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A note on products involving \(\zeta (3)\) and Catalan’s constant

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Abstract

In a recent paper Kachi and Tzermias give elementary proofs of four product formulas involving \(\zeta (3)\), \(\pi \), and Catalan’s constant. They indicate that they were not able to deduce these products directly from the values of a function introduced in 1993 by Borwein and Dykshoorn. We provide here such a proof for two of these formulas. We also give a direct proof for the other two formulas, using a generalization of the Borwein–Dykshoorn function due to Adamchik. Finally we give an expression of the Borwein–Dykshoorn function in terms of the “parameterized-Euler-constant function” introduced by Xia in 2013, which happens to be a particular case of the “generalized Euler constant function” introduced by K. Hessami Pilehrood and T. Hessami Pilehrood in 2010.

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Acknowledgments

We would like to thank Khodabakhsh Hessami Pilehrood, Jia-Yan Yao, and the referee for their useful comments on a previous version of this paper.

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Authors and Affiliations

  1. CNRS, Institut de Mathématiques de Jussieu-PRG, Équipe Combinatoire et Optimisation, Université Pierre et Marie Curie, Case 247, 4 Place Jussieu, 75252, Paris Cedex 05, France

    J.-P. Allouche

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  1. J.-P. Allouche
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Correspondence to J.-P. Allouche.

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The author was partially supported by the ANR project “FAN” (Fractal and Numeration)

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Allouche, JP. A note on products involving \(\zeta (3)\) and Catalan’s constant. Ramanujan J 37, 79–88 (2015). https://doi.org/10.1007/s11139-014-9604-5

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  • DOI: https://doi.org/10.1007/s11139-014-9604-5

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