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Euler sums of generalized hyperharmonic numbers. (English) Zbl 1397.11059

Summary: The generalized hyperharmonic numbers \(h_n^{(m)}(k)\) are defined by means of the multiple harmonic numbers. We show that the hyperharmonic numbers \(h_n^{(m)}(k)\) satisfy certain recurrence relation which allow us to write them in terms of classical harmonic numbers. Moreover, we prove that the Euler-type sums with hyperharmonic numbers: \[ S({k,m;p}):=\sum\limits_{n = 1}^\infty\frac{h_n^{(m)}(k)}{n^p}\;(p\geq m+1,\;k=1,2,3) \] can be expressed as a rational linear combination of products of Riemann zeta values and harmonic numbers. This is an extension of the results of A. Dil and K. N. Boyadzhiev [J. Number Theory 147, 490–498 (2015; Zbl 1311.11019)] and I. Mező and A. Dil [ibid. 130, No. 2, 360–369 (2010; Zbl 1225.11032)]. Some interesting new consequences and illustrative examples are considered.

MSC:

11B73 Bell and Stirling numbers
11B83 Special sequences and polynomials
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M32 Multiple Dirichlet series and zeta functions and multizeta values
11M99 Zeta and \(L\)-functions: analytic theory
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References:

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