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Ramanujan-type \(1/\pi \)-series from bimodular forms

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Abstract

We develop an approach to establish \(1/\pi \)-series from bimodular forms. Utilizing this approach, we obtain new families of 2-variable \(1/\pi \)-series associated to Zagier’s sporadic Apéry-like sequences. We first establish a general form of \(1/\pi \)-series involving constants related to the evaluation of modular forms at CM-points. Then we discuss situations when the series have rational terms. In addition, we discuss strategies for evaluating the constants involved and present a number of explicit \(1/\pi \)-series.

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Acknowledgements

The authors would like to thank Wadim Zudilin for many insightful comments on the earlier draft of the paper, especially those about Brafman’s works.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, Hubei, People’s Republic of China

    Liuquan Wang

  2. Department of Mathematics, National Taiwan University and National Center for Theoretical Sciences, Taipei, 10617, Taiwan

    Yifan Yang

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  1. Liuquan Wang
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  2. Yifan Yang
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Correspondence to Liuquan Wang.

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The first author was supported by the National Natural Science Foundation of China (11801424) and a start-up research grant from Wuhan University. The second author was partially supported by Grant 106-2115-M-002-009-MY3 from the Ministry of Science and Technology, Taiwan (R.O.C.)

Appendix A: 2-Variable \(1/\pi \)-series for sporadic Apéry sequences

Appendix A: 2-Variable \(1/\pi \)-series for sporadic Apéry sequences

See Tables 2, 3, 4, 5, 6, and 7.

Table 2 \(1/\pi \)-Series for \((a,b,c)=(7,2,-8)\)
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Table 3 \(1/\pi \)-Series for \((a,b,c)=(10,3,9)\)
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Table 4 \(1/\pi \)-Series for \((a,b,c)=(-17,-6,72)\)
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Table 5 \(1/\pi \)-Series for \((a,b,c)=(-9,-3,27)\)
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Table 6 \(1/\pi \)-Series for \((a,b,c)=(11,3,-1)\)
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Table 7 \(1/\pi \)-Series for \((a,b,c)=(12,4,32)\)
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Wang, L., Yang, Y. Ramanujan-type \(1/\pi \)-series from bimodular forms. Ramanujan J 59, 831–882 (2022). https://doi.org/10.1007/s11139-021-00532-6

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