Euler sums of generalized hyperharmonic numbers
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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: \[S\left( {k,m;p} \right): = \sum\limits_{n = 1}^\infty {\frac{{h_n^{\left( m \right)}\left( k \right)}}{{{n^p}}}} \;\;\left(p\geq m+1,\ {k = 1,2,3} \right)\] 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 Dil \cite{AD2015} and Mez\H{o} \cite{M2010}. Some interesting new consequences and illustrative examples are considered.
Abstract
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: \[S\left( {k,m;p} \right): = \sum\limits_{n = 1}^\infty {\frac{{h_n^{\left( m \right)}\left( k \right)}}{{{n^p}}}} \;\;\left(p\geq m+1,\ {k = 1,2,3} \right)\] 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 Dil \cite{AD2015} and Mez\H{o} \cite{M2010}. Some interesting new consequences and illustrative examples are considered.
참고문헌 (26)
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