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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.
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참고문헌 (26)
참고문헌 신청
D. H. Bailey / 1994 / Experimental evaluation of Euler sums / Experiment. Math. 3(1):17~30
A. T. Benjamin / 2003 / A combinatorial approach to hyperharmonic numbers / Integers 3:A15
D. Borwein / 1995 / Explicit evaluation of Euler sums / Proc. Edinburgh Math. Soc. 38(2):277~294
J. Borwein / 1996 / Making sense of experimental mathematics / Math. Intelligencer 18(4):12~18
J. M. Borwein / 2001 / Special values of multiple polylogarithms / Trans. Amer. Math. Soc. 353(3):907~941
A. T. Benjamin / 2003 / A combinatorial approach to hyperharmonic numbers / Integers 3:A15
D. Borwein / 1995 / Explicit evaluation of Euler sums / Proc. Edinburgh Math. Soc. 38(2):277~294
J. Borwein / 1996 / Making sense of experimental mathematics / Math. Intelligencer 18(4):12~18
J. M. Borwein / 2001 / Special values of multiple polylogarithms / Trans. Amer. Math. Soc. 353(3):907~941
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