Affiliation:
1. Bomaderry NSW , Australia
Abstract
Abstract
In this paper a number of new explicit expressions for quadratic Euler-type sums containing double-index harmonic numbers H
2
n
are given. These are obtained using ordinary generating functions containing the square of the harmonic numbers H
n
. As a by-product of the generating function approach used new proofs for the remarkable quadratic series of Au-Yeung
∑
n
=
1
∞
(
H
n
n
)
2
=
17
π
4
360
\sum\limits_{n = 1}^\infty {{{\left( {{{{H_n}} \over n}} \right)}^2} = {{17{\pi ^4}} \over {360}}}
together with its closely related alternating cousin are given. New proofs for other closely related quadratic Euler-type sums that are known in the literature are also obtained.
Reference45 articles.
1. [1] BAILEY, D. H.—BORWEIN, J. M.—GIRGENSOHN, R.: Experimental evaluation of Euler sums, Experiment. Math. 3 (1994), no. 1, 17–30.
2. [2] BASTIEN, G.: Elementary methods for evaluating Jordan’s sums∑n⩾1(1+13+⋯⋯+12n-1)\sum {_{n \geqslant 1}\left( {1 + {1 \over 3} + \cdots \cdots + {1 \over {2n - 1}}} \right)} and ∑n⩾1(1+13+⋯+12n-1)1(2n-1)2α\sum {_{n \geqslant 1}\left( {1 + {1 \over 3} + \cdots + {1 \over {2n - 1}}} \right){1 \over {{{\left( {2n - 1} \right)}^{2\alpha }}}}}and analogous Euler’s type sums and for setting a σ-sum theorem, arXiv:1301.7662 (2013).
3. [3] BASU, A.—APOSTOL, T. M.: A new method for investigating Euler sums, Ramanujan J. 4 (2000), no. 4, 397–419.
4. [4] BORWEIN, D.—BORWEIN, J. M.: On an intriguing integral and some series related to ζ(4), Proc. Amer. Math. Soc. 123 (1995), no. 4, 1191–1198.
5. [5] BORWEIN, D.—BORWEIN, J. M.—GIRGENSOHN, R.: Explicit evaluation of Euler sums,Proc. Edinburgh Math.Soc. 38 (1995), no. 2, 277–294.
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