Monday, 19 August 2013

sequences and series - Evaluation of two Euler type sums

We know that the harmonic number sum (also called Euler type sum) enter link description here
n=1H(2)nn22n=Li4(12)+116ζ(4)+14ζ(3)log214ζ(2)log22+124log42,
How to calculate the closed form of the following Euler type Sums
n=1H(2)nn32n,n=1H(2)nn42n.
Here the harmonic numbers are defined by
H(k)n:=nj=11jkandH(k)0:=0.

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