[seqfan] Re: Sum of expression with harmonic numbers should simplify to Zeta?

[allouche at math.jussieu.fr]

Hi,

Without being a specialist I think you might find some ideas in the paper
http://www.sciencedirect.com/science/article/pii/S0096300301001722
(by Rassias and Srivastava)
(look in particular at formulas 4.14 and 4.15 which
do not answer your question completely but may give some
ideas and references to go further)

best wishes
jpa


Jean-François Alcover <jf.alcover at gmail.com> a écrit :

> Dear Seqfans,
>
> Thanks in advance to the (kind) specialist
> who could explain how can be proved that this expression:
>
> Sum[ HarmonicNumber[k]/k^n - (PolyGamma[k+1] + EulerGamma)/(k+1)^n, {k, 1,
> Infinity}]
>
> always simplifies to Zeta[n+1].
>
> J.-F. Alcover
>
> P.s.
> FullSimplify returns:
> Sum[ (-(k*(1+k))^(-n))*(k^n - (1+k)^n)*HarmonicNumber[k], {k, 1, Infinity}]
>
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