Infinite product for e^{6ζ(3)}
Abstract
The author uses the summation of rational series using the properties of the digamma function Ψ(x) and the methods of the residue calculus to evaluate the function Hα(x) for α = 1 and x = a−1(N), N∈ℕ (see Theorem 1) which is called the function generating the generalized harmonic numbers of order 1 (see Definition 1). The relation between the functions H1(x), x > 0, and Ψ(x) is used to find the approximations of the constant e6ζ(3) in the form of the infinite product which contains only the numbers e, π and the roots of unity, where ζ(3) is the Apéry constant.
Keywords
summation methods; infinite series; infinite products
References
2. Corvaja P., Hančl J., A transcendence criterion for infinite products. Preprint (2006).
3. Guillera J., Sondow J., Double integrals and infinite products for some classical constants via analytic continuations of Lerch’s transcendent. Available at http://arxiv.org/abs/math.NT/0506319 (2005).
4. Hančl J., Tijdeman R., On the irrationality of Cantor series, J. Reine Angew. Math. 571 (2004), 145–158.
5. Hančl J., Tijdeman R., On the irrationality of factorial series, Acta Arith. 118 (2005), no. 4, 383–401.
6. Sondow J., An infinite product for e^γ via hypergeometric formulas for Euler’s constant γ. Available at http://arXiv.org/abs/math/0306008 (2003).
Department of Mathematics, University of Ostrava, Czech Republic Czechia
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
