<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper a number of new explicit expressions for quadratic Euler-type sums containing double-index harmonic numbers <italic>H</italic>2<italic>n</italic> are given. These are obtained using ordinary generating functions containing the square of the harmonic numbers <italic>H</italic><italic>n</italic>. As a by-product of the generating function approach used new proofs for the remarkable quadratic series of Au-Yeung
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2020-0034_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></munderover><mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mrow><mi>H</mi></mrow><mi>n</mi></msub></mrow><mi>n</mi></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mn>17</mn><msup><mrow><mi>π</mi></mrow><mn>4</mn></msup></mrow><mrow><mn>360</mn></mrow></mfrac></mrow></mrow></math><tex-math>\sum\limits_{n = 1}^\infty {{{\left( {{{{H_n}} \over n}} \right)}^2} = {{17{\pi ^4}} \over {360}}}</tex-math></alternatives></disp-formula>
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.</abstract>

In this paper a number of new explicit expressions for quadratic Euler-type sums containing double-index harmonic numbers H2n are given. These are obtained using ordinary generating functions containing the square of the harmonic numbers Hn. As a by-product of the generating function approach used new proofs for the remarkable quadratic series of Au-Yeung
∑n=1∞(Hnn)2=17π4360\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.

Explicit Evaluation of Some Quadratic Euler-Type Sums Containing Double-Index Harmonic Numbers

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

In this paper a number of new explicit expressions for quadratic Euler-type sums containing double-index harmonic numbers H2n are given. These are obtained...

Explicit Evaluation of Some Quadratic Euler-Type Sums Containing Double-Index Harmonic Numbers

Published Online: Dec 31, 2020

Page range: 73 - 98

Received: Mar 15, 2020

DOI: https://doi.org/10.2478/tmmp-2020-0034

Keywords
Euler sums, Harmonic number, Generating function, Polylogarithm function, Riemann zeta function

© 2020 Seán Mark Stewart, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Explicit Evaluation of Some Quadratic Euler-Type Sums Containing Double-Index Harmonic Numbers

Published Online: Dec 31, 2020

Page range: 73 - 98

Received: Mar 15, 2020

DOI: https://doi.org/10.2478/tmmp-2020-0034

KeywordsEuler sums, Harmonic number, Generating function, Polylogarithm function, Riemann zeta function

© 2020 Seán Mark Stewart, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Euler sums, Harmonic number, Generating function, Polylogarithm function, Riemann zeta function