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arXiv · 2112.14047

Stieltjes constants appearing in the Laurent expansion of the hyperharmonic zeta function

Mümün Can·Ayhan Dil·Levent Kargın
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Abstract

In this paper, we consider meromorphic extension of the function \[ ζ_{h^{\left( r\right) }}\left( s\right) =\sum_{k=1}^{\infty} \frac{h_{k}^{\left( r\right) }}{k^{s}},\text{ }\operatorname{Re}\left( s\right) >r, \] (which we call \textit{hyperharmonic zeta function}) where $h_{n}^{(r)}$ are the hyperharmonic numbers. We establish certain constants, denoted $γ_{h^{\left( r\right) }}\left( m\right) $, which naturally occur in the Laurent expansion of $ζ_{h^{\left( r\right) }}\left( s\right) $. Moreover, we show that the constants $γ_{h^{\left( r\right) }}\left( m\right) $ and integrals involving generalized exponential integral can be written as a finite combination of some special constants.

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Mümün Can, Ayhan Dil, Levent Kargın. 2021-12-28. Stieltjes constants appearing in the Laurent expansion of the hyperharmonic zeta function. https://doi.org/10.1007/s11139-022-00676-z

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