On restricted approximation measures of Jacobi’s triple product
Leinonen, Leena; Leinonen, Marko (2023-09-11)
Springer
11.09.2023
Leinonen, L., Leinonen, M. On restricted approximation measures of Jacobi’s triple product. Ramanujan J 63, 1–12 (2024). https://doi.org/10.1007/s11139-023-00776-4
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© The Author(s) 2023. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
https://creativecommons.org/licenses/by/4.0/
© The Author(s) 2023. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi:oulu-202402231971
https://urn.fi/URN:NBN:fi:oulu-202402231971
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Abstract
We obtain rational approximations for Jacobi’s triple product \[ \Pi _q(t):= \prod _{m=1}^\infty (1-q^{2m})(1+q^{2m-1}t)(1+q^{2m-1}t^{-1}), \] when \(t=a/b\in {\mathbb {Q}}\) is non-zero and \(q=1/d\) with \(d\in {\mathbb {Z}}{\setminus }\{0, \pm 1 \}\). Especially we give effective and restricted approximation for the values of Jacobi’s triple product and for the values of Euler’s infinite product.
We obtain rational approximations for Jacobi’s triple product \[ \Pi _q(t):= \prod _{m=1}^\infty (1-q^{2m})(1+q^{2m-1}t)(1+q^{2m-1}t^{-1}), \] when \(t=a/b\in {\mathbb {Q}}\) is non-zero and \(q=1/d\) with \(d\in {\mathbb {Z}}{\setminus }\{0, \pm 1 \}\). Especially we give effective and restricted approximation for the values of Jacobi’s triple product and for the values of Euler’s infinite product.
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