | Bell and Stirling numbers |
| 05A19 | Combinatorial identities, bijective combinatorics |
Citations:
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